# THEORY OF HYPERSURFACES OF A FINSLER SPACE WITH THE GENERALIZED SQUARE
METRIC

Sonia Rani  Vinod Kumar  Mohammad Rafee Corresponding authour: Mohammad Rafee;
Email Id<EMAIL_ADDRESS>
###### Abstract

The emergence of generalized square metrics in Finsler geometry can be
attributed to various classification concerning $(\alpha,\beta)$-metrics. They
have excellent geometric properties in Finsler geometry. Within the scope of
this research paper, we have conducted an investigation into the generalized
square metric denoted as
$F(x,y)=\frac{[\alpha(x,y)+\beta(x,y)]^{n+1}}{[\alpha(x,y)]^{n}}$, focusing
specifically on its application to the Finslerian hypersurface. Furthermore,
the classification and existence of first, second, and third kind of
hyperplanes of the Finsler manifold has been established.

AMS Subject Classification: 53B40, 53C60

Key Words and Phrases: Finsler Space, Finsler metric, Generalized square
metric, Hypersurface of Finsler space, Hyperplane.

##  1 Introduction

”Let $M$ be an $n$-dimensional differential manifold. Define a Finsler metric
on the differentiable manifold $M$. This Finsler metric is known as Finsler
fundamental fuction on the manifold $M$. Let us first define what is exactly
mean by Finsler fundamental function.

######  Definition 1.1 (Finsler metric).

We say a function $F:TM\rightarrow R$ is a Finsler metric or Finsler
fundamental function on the manifold $M$ if $F$ satisfy the following
conditions:

  1. 1.

$F$ is $C^{\infty}$ away from zero vectors of the tangent spaces:
That is, $F$ is smooth on $TM\backslash\\{0\\}=\left\\{(x,y)|x\in M,y\in
T_{x}M,y\neq 0\right\\}$. The smoothness property is desired so that we can
apply differential calculus on the Finsler metric $F$.

  2. 2.

Positivity of function $F$:
$F(x,y)\geq 0$ for $x\in M$ and all $y\in T_{x}M$.
This property ensures that the length of a tangent vector $y\in T_{x}M$ is
either positive or zero. In terms of arc length this property also ensures
that arc length defined by the integral 1.1 is either positive or zero.

  3. 3.

Positive homogeneity of function $F$:
$F(x,\lambda y)=\lambda F(x,y)$, $\forall$ $\lambda>0$; for $x\in M$ and all
$y\in T_{x}M$.
That is, $F$ is +ve 1-homogeneous of first degree in the directional argument
$y$. This property ensures that, length of a tangent vector $\lambda y\in
T_{x}M$ is nothing but $\lambda\times F(x,y)\in R$.

Figure 1: This figure shows the tangent vector $\lambda y\in T_{x}M$,
$\lambda>0$, is $\lambda$ times the tangent vector $y\in T_{x}M$.

  4. 4.

Strict convexity of the function $F$:
$F:TM\rightarrow R$ is strictly convex over the tangent bundle $TM$.

It is important to note that strictly convex condition is equivalent to the
hessian matrix $[g_{ij}]$, where $i,j\in\left\\{1,2,3,......dim(M)\right\\}$,
defined by $\frac{1}{2}\frac{\partial^{2}F^{2}}{\partial y^{i}\partial
y^{j}}(x,y)=g_{ij}(x,y)$ is positive definite for any $(x,y)\in{TM}$. It is
the convexity condition on the Finsler metric $F$ that guaranties for the arc
length minimization, given by the following formula, of the admissible curves
$\gamma:[a,b]\rightarrow M$ belonging to the set $C^{\infty}[a,b]$

$s[\gamma(t)]=\int_{a}^{b}F(\gamma(t),\dot{\gamma}(t))dt$ (1.1)

can be achieved.

Figure 2: This figure shows the curves $\gamma:[a,b]\rightarrow M$ over the
manifold $M$ such that $\gamma(a)=P$ and $\gamma(b)=Q$.

In other words, the convexity condition on the Finsler metric is a geometric
requirement that makes sure that the length of a curve can be defined and that
the arc length functional $s[\gamma(t)]$ is well-behaved, particularly for the
purpose of minimizing or finding geodesics with respect to the given Finsler
metric $F$. Without convexity, the concept of length and the corresponding
optimization problems will not make sense or may not have unique solutions.

In the definition provided above, $F(x,y)$ signifies the magnitude of the
vector $y$ within the tangent space $T_{x}M$ originating from an arbitrary
point $x$ within the manifold $M$. This quantity is commonly referred to as
the ”F-length” or simply the ”Finslerian length” of the tangent vector $y$ in
$T_{x}M$. If we fix the point $x\in M$ in the manifold $M$, then $F(x,.)$ can
eat every tangent vector $y\in T_{x}M$ and spits a real number.
We can think of while considering physical problems involving position and
direction, the Finsler metric $F(x,y)$ as having two arguments $x$ and $y$
representing location and velocity at any point $p\in M$ respectively.

######  Definition 1.2.

(Finsler Manifold)
A differentiable manifold denoted as $M$, when equipped with a Finsler metric
$F(x,y)$, is referred to as a Finsler manifold or Finsler space. This is
typically denoted as $(M,F)$.

######  Definition 1.3.

(Finsler Geometry)
Consider a Finsler manifold denoted as $(M,F)$, where $F(x,y)$ represents a
Finsler metric. The branch of geometry that relies on the Finsler metric
$F(x,y)$ defined on the manifold $M$ is termed Finsler geometry.

In Finsler geometry, a significant and distinct class of Finsler metric known
as the $(\alpha,\beta)$-metric is of particular importance. This metric is
defined as follows:

######  Definition 1.4 ($(\alpha,\beta)$-metric).

Consider a Finsler space denoted as $(M,F(x,y))$, where $F(x,y)$ represents
the Finsler fundamental function. This space is said to possess an
$(\alpha,\beta)$-metric if the fundamental function $F(x,y)$ can be expressed
in the following manner:

$\displaystyle F(x,y)=F\left(\alpha(x,y),\beta(x,y)\right)$

In this expression, $F\left(\alpha(x,y),\beta(x,y)\right)$ is a differentiable
function of two variables $\alpha(x,y)$ and $\beta(x,y)$. Here, $\alpha(x,y)$
is the Riemannian fundamental function defined as
$\sqrt{a_{ij}(x)y^{i}y^{j}}$, where $a_{ij}(x)$ represents a Riemannian metric
tensor, and $\beta(x,y)$ is a differential 1-form defined on the tangent
bundle $TM$, with $b_{i}(x)$ representing a covariant vector field.

The class of $(\alpha,\beta)$-metrics was originally introduced by the
renowned geometer M. Matsumoto [9].

Some important examples of $(\alpha,\beta)$-metric are:

######  Example 1.5.

The metric defined by

$\displaystyle F(x,y)=\alpha(x,y)+\beta(x,y)$

is called Randers metric and the space
$\left(M,F(x,y)=\alpha(x,y)+\beta(x,y)\right)$ constructed with Randers metric
is called Randers space. This metric was first introduced by Physicist G.
Randers in 1941 [14], in his study of general relativity.

######  Example 1.6.

The metric defined by

$\displaystyle F=\frac{\alpha^{2}(x,y)}{\beta(x,y)},\beta>0$

is called Kropina metric. A space
$\left(M,F=\frac{\alpha^{2}(x,y)}{\beta(x,y)}\right)$ constructed with Kropina
metric is called Kropina space. This metric was introduced by the Russian
physicist V.K. Kropina [7]. It has many important and interesting applications
in physics, electron optics with a magnetic field, dissipative mechanics and
irreversible thermodynamics, relativistic field theory, control theory,
evolution and developmental biology [11].

######  Example 1.7.

The metric defined by

$\displaystyle F=\frac{\alpha^{n+1}(x,y)}{\beta^{n}(x,y)},\beta>0$

is called generalized Kropina metric. A space
$\left(M,F=\frac{\alpha^{n+1}(x,y)}{\beta^{n}(x,y)}\right)$ constructed with
Kropina metric is called generalized Kropina space.

######  Example 1.8.

The metric defined by

$\displaystyle
F=\frac{\alpha^{2}(x,y)}{\alpha(x,y)-\beta(x,y)},\alpha-\beta>0$

is called Matsumoto metric. A space
$\left(M,F=\frac{\alpha^{2}(x,y)}{\alpha(x,y)-\beta(x,y)}\right)$ constructed
with Matsumoto metric is called Matsumoto space. This metric was first
introduced by M. Matsumoto [8] while investigating the model of a Finsler
space. This metric is also named as slope metric.

######  Example 1.9.

The metric defined by

$\displaystyle F(x,y)=\frac{[\alpha(x,y)+\beta(x,y)]^{2}}{\alpha(x,y)}$

is called square metric. We say the space
$\left(M,F(x,y)=\frac{[\alpha(x,y)+\beta(x,y)]^{2}}{\alpha(x,y)}\right)$
constructed with square metric Shen’s square space.

######  Example 1.10.

The metric defined by

$\displaystyle
F(x,y)=\frac{[\alpha(x,y)+\beta(x,y)]^{n+1}}{[\alpha(x,y)]^{n}}$

is called generalized square metric. We say the space
$\left(M,F(x,y)=\frac{[\alpha(x,y)+\beta(x,y)]^{p+1}}{[\alpha(x,y)]^{p}}\right)$
constructed with generalized square metric the1.generalized square space or
the Shen’s generalized square space. If we put $n=1$ in above metric, i.e., in
generalized square metric, we get the square metric of definition (1.9).

Among these Finsler metrics this was the “generalized square metric” that
draws our attention to work with hypersurface of a Finsler space $(M,F(x,y))$.
Let us first define the meaning of a hypersurface:

######  Definition 1.11.

A hypersurface is an embedded submanifold of codimention 1. That is, a
submanifold of dimension less that 1 of a given manifold is called
hypersuraface of the underlying manifold.

This means, if we have a manifold $M$ such that dimension of $M$ is $n$, then
the submanifold having dimension $n-1$ will be understood as hypersurface of
the given manifold.
A submanifold of dimension $n-1$ is generally denoted by the symbol $M^{n-1}$.
In this paper, we will also denote a hupersurafce of a manifold $M$ by the
symbol $M^{n-1}$.

######  Example 1.12.

Let $M=R^{2}$ be a manifold. Then a straight line in Figure 3 denoted by
$M^{n-1}$ and defined by linear equation in two variables $x$ and $y$, i.e.,
$ax+by+c=0$ is a hypersuarface of the underlying manifold $M=R^{2}$, because
dim($M$)=dim($R^{2}$)=2 while the dim($M^{n-1}$)=$n-1=2-1=1$.

Figure 3: The line represented by eqution ax+by+c=0 is a hypersurface of the
2-dimensional manifold $R^{2}$.

######  Example 1.13.

Let $M=R^{3}$ be a manifold. Then the unit sphere in Figure 4 denoted by
$M^{n-1}$ and defined by $x^{2}+y^{2}+z^{2}=1$ is a hypersuarface of the
underlying manifold $M=R^{3}$, because dim($M$)=dim($R^{3}$)=3 while the
dim($M^{n-1}$)=$n-1=3-1=2$.

Figure 4: The sphere represented by equation $x^{2}+y^{2}+z^{2}=16$ is a
hypersurface of the 3-dimensional manifold $R^{3}$.

Matsumoto [10], a prominent Finslerian, was the first person who studies the
hypersurfaces and charecterised the special hypersurfaces $M^{n-1}$ of a
Finsler manifold. He, specifically, characterised the properties of
hypersurface $M^{n-1}$ of Randers space [14]. After this, the number of
Finslerians dramatically increased to show their interest in Finsler
hypersurface $M^{n-1}$. Many authors around the world ( [6], [5], [3], [15],
[16], [18], [2], [12], [1], [4]) did study the properties of special
hypersurface $M^{n-1}$ and derived the conditions under which a Finsler
hypersurface $M^{n-1}$ of a Finsler manifold $(M,F(x,y))$ becomes a hyperplane
of first kind, second kind but not of the third kind. Aim of the present paper
is to investigate the hypersurface $M^{n-1}$ of Finsler space using
generalized square metric
$F(x,y)=\frac{[\alpha(x,y)+\beta(x,y)]^{n+1}}{[\alpha(x,y)]^{n}}$.

##  2 Preliminaries

We consider the Finsler space $(M,F)$, where $F$ is the generalized square
metric, that is given by

$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$ (2.1)

Calculate all the partial derivatives of equation (2.1) up to second order, we
get

$\displaystyle F_{\alpha}$
$\displaystyle=\frac{(\alpha-n\beta)(\alpha+\beta)^{n}}{\alpha^{{n+1}}}$ (2.2)
$\displaystyle F_{\beta}$
$\displaystyle=\frac{(n+1)(\alpha+\beta)^{n}}{\alpha^{n}}$ (2.3)
$\displaystyle F_{\alpha\alpha}$
$\displaystyle=\frac{n(n+1)\beta^{2}(\alpha+\beta)^{n-1}}{\alpha^{n+2}}$ (2.4)
$\displaystyle F_{\beta\beta}$
$\displaystyle=\frac{n(n+1)(\alpha+\beta)^{n-1}}{\alpha^{n}}$ (2.5)
$\displaystyle F_{\alpha\beta}$
$\displaystyle=-\frac{n(n+1)\beta(\alpha+\beta)^{n-1}}{\alpha^{n+1}}$ (2.6)

We already know that, in a general Finsler manifold $(M,F)$, the normalized
element of support $l_{i}=\frac{\partial F}{\partial y_{i}}$ and the angular
metric tensor $h_{ij}$ [14] are evaluated by the following formula:

$\displaystyle l_{i}=\frac{F_{\alpha}y_{i}}{\alpha}+F_{\beta}b_{i}$ (2.7)
$\displaystyle
h_{ij}=pa_{ij}+q_{0}b_{i}b_{j}+q_{1}(b_{i}y_{j}+b_{j}y_{i})+q_{2}y_{i}y_{j},$
(2.8)

and the coefficients are defined and calculated as follows:

$\displaystyle y_{i}=a_{ij}y^{j}$

$\displaystyle p$
$\displaystyle=\frac{FF_{\alpha}}{\alpha}=\frac{(\alpha-n\beta)(\alpha+\beta)^{2n+1}}{\alpha^{2n+2}}$
(2.9) $\displaystyle q_{0}$
$\displaystyle=FF_{\beta\beta}=\frac{n(n+1)(\alpha+\beta)^{2n}}{\alpha^{2n}}$
(2.10) $\displaystyle q_{1}$
$\displaystyle=\frac{FF_{\alpha\beta}}{\alpha}=-\frac{n(n+1)\beta(\alpha+\beta)^{2n}}{\alpha^{2n+2}}$
(2.11) $\displaystyle q_{2}$
$\displaystyle=\frac{F(F_{\alpha\alpha}-\frac{F_{\alpha}}{\alpha})}{\alpha^{2}}$
$\displaystyle=\frac{(\alpha+\beta)^{2n}\left\\{n\beta(n+2\beta+n\beta)-\alpha(\alpha+\beta)\right\\}}{\alpha^{2n+4}}$
(2.12)

We also know that, in a general Finsler manifold $(M,F)$, the fundamental
metric tensor $g_{ij}=\frac{1}{2}\frac{\partial^{2}F^{2}}{\partial
y^{i}\partial y^{j}}$ is evaluated by [14] the following formula:

$\displaystyle
g_{ij}=pa_{ij}+p_{0}b_{i}b_{j}+p_{1}(b_{i}y_{j}+b_{j}y_{i})+p_{2}y_{i}y_{j}$
(2.13)

whereas its coefficients $p,p_{0},p_{1}$ and $p_{2}$ are defined and
calculated as follows:

$\displaystyle p$ $\displaystyle=\frac{FF_{\alpha}}{\alpha}$
$\displaystyle=\frac{(\alpha-n\beta)(\alpha+\beta)^{2n+1}}{\alpha^{2n+2}}$
(2.14) $\displaystyle p_{0}$ $\displaystyle=q_{0}+F_{\beta}^{2}$
$\displaystyle=\frac{(n+1)(2n+1)(\alpha+\beta)^{2n}}{\alpha^{2n}}$ (2.15)
$\displaystyle p_{1}$ $\displaystyle=q_{1}+\frac{pF_{\beta}}{F}$
$\displaystyle=\frac{(n+1)(\alpha+\beta)^{2n}(\alpha-2n\beta)}{\alpha^{2n+2}}$
(2.16) $\displaystyle p_{2}$ $\displaystyle=q_{2}+\frac{p^{2}}{F^{2}}$
$\displaystyle=\frac{\beta(\alpha+\beta)^{2n}\left\\{2n^{2}\beta+2n\beta-n\alpha-\alpha\right\\}}{\alpha^{2}{(n+2)}}$
(2.17)

We know that, in a Finsler manifold $(M,F)$, reciprocal metric tensor of a
fundamental metric tensor $g_{ij}=\frac{1}{2}\frac{\partial\partial
F^{2}}{\partial y^{i}\partial y^{j}}$ is denoted by $g^{ij}$ and is evaluated
by the formula [14]

$\displaystyle
g^{ij}=\frac{a^{ij}}{p}-S_{0}b^{i}b^{j}-S_{1}(b^{i}y^{j}+b^{j}y^{i})-S_{2}y^{i}y^{j}$
(2.18)

whereas its coefficients $b^{i}$, $S_{0}$, $S_{1}$ and $S_{2}$ are evaluated
by the following formulae:

$\displaystyle b^{i}$ $\displaystyle=a^{ij}b_{j}$
$S_{0}=\frac{pp_{0}+(p_{0}p_{2}-p_{1}^{2})\alpha^{2}}{p\zeta}$ (2.19)
$S_{1}=\frac{pp_{1}+(p_{0}p_{2}-p_{1}^{2})\beta}{p\zeta}$ (2.20)
$S_{2}=\frac{pp_{2}+(p_{0}p_{2}-p_{1}^{2})b^{2}}{p\zeta}$ (2.21)
$\zeta=p(p+p_{0}b^{2}+p_{1}\beta)+(p_{0}p_{2}-p_{1}^{2})(\alpha^{2}b^{2}-\beta^{2})$
(2.22)

where $b^{2}=a_{ij}b^{i}b^{j}$.
Let us define the $hv$-torsion tensor $C_{ijk}=\frac{1}{2}\frac{\partial
g_{ij}}{\partial y^{k}}$ as follows [17]:

$\displaystyle
C_{ijk}=\frac{p_{1}(h_{ij}m_{k}+h_{jk}m_{i}+h_{ki}m_{j})+\gamma_{1}m_{i}m_{j}m_{k}}{2p}$
(2.23)

and its coefficients $\gamma_{1}$ and $m_{i}$ are evaluated by the formulae

$\displaystyle\gamma_{1}=p\frac{\partial
p_{0}}{\partial\beta}-3p_{1}q_{0},m_{i}=b_{i}-\frac{y_{i}\beta}{\alpha^{2}}$
(2.24)

Here $m_{i}$ is known as non-zero covariant vector orthogonal to element of
support $y^{i}$. Let $\Gamma_{jk}^{i}$be the components of Christoffel symbol
of the associated Riemannian space $R^{n}$ and $\nabla_{k}$ be the covariant
differentiation with respect to $x^{k}$ relative to Christoffel symbol. Now we
put

$\displaystyle 2E_{ij}$ $\displaystyle=b_{ij}+b_{ji}$ (2.25) $\displaystyle
2F_{ij}$ $\displaystyle=b_{ij}-b_{ji}$ (2.26)

where $b_{ij}=\nabla_{j}b_{i}$. Let
$C\Gamma=(\Gamma_{jk}^{*i},\Gamma_{0k}^{*i},C_{jk}^{i})$ be Cartan connection
of $(M,F)$. The difference tensor
$D_{jk}^{i}=\Gamma_{jk}^{*i}-\Gamma_{jk}^{i}$ of the special Finsler manifold
$(M,F)$ is given by [17]

$\displaystyle D_{jk}^{i}=$ $\displaystyle
B^{i}E_{jk}+F_{k}^{i}B_{j}+F_{j}^{i}B_{k}+B_{j}^{i}b_{0k}+B_{k}^{i}b_{0j}-b_{0m}g^{im}B_{jk}-C_{jm}^{i}A_{k}^{m}$
$\displaystyle-
C_{km}^{i}A_{j}^{m}+C_{jkm}A_{s}^{m}g^{is}+\lambda^{s}(C_{jm}^{i}C{sk}^{m}+C_{km}^{i}C_{sj}^{m}-C_{jk}^{m}C_{ms}^{i})$
(2.27)

where

$\displaystyle B_{k}$ $\displaystyle=p_{0}b_{k}+p_{1}y_{k}$ (2.28)
$\displaystyle B^{i}$ $\displaystyle=g^{ij}B_{j}$ (2.29) $\displaystyle
B_{ij}$
$\displaystyle=\frac{p_{1}(a_{ij}-\frac{y_{i}y_{j}}{\alpha^{2}})+\frac{\partial
p_{0}}{\partial\beta}m_{i}m_{j}}{2}$ (2.30) $\displaystyle B_{i}^{k}$
$\displaystyle=g^{kj}B_{ji}$ (2.31) $\displaystyle A_{k}^{m}$
$\displaystyle=B_{k}^{m}E_{00}+B^{m}E_{k0}+B_{k}F_{0}^{m}+B_{0}F_{k}^{m}$
(2.32) $\displaystyle\lambda^{m}$ $\displaystyle=B^{m}E_{00}+2B_{0}F_{0}^{m}$
(2.33) $\displaystyle F_{i}^{k}$ $\displaystyle=g^{kj}F_{ji}$ (2.34)
$\displaystyle B_{0}$ $\displaystyle=B_{i}Y^{i}$ (2.35)

Here as well as henceforward ’0’ denotes tensorial contraction with $y^{i}$
besides $p_{0}$,$q_{0}$ and $S_{0}.$”

##  3 Induced Cartan Connection

Let $(M,F)$ be a Finsler manifold, where
$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$ is generalized
square metric. Also, let $M^{n-1}$ be a hypersurface of the Finsler manifold
$(M,F)$ whose hypothetical picture is depicted in the Figure 4.

Figure 5: A hypothetical picture of hypersurface $M^{n-1}$ of the
3-dimensional Finsler manifold $(M,F)$.

We will describe this hypersurface $M^{n-1}$ by following parametric
equations:

$\displaystyle
x^{i}=x^{i}(u^{\alpha}),(i=1,2,3,........n;\alpha=1,2,3,........n-1)$ (3.1)

where $u^{\alpha}$ is a parameter that represents coordinates on the
hypersurface $M^{n-1}$. Do you know why we are considering parametric equation
of the hypersurface? We considered parametric equation only because of the
fact that parametrization always makes our life simpler. Remember those days
when we used to consider parametric equations of curves and surfaces.
Moreover, using parametrizations we were able to calculate speed of the curves
as well as tangent planes and normal lines of the sphere. In elementary
differential geometry we also have learnt that parametrization helps us to
determine the shape, that is, curvature and torsion of a curves and surfaces.
Thus a parametrization has more information than the set of points
constituting the hypersurface $M^{n-1}$.

Now differentiating the equation (3.1) of the hypersurface with respect to
parameters $u^{\alpha}$, we get $B_{\alpha}^{i}=\frac{\partial x^{i}}{\partial
u^{\alpha}}$. Here each $B_{\alpha}^{i}=\frac{\partial x^{i}}{\partial
u^{\alpha}}$ for ($\alpha$=1,2,3,……..n-1) represents components of tangent
vectors and these tangent vectors $B_{\alpha}^{i}$ represent a tangent space
at a point $p$ of the hypersurface $M^{n-1}$ . Let the matrix corresponding to
first derivative $B_{\alpha}^{i}=\frac{\partial x^{i}}{\partial u^{\alpha}}$
be $[B_{\alpha}^{i}]=[\frac{\partial x^{i}}{\partial u^{\alpha}}]$, and it has
maximal rank , namely, (n-1). The maximal rank required here is to ensure that
tangent vectors forms linearly independent set so that any generic vector
tangent to $M^{n-1}$ is linearly expressible in terms of these linearly
independent tangent vectors. To introduce a Finsler structure in the
hypersurface $M^{n-1}$ , the supporting element $y^{i}$ at a point
$u^{\alpha}$ of $M^{n-1}$ is assumed to be tangential to $M^{n-1}$, so that we
may write

$\displaystyle y^{i}=B_{\alpha}^{i}(u)v^{\alpha}$ (3.2)

Therefore $v^{\alpha}$ is the element of support of hypersurface $M^{n-1}$ at
the point $u^{\alpha}$. The metric tensor $g_{\alpha\beta}$ and hv-torsion
tensor $C_{\alpha\beta\gamma}$ of hypersurface $M^{n-1}$ are defined by

$\displaystyle
g_{\alpha\beta}=g_{ij}B_{\alpha}^{i}B_{\beta}^{j},C_{\alpha\beta\gamma}=C_{ijk}B_{\alpha}^{i}B_{\beta}^{j}B_{\gamma}^{k}$
(3.3)

Now the unit normal vector $N^{i}(u,v)$ at an arbitrary point $u^{\alpha}$ of
the hypersurface $M^{n-1}$ is defined as follows:

######  Definition 3.1.

A vector $N^{i}(u,v)$ at a point $u^{\alpha}$ of the hypersurface $M^{n-1}$ is
said to be unit normal vector if

$\displaystyle
g_{ij}(x(u,v),y(u,v))B_{\alpha}^{i}N^{j}=0,g_{ij}(x(u,v),y(u,v))N^{i}N^{j}=1$
(3.4)

Let us define angular metric tensor $h_{ij}$ as follows:

######  Definition 3.2.

We say the tensor $h_{ij}$ an angular metric tensor, if $h_{ij}$ satisfies the
following conditions:

$\displaystyle
h_{\alpha\beta}=h_{ij}B_{\alpha}^{i}B_{\beta}^{j},h_{ij}B_{\alpha}^{i}N^{j}=0,h_{ij}N^{i}N^{j}=1$
(3.5)

Let $(B_{i}^{\alpha},N_{i})$ be the inverse of $(B_{\alpha}^{i},N^{i}),$ then
we have

$\displaystyle
B_{i}^{\alpha}=g^{\alpha\beta}g_{ij}B_{\beta}^{j},B_{\alpha}^{i}B_{i}^{\beta}=\delta_{\alpha}^{\beta},B_{i}^{\alpha}N^{i}=0,$
$\displaystyle
B_{\alpha}^{i}N_{i}=0,N_{i}=g_{ij}N^{i},B_{i}^{k}=g^{kj}B_{ji},$
$\displaystyle B_{\alpha}^{i}B_{j}^{\alpha}+N^{i}N_{j}=\delta_{j}^{i}$

We shall denote by $\mathcal{I}$$C\Gamma$ the connection of a hypersurface
$M^{n-1}$ induced from the Cartan connection $C\Gamma$. The induced Cartan
connection
$\mathcal{I}$$C\Gamma=(\Gamma_{\beta\gamma}^{*\alpha},G_{\beta}^{\alpha},C_{\beta\gamma}^{\alpha})$
on hypersurface $M^{n-1}$ induced from the Cartan’s connection
$C\Gamma=(\Gamma_{jk}^{*i},\Gamma_{0k}^{*i},C_{jk}^{i})$ is given by [10]

$\displaystyle\Gamma_{\beta\gamma}^{*\alpha}$
$\displaystyle=B_{i}^{\alpha}(B_{\beta\gamma}^{i}+\Gamma_{jk}^{*i}B_{\beta}^{j}B_{\gamma}^{k})+M_{\beta}^{\alpha}H_{\gamma}$
$\displaystyle G_{\beta}^{\alpha}$
$\displaystyle=B_{i}^{\alpha}(B_{0\beta}^{i}+\Gamma_{0j}^{*i}B_{\beta}^{j})$
$\displaystyle C_{\beta\gamma}^{\alpha}$
$\displaystyle=B_{i}^{\alpha}C_{jk}^{i}B_{\beta}^{j}B_{\gamma}^{k}$ (3.6)

where second fundamental $v$-tensor $M_{\beta\gamma}$ is defined by

$\displaystyle M_{\beta\gamma}$
$\displaystyle=N_{i}C_{jk}^{i}B_{\beta}^{j}B_{\gamma}^{k}$
$\displaystyle=N_{i}g^{li}C_{ljk}B_{\beta}^{j}B_{\gamma}^{k}\quad\text{(using
$C_{jk}^{i}=g^{li}C_{ljk}$)}$
$\displaystyle=N^{l}C_{ljk}B_{\beta}^{j}B_{\gamma}^{k}\quad\text{(using
$N^{i}g^{li}=N^{l}$)}$
$\displaystyle=C_{ijk}B_{\beta}^{i}B_{\gamma}^{j}N^{k}\quad\text{(adjusting
the indices $j$, $k$ and $l$)}$ (3.7) $\displaystyle M_{\beta}^{\alpha}$
$\displaystyle=g^{\alpha\gamma}M_{\beta\gamma}$

and normal curvature vector $H_{\beta}$ is defined by

$\displaystyle H_{\beta}$
$\displaystyle=N_{i}(B_{0\beta}^{i}+\Gamma_{0j}^{*i}B_{\beta}^{j}),$

where

$\displaystyle B_{\beta\gamma}^{i}$ $\displaystyle=\frac{\partial
B_{\beta}^{i}}{\partial U^{\gamma}}$ $\displaystyle B_{0\beta}^{i}$
$\displaystyle=B_{\alpha\beta}^{i}v^{\alpha}$

The quantities $M_{\beta\gamma}$ and $H_{\beta}$ appeared in above equations
are called the second fundamental $v$-tensor and normal curvature vector
respectively [10]. The second fundamental $h$-tensor $H_{\beta\gamma}$ is
defined as [10]

$\displaystyle
H_{\beta\gamma}=N_{i}(B_{\beta\gamma}^{i}+\Gamma_{jk}^{*i}B_{\beta}^{j}B_{\gamma}^{k})+M_{\beta}H_{\gamma}$
(3.8)

where

$\displaystyle
M_{\beta}=C_{jk}^{i}B_{\beta}^{j}N_{i}N^{k}=C_{ijk}B_{\beta}^{i}N^{j}N^{k}$
(3.9)

The relative $h$-covariant derivative and $v$-covariant derivative of
projection factor $B_{\alpha}^{i}$ with respect to induced Cartan connection
$\mathcal{I}$$C\Gamma$ are respectively given by

$\displaystyle B_{\alpha|\beta}^{i}=H_{\alpha\beta}N^{i}$ (3.10)
$\displaystyle B_{\alpha|\beta}^{i}=M_{\alpha\beta}N^{i}$ (3.11)

The equation (3.8) shows that $H_{\beta\gamma}$ is not always symmetric and

$\displaystyle
H_{\beta\gamma}-H_{\gamma\beta}=M_{\beta}H_{\gamma}-M_{\gamma}H_{\beta}$
(3.12)

Thus the above equation simplifies to

$\displaystyle H_{0\gamma}=H_{\gamma},H_{\gamma 0}=H_{\gamma}+M_{\gamma}H_{0}$
(3.13)

######  Definition 3.3 ( [10]).

A hypersurface $M^{n-1}$ of a Finsler manifold $(M,F)$ is said to be a
hyperplane of first kind if each path connecting two different points of the
hypersurface $M^{n-1}$ with respect to the induced Cartan connection
$\mathcal{I}$$C\Gamma$ is also becomes the path of the ambient Finsler
manifold $(M,F)$ with respect to Cartan connection $C\Gamma$.

######  Definition 3.4 ( [10]).

A hypersurface $M^{n-1}$ of a Finsler manifold $(M,F)$ is said to be a
hyperplane of second kind if each $h$-path of the hypersurface $M^{n-1}$ with
respect to the induced Cartan connection $\mathcal{I}$$C\Gamma$ is also the
$h$-path of the ambient Finsler manifold $(M,F)$ with respect to Cartan
connection $C\Gamma$.

######  Definition 3.5 ( [10]).

A hypersurface $M^{n-1}$ of a Finsler manifold $(M,F)$ is said to be a
hyperplane of third kind if unit normal vector $B^{i}$ of the hypersurface
$M^{n-1}$ with respect to the metric $F$ is parallel along each curve
$(u^{\alpha},v^{\alpha})$.

If one wants to prove a hypersurface a hyperplane of first kind, hyperplane of
second kind and hyperplane of third kind, it is very difficult to prove it
only with the help of definitions mentioned above, in that situation one
should incorporate below sufficient conditions given by M. Matsumoto to prove
the same:

######  Lemma 3.6 ( [10]).

The normal curvature $H_{0}=H_{\beta}v^{\beta}$ vanishes if and only if normal
curvature vector $H_{\beta}$ vanishes.

######  Lemma 3.7 ( [10]).

A hypersurface $M^{n-1}$is a hyperplane of first kind if and only if
$H_{\alpha}=0$.

######  Lemma 3.8 ( [10]).

A hypersurface $M^{n-1}$ is a hyperplane of second kind with respect to Cartan
connection $C\Gamma$ if and only if $H_{\alpha}=0$ and $H_{\alpha\beta}=0$.

######  Lemma 3.9 ( [10]).

A hypersurface $M^{n-1}$ is a hyperplane of third kind with respect to Cartan
connection $C\Gamma$ if and only if $H_{\alpha}=0,$ $H_{\alpha\beta}=0$ and
$M_{\alpha\beta}=0$.

##  4 Hypersurface $M^{n-1}$ of the special Finsler space

We know that a hypersurface is an embedded subspace of codimension 1. That is,
a subspace of dimension less that 1 than the dimension of a given manifold is
called hypersuraface of the given manifold or space. For the sake of our study
we are dealing with hypersurfaces rather than the subspaces of arbitrary
dimensions. In this paper we are specifically confined to Finslerian
hypersurfaces $M^{n-1}$.
Let us proof the following propositions in context of Finslerian
hypersurfaces.

######  Proposition 4.1.

Let $(M,F)$ be a Finsler manifold, where
$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n\in N$, is a
generalized square metric and $M^{n-1}$ be its hypersurface. Then fundamental
function of the hypersurface $M^{n-1}$ induced from the Finsler manifold
$(M,F)$ is a Riemannian metric.

###### Proof.

It is given that $(M,F)$ be a Finsler manifold, where
$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n\in N$, is a
generalized square metric. Let level equation of the hypersurface $M^{n-1}$ be
given by

$b(x)=c$

where c is a real number.
Take the gradient of the above level equation representing hypersurface
$M^{n-1}$, we get

$b_{i}(x)=\partial_{i}b$

Again consider the parametric equation of the same hypersurface $M^{n-1}$ as

$x^{i}=x^{i}(u^{\alpha})$

Differentiating the equation of hypersurface $b(x(u))=c$ with respect to
parameter $u^{\alpha}$, we get

$\displaystyle\frac{\partial b(x(u))}{\partial x^{i}}\frac{\partial
x^{i}}{\partial u^{\alpha}}=0$ $\displaystyle b_{i}(x)B^{i}_{\alpha}=0$

where $b_{i}(x)=\frac{\partial b(x(u))}{\partial x^{i}}$ and
$B^{i}_{\alpha}=\frac{\partial x^{i}}{\partial u^{\alpha}}$

This implies that $b_{i}(x)$ are normal vector field (covariant component) of
hypersurface $M^{n-1}$.

Thus at any point of the hypersurface $M^{n-1}$ we now have

$\displaystyle b_{i}B_{\alpha}^{i}=0$ (4.1) $\displaystyle
b_{i}y^{i}=0,i.e.,\beta=0$ (4.2)

Now, we will see how generalized square metric
$F=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n\in N$, induces a metric on the
hypersurface $M^{n-1}$. In this case we will denote induced metric by
$\bar{F}$. First consider the generalized square metric

$\displaystyle F$ $\displaystyle=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$
$\displaystyle=\frac{(\sqrt{a_{ij}y^{i}y^{j}}+b_{i}y^{i})^{n+1}}{(\sqrt{a_{ij}y^{i}y^{j}})^{n}}$
$\displaystyle=\frac{\left(\sqrt{a_{ij}B_{\alpha}^{i}(u)v^{\alpha}B_{\beta}^{j}(u)v^{\beta}}+b_{i}y^{i}\right)^{n+1}}{\left(\sqrt{a_{ij}B_{\alpha}^{i}(u)v^{\alpha}B_{\beta}^{j}(u)v^{\beta}}\right)^{n}}$
$\displaystyle=\frac{\left(\sqrt{a_{ij}B_{\alpha}^{i}(u)B_{\beta}^{j}(u)v^{\alpha}v^{\beta}}+b_{i}y^{i}\right)^{n+1}}{\left(\sqrt{a_{ij}B_{\alpha}^{i}(u)B_{\beta}^{j}(u)v^{\alpha}v^{\beta}}\right)^{n}}$

which is the general induced metric on the corresponding hypersurface
$M^{n-1}$. Using equation (4.2), general induced metric of the hypersurface
becomes

$\displaystyle F(u,v)$
$\displaystyle=\frac{\left(a_{ij}B_{\alpha}^{i}(u)B_{\beta}^{j}(u)v^{\alpha}v^{\beta}\right)^{\frac{n+1}{2}}}{\left(a_{ij}B_{\alpha}^{i}(u)B_{\beta}^{j}(u)v^{\alpha}v^{\beta}\right)^{\frac{n}{2}}}$
$\displaystyle=\sqrt{a_{ij}B_{\alpha}^{i}(u)B_{\beta}^{j}(u)v^{\alpha}v^{\beta}}$
$\displaystyle\bar{F}(u,v)$
$\displaystyle=\sqrt{a_{\alpha\beta}v^{\alpha}v^{\beta}}$ (4.3)

where $a_{\alpha\beta}=a_{ij}B_{\alpha}^{i}(u)B_{\beta}^{j}(u)$.
Thus function represented by equation (4.3) is fundamental function or the
metric of the hypersurface $M^{n-1}$ induced from the ambient Finsler manifold
$(M,F)$.

The fundamental function of the hypersurface $M^{n-1}$ represented by equation
(4.3) do not have $\beta$ component as $\beta=b_{i}y^{i}=0$ over the
hypersurface $M^{n-1}$ therefore fundamental function of the hypersurface
$M^{n-1}$ induced from the Finsler manifold $(M,F)$ is a Riemannian metric. ∎

######  Proposition 4.2.

Let $(M,F)$ be a Finsler manifold, where
$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n\in N$, is a
generalized square metric and $M^{n-1}$ be its associated hypersurface. Then
the covariant and contravariant components of normal vector field on the
hypersurface $M^{n-1}$ are given by

  1. 1.

$b_{i}=\sqrt{\frac{b^{2}}{1+n(n+1)}}N_{i}$

  2. 2.

$b^{i}=\sqrt{b^{2}\left\\{1+n(n+1)\right\\}}N^{i}+\frac{b^{2}}{\alpha}y^{i}$

###### Proof.

It is given that $M^{n-1}$ is a hypersurface of the manifold $(M,F)$, where
$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n\in N$, is a
generalized square metric.

Moreover, we know from equation (4.2) that $\beta=0$ over the hypersurface
$M^{n-1}$. Let us calculate the value of $p$, $p_{0}$, $p_{1}$ and $p_{2}$.
For that, substitute the value of $\beta=0$ into equations (2.14), (2.15),
(2.16), and (2.17), we get

$\displaystyle p=1,p_{0}=(n+1)(2n+1),p_{1}=\frac{n+1}{\alpha},p_{2}=0$ (4.4)

Now put the values of $p$, $p_{0}$, $p_{1}$, $p_{2}$ into equations (2.19),
(2.20), (2.21) and (2.22), we get

$\displaystyle S_{0}$ $\displaystyle=\frac{n(n+1)}{1+n(n+1)b^{2}}$ (4.5)
$\displaystyle S_{1}$
$\displaystyle=\frac{n+1}{\alpha\left\\{1+n(n+1)b^{2}\right\\}}$ (4.6)
$\displaystyle S_{2}$
$\displaystyle=-\frac{(n+1)^{2}b^{2}}{\alpha^{2}\left\\{1+n(n+1)b^{2}\right\\}}$
(4.7) $\displaystyle\zeta$ $\displaystyle=1+n(n+1)b^{2}$ (4.8)

Substituting the values of $p,S_{0},S_{1},S_{2}$ from the equations (4.4),
(4.5), (4.6) and (4.7) into equation (2.18), we have

$\displaystyle g^{ij}$
$\displaystyle=\frac{a^{ij}}{1}-\frac{n(n+1)}{1+n(n+1)b^{2}}\times
b^{i}b^{j}-\frac{n+1}{\alpha\left\\{1+n(n+1)b^{2}\right\\}}\times(b^{i}y^{j}+b^{j}y^{i})+$
$\displaystyle\frac{(n+1)^{2}b^{2}}{\alpha^{2}\left\\{1+n(n+1)b^{2}\right\\}}\times
y^{i}y^{j}$ (4.9)

Multiplying equation 4.9 by $b_{i}b_{j}$ and using $\beta=b_{i}y^{i}=0$, over
the hypersurface $M^{n-1}$, it becomes

$\displaystyle g^{ij}$ $\displaystyle=a^{ij}\times
b_{i}b_{j}-\frac{n(n+1)}{1+n(n+1)b^{2}}\times b^{i}b^{j}\times
b_{i}b_{j}-\frac{n+1}{\alpha\left\\{1+n(n+1)b^{2}\right\\}}\times(b^{i}y^{j}+b^{j}y^{i})\times
b_{i}b_{j}+$
$\displaystyle\frac{(n+1)^{2}b^{2}}{\alpha^{2}\left\\{1+n(n+1)b^{2}\right\\}}\times
y^{i}y^{j}\times b_{i}b_{j}$
$\displaystyle=(a^{ij}b_{i})b_{j}-\frac{n(n+1)}{1+n(n+1)b^{2}}\times(b^{i}b_{i})(b^{j}b_{j})-$
$\displaystyle\frac{n+1}{\alpha\left\\{1+n(n+1)b^{2}\right\\}}\times\left\\{(b^{i}b_{i})(b_{j}y^{j})+(b_{i}b^{j})(b_{i}y^{i})\right\\}+\frac{(n+1)^{2}b^{2}}{\alpha^{2}\left\\{1+n(n+1)b^{2}\right\\}}\times\left\\{(b_{i}y^{i})(b_{j}y^{j})\right\\}$
$\displaystyle=b^{j}b_{j}-\frac{n(n+1)}{1+n(n+1)b^{2}}\times(b^{2})(b^{2})-$
$\displaystyle\frac{n+1}{\alpha\left\\{1+n(n+1)b^{2}\right\\}}\times\left\\{(b^{2})(0)+(b^{2})(0)\right\\}+\frac{(n+1)^{2}b^{2}}{\alpha^{2}\left\\{1+n(n+1)b^{2}\right\\}}\times\left\\{(0)(0)\right\\}$
$\displaystyle=b^{2}-\frac{n(n+1)}{1+n(n+1)b^{2}}\times b^{4}$ $\displaystyle
g^{ij}b_{i}b_{j}$ $\displaystyle=\frac{b^{2}}{1+n(n+1)}$

Thus at any generic point of the hypersurface $M^{n-1}$, we have

$\displaystyle g^{ij}b_{i}b_{j}$ $\displaystyle=\frac{b^{2}}{1+n(n+1)}$

Now from the above equation and using the equation (3.4), we get

$\displaystyle b_{i}=\sqrt{\frac{b^{2}}{1+n(n+1)}}N_{i}$ (4.10)

which is the covariant component of the normal vector field on the
hypersurface $M^{n-1}$. Now from (4.9) and (4.10) we get

$\displaystyle b^{i}$ $\displaystyle=a^{ij}b_{j}$
$\displaystyle=\sqrt{b^{2}\left\\{1+n(n+1)\right\\}}N^{i}+\frac{b^{2}}{\alpha}y^{i}$
(4.11)

which is the contravariant component of the normal vector field on the
hypersurface $M^{n-1}$. ∎

######  Proposition 4.3.

Let $(M,F)$ be a Finsler manifold, where
$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n\in N$, is a
generalized square metric and $M^{n-1}$ be its associated hypersurface. Then
second fundamental v-tensor of hypersurface $M^{n-1}$ is given by

$M_{\alpha\beta}=\frac{(n+1)}{2\alpha}\left(\sqrt{\frac{b^{2}}{1+n(n+1)}}\right)h_{\alpha\beta}$

and second fundamental h-tensor $H_{\alpha\beta}$ is symmetric, i.e.,
$H_{\alpha\beta}=H_{\beta\alpha}$.

###### Proof.

It is given that $M^{n-1}$ is a hypersurface of the manifold $(M,F)$, where
$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n\in N$, is a
generalized square metric.

Moreover, we know from equation (4.2) that $\beta=0$ over the hypersurface
$M^{n-1}$. Put the value of $\beta=0$ into equations (2.14), (2.15), (2.16),
and (2.17), we get

$\displaystyle p=1,p_{0}=(n+1)(2n+1),p_{1}=\frac{n+1}{\alpha},p_{2}=0$

Now, put the values of $p,p_{0},p_{1}$ and $p_{2}$ obtained above into
equation 2.13, we get fundamental metric tensor of the hypersurace $M^{n-1}$

$\displaystyle
g_{ij}=a_{ij}+(n+1)(2n+1)b_{i}b_{j}+\frac{(n+1)}{\alpha}(b_{i}y_{j}+b_{j}y_{i})$
(4.12)

Let us calculate the value of $q_{0}$, $q_{1}$ and $q_{2}$. For that,
substitute the value of $\beta=0$ into equations (2.10), (2.11), and (2.12),
we get

$\displaystyle q_{0}=n(n+1),q_{1}=0,q_{2}=-\frac{1}{\alpha^{2}}$

Substituting the values of $p,q_{0},q_{1}$ and $q_{2}$ in equation 2.8, we get
angular metric tensor of the hypersurface $M^{n-1}$

$\displaystyle h_{ij}=a_{ij}+n(n+1)b_{i}b_{j}-\frac{1}{\alpha^{2}}y_{i}y_{j}$
(4.13)

Differentiating equation 2.15 with respect to $\beta$, we have

$\displaystyle\frac{\partial
p_{0}}{\partial\beta}=\frac{2n(n+1)(2n+1)(\alpha+\beta)^{2n-1}}{\alpha^{2n}}.$

We know from equation (4.2) that $\beta=0$ over the hypersurface $M^{n-1}$ so
put the value of $\beta=0$ into above equation and equation 2.24, we get

$\displaystyle\frac{\partial p_{0}}{\partial\beta}$
$\displaystyle=\frac{2n(n+1)(2n+1)}{\alpha}$ $\displaystyle\gamma_{1}$
$\displaystyle=\frac{n(n^{2}-1)}{\alpha}$ $\displaystyle m_{i}$
$\displaystyle=b_{i}$

Using the values of $p,p_{1},\gamma_{1}$ and $m_{i}$ in equation 2.23, hv-
torsion tensor on the hypersurface $M^{n-1}$, becomes

$\displaystyle
C_{ijk}=\frac{(n+1)\left[(h_{ij}b_{k}+h_{jk}b_{i}+h_{ki}b_{j})+n(n^{2}-1)b_{i}b_{j}b_{k}\right]}{2\alpha}$
(4.14)

Substituting the value of $C_{ijk}$ from equation 4.14 in equation 3.7 as
follows:

$\displaystyle M_{\beta\gamma}=C_{ijk}B_{\beta}^{i}B_{\gamma}^{j}N^{k}$
$\displaystyle\therefore
M_{\alpha\beta}=C_{ijk}B_{\alpha}^{i}B_{\beta}^{j}N^{k}$
$\displaystyle=\left[\frac{(n+1)\left(h_{ij}b_{k}+h_{jk}b_{i}+h_{ki}b_{j})+n(n^{2}-1)b_{i}b_{j}b_{k}\right]}{2\alpha}\right]B_{\alpha}^{i}B_{\beta}^{j}N^{k}$
$\displaystyle=\left[\frac{(n+1)\left\\{h_{ij}B_{\alpha}^{i}B_{\beta}^{j}b_{k}+h_{jk}B_{\beta}^{j}(b_{i}B_{\alpha}^{i})+h_{ki}B_{\alpha}^{j}(b_{j}B_{\beta}^{j})\right\\}+n(n^{2}-1)(b_{i}B_{\alpha}^{i})(b_{j}B_{\beta}^{j})b_{k}}{2\alpha}\right]N^{k}$

Using the equation 4.1 in above expression, we get

$\displaystyle=\left[\frac{(n+1)\left\\{h_{ij}B_{\alpha}^{i}B_{\beta}^{j}b_{k}+h_{jk}B_{\beta}^{j}(0)+h_{ki}B_{\alpha}^{j}(0)\right\\}+n(n^{2}-1)(0)(0)b_{k}}{2\alpha}\right]N^{k}$
$\displaystyle=\left[\frac{(n+1)h_{ij}B_{\alpha}^{i}B_{\beta}^{j}b_{k}}{2\alpha}\right]N^{k}$
$\displaystyle=\left[\frac{(n+1)(h_{ij}B_{\alpha}^{i}B_{\beta}^{j})b_{k}N^{k}}{2\alpha}\right]$

Using the equation 3.5, we get

$\displaystyle=\left[\frac{(n+1)h_{\alpha\beta}b_{k}N^{k}}{2\alpha}\right]$

Using the equation 4.10 in above expression, we get

$\displaystyle=\frac{(n+1)}{2\alpha}\left(\sqrt{\frac{b^{2}}{1+n(n+1)}}N_{k}\right)h_{\alpha\beta}N^{k}$
$\displaystyle=\frac{(n+1)}{2\alpha}\left(\sqrt{\frac{b^{2}}{1+n(n+1)}}\right)h_{\alpha\beta}N_{k}N^{k}$

We know that $N_{k}N^{k}=1$. Use this fact in above expression, we get

$\displaystyle M_{\alpha\beta}$
$\displaystyle=\frac{(n+1)}{2\alpha}\left(\sqrt{\frac{b^{2}}{1+n(n+1)}}\right)h_{\alpha\beta}$
(4.15)

Again, substituting the value of $C_{ijk}$ from equation 4.14 into equation
3.9 as follows:

$\displaystyle M_{\beta}=C_{ijk}B_{\beta}^{i}N^{j}N^{k}$
$\displaystyle\therefore M_{\alpha}=C_{ijk}B_{\alpha}^{i}N^{j}N^{k}$
$\displaystyle=\left[\frac{(n+1)\left(h_{ij}b_{k}+h_{jk}b_{i}+h_{ki}b_{j})+(n^{2}-1)b_{i}b_{j}b_{k}\right]}{2\alpha}\right]B_{\alpha}^{i}N^{j}N^{k}$
$\displaystyle=\left[\frac{(n+1)\left(h_{ij}b_{k}B_{\alpha}^{i}N^{j}N^{k}+h_{jk}b_{i}B_{\alpha}^{i}N^{j}N^{k}+h_{ki}b_{j}B_{\alpha}^{i}N^{j}N^{k})+n(n^{2}-1)b_{i}B_{\alpha}^{i}b_{j}b_{k}N^{j}N^{k}\right]}{2\alpha}\right]$

Using equations 4.1 and 3.5 in above expression, we get

$\displaystyle M_{\alpha}=0$ (4.16)

Substituting the value of $M_{\alpha}$ from the equation 4.16 in equation 3.12
as follows:

$\displaystyle H_{\beta\gamma}-H_{\gamma\beta}$
$\displaystyle=M_{\beta}H_{\gamma}-M_{\gamma}H_{\beta}$ $\displaystyle
H_{\beta\gamma}-H_{\gamma\beta}$ $\displaystyle=0\times H_{\gamma}-0\times
H_{\beta}$ $\displaystyle H_{\beta\gamma}-H_{\gamma\beta}$ $\displaystyle=0$
$\displaystyle H_{\beta\gamma}=H_{\gamma\beta}$ $\displaystyle\therefore
H_{\alpha\beta}=H_{\beta\gamma}$

which shows that $H_{\alpha\beta}$ is symmetric. ∎

######  Theorem 4.4.

Let $(M,F)$ be a Finsler manifold, where
$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n\in N$, is a
generalized square metric and $M^{n-1}$ be its associated hypersurface. Then
the hypersurface $M^{n-1}$ will be hyperplane of first kind if and only if

$\displaystyle 2b_{ij}=b_{i}c_{j}+b_{j}c_{i}$

Moreover we show that second fundamental tensor $H_{\alpha\beta}$ of $M^{n-1}$
is proportional to it’s angular metric tensor $h_{\alpha\beta}$. That is,
$H_{\alpha\beta}=\frac{c_{0}b}{\sqrt{1+n(n+1)}}h_{\alpha\beta}$.

###### Proof.

Let us differentiate equation 4.1 with respect to $\beta$, we get

$\displaystyle b_{i|\beta}B_{\alpha}^{i}+b_{i}B_{\alpha|\beta}^{i}=0$ (4.17)

Put the value of $B_{\alpha|\beta}^{i}$ from equation 3.10 and
$b_{i|\beta}=b_{i|j}B_{\beta}^{j}+b_{i|j}N^{j}H_{\beta}$ into equation 4.17,
we get

$\displaystyle
b_{i|j}B_{\beta}^{j}B_{\alpha}^{i}+b_{i|j}N^{j}H_{\beta}B_{\alpha}^{i}+b_{i}H_{\alpha\beta}N^{i}=0$
(4.18)

We know that

$\displaystyle b_{i|j}=-b_{h}C_{ij}^{h}$

Put the value of $b_{h}$ from equation 4.10 into above expression as follows:

$\displaystyle b_{i|j}$ $\displaystyle=-b_{h}C_{ij}^{h}$ $\displaystyle
b_{i|j}$ $\displaystyle=-\sqrt{\frac{b^{2}}{1+n(n+1)}}N_{h}C_{ij}^{h}$
$\displaystyle b_{i|j}B_{\alpha}^{i}N^{j}$
$\displaystyle=-\sqrt{\frac{b^{2}}{1+n(n+1)}}N_{h}C_{ij}^{h}B_{\alpha}^{i}N^{j}$
$\displaystyle=-\sqrt{\frac{b^{2}}{1+n(n+1)}}M_{\alpha}$
$\displaystyle=-\sqrt{\frac{b^{2}}{1+n(n+1)}}\times 0\quad\text{(using
Equation ~{}\eqref{eq4.16} )}$ $\displaystyle=0$

Using $b_{i|j}B_{\alpha}^{i}N^{j}=0$ and equation 4.10 in the equation 4.18
and then using the fact that $N_{i}N^{i}=1$, we get

$\displaystyle
b_{i|j}B_{\beta}^{j}B_{\alpha}^{i}+\sqrt{\frac{b^{2}}{1+n(n+1)}}H_{\alpha\beta}=0$
(4.19)

It is obvious that $b_{i|j}$ is symmetric. Now contracting 4.19 with
$v^{\beta}$ first and then with $v^{\alpha}$ respectively and using the
equations 3.2, 3.13 and 4.16, we get

$\displaystyle
b_{i|j}B_{\alpha}^{i}y^{j}+\sqrt{\frac{b^{2}}{1+n(n+1)}}H_{\alpha}=0$ (4.20)
$\displaystyle b_{i|j}y^{i}y^{j}+\sqrt{\frac{b^{2}}{1+n(n+1)}}H_{0}=0$ (4.21)

We know from the Lemma 3.6 and Lemma 3.7, a hypersurface $M^{n-1}$ is a
hyperplane of first kind if and only if normal curvature vanishes, i.e.,
$H_{0}=0$. Using the value $H_{0}=0$ in equation 4.21 we find that
hypersurface $M^{n-1}$ is a hyperplane of first kind if and only if
$b_{i|j}y^{i}y^{j}=0$. This $b_{i|j}$ is the covariant derivative of with
respect to Cartan connection $C\Gamma$ of Finsler space $F$, it may depend on
$y^{i}$. Moreover $\nabla_{j}b_{i}=b_{ij}$ is the covariant derivative of
$b_{i}$ with respect to Riemannian connection $\Gamma_{jk}^{i}$ constructed
from $a_{ij}(x)$, therefore $b_{ij}$ dose not depend on $y^{i}$. We shall
consider the difference $b_{i|j}-b_{ij}$ of above covariant derivatives in
further discussion. The difference tensor
$D_{jk}^{i}=\Gamma_{jk}^{*i}-\Gamma_{jk}^{i}$ is given by equation THEORY OF
HYPERSURFACES OF A FINSLER SPACE WITH THE GENERALIZED SQUARE METRIC. Since
$b_{i}$ is a gradient vector, from equations 2.25 and 2.26 we have

$\displaystyle E_{ij}=b_{ij},F_{ij}=0,F_{j}^{i}=0$ (4.22)

Using equation 4.22 into equation THEORY OF HYPERSURFACES OF A FINSLER SPACE
WITH THE GENERALIZED SQUARE METRIC, we get

$\displaystyle
D_{jk}^{i}=b_{jk}B^{i}+b_{0k}B_{j}^{i}+b_{0j}B_{k}^{i}-b_{0m}g^{im}B_{jk}$
$\displaystyle-A_{k}^{m}C_{jm}^{i}-A_{j}^{m}C_{km}^{i}+A_{s}^{m}C_{jkm}g^{is}$
(4.23)
$\displaystyle+\lambda^{s}(C_{sk}^{m}C_{jm}^{i}+C_{sj}^{m}C_{km}^{i}-C_{ms}^{i}C_{jk}^{m})$

Using the equations 4.2, 4.4, 4.5 and 4.6 into equations 2.28 to 2.23, we get

$\displaystyle B_{k}$
$\displaystyle=(n+1)(2n+1)b_{k}+\frac{n+1}{\alpha}y_{k},B^{i}=bb^{i}+by^{i}$
(4.24) $\displaystyle B_{ij}$
$\displaystyle=\frac{(n+1)\left\\{a_{ij}\alpha^{2}-y_{i}y_{j}+2n(n+1)b_{i}b_{j}\alpha\right\\}}{2\alpha^{3}}$
(4.25) $\displaystyle B_{j}^{i}$ $\displaystyle=0$ (4.26) $\displaystyle
A_{k}^{m}$ $\displaystyle=0,\lambda^{m}=B^{m}b_{00}.$ (4.27)

Using tensor contraction operation with equations 4.25 and 4.26 by $y^{j}$, we
get $B_{i0}=0,$ $B_{0}^{i}=0.$ Further contracting equation 4.27 by $y^{k}$
and using the fact that $B_{0}^{i}=0$, we get $A_{0}^{m}=B^{m}b_{00}.$
Contracting equation 4.23 by $y^{k}$ and using the facts $B_{i0}=0$,
$B_{0}^{i}=0$, $A_{0}^{m}=B^{m}b_{00}$ and $C_{s0}^{m}=0$, $C_{0m}^{i}=0$,
$C_{j0}^{m}=0$ obtained by contracting equations 4.25, 4.26, 4.27 and 3.6, we
get

$\displaystyle D_{j0}^{i}$
$\displaystyle=B^{i}b_{j0}+B_{j}^{i}b_{00}-b_{00}B^{m}C_{jmi}^{i}$ (4.28)
$\displaystyle D_{00}^{i}$ $\displaystyle=bb^{i}b_{00}+by^{i}b_{00}$ (4.29)

Multiplying equation 4.25 by $b_{i}$ and then using equations 4.2, 4.21 and
4.23, we get

$\displaystyle
b_{i}D_{j0}^{i}=bb_{j0}+bb_{j}b_{00}-bb_{i}b^{m}C_{jm}^{i}b_{00}$ (4.30)

Now multiplying equation 4.26 by $b_{i}$ and then using equation 4.2 we get

$\displaystyle b_{i}D_{00}^{i}=\frac{b^{2}}{1+n(n+1)}b_{00}$ (4.31)

From equations 4.14 and 4.16 it is clear that

$\displaystyle
b^{m}b_{i}C_{jm}^{i}B_{\alpha}^{j}=\frac{b^{2}}{1+n(n+1)}M_{\alpha}=0$ (4.32)

Contracting the expression $b_{i|j}=b_{ij}-b_{r}D_{ij}^{r}$ by $y^{i}$ and
$y^{j}$ respectively and then using equation 4.31 we get

$\displaystyle
b_{i|j}y^{i}y^{j}=b_{00}-b_{r}D_{00}^{r}=\frac{b^{2}}{1+n(n+1)}b_{00}$

Put $b_{i|j}=b_{ij}-b_{r}D_{ij}^{r}$ in equations 4.17 and 4.18 and then using
equations 4.27, 4.1 and 4.29 and the value of $b_{i|j}y^{i}y^{j}$ above,
equations 4.20 and 4.21 can be written as

$\displaystyle\sqrt{\frac{b^{2}}{1+n(n+1)}}b_{i0}B_{\alpha}^{i}+bH_{\alpha}=0$
(4.33) $\displaystyle\sqrt{\frac{b^{2}}{1+n(n+1)}}b_{00}+bH_{0}=0$ (4.34)

From the equation 4.31 it is clear that the condition $H_{0}=0$ is equivalent
to $b_{00}=0,$ where $b_{ij}$ is independent of $y^{i}$. Since $y^{i}$ satisfy
equation 4.2, the condition can be written as
$b_{ij}y^{i}y^{j}=(b_{i}y^{i})(c_{j}y^{j})$ for some $c_{j}(x)$, so that we
have

$\displaystyle 2b_{ij}=b_{i}c_{j}+b_{j}c_{i}$ (4.35)

Thus we shown that a Finslerian hypersurface $M^{n-1}$ will be hyperplane of
first kind if and only if $2b_{ij}=b_{i}c_{j}+b_{j}c_{i}$.

Now we try to show that second fundamental tensor $H_{\alpha\beta}$ of the
hypersurface $M^{n-1}$ is proportional to its angular metric tensor
$h_{\alpha\beta}$.

For that, contracting equation 4.35 and using the fact that $b_{i}y^{j}=0$, we
get $b_{00}=0$. This implies that the condition $b_{00}=0$ and
$2b_{ij}=b_{i}c_{j}+b_{j}c_{i}$ are equivalent.

Multiplying equation 4.35 by $B_{\alpha}^{i}$ and then $B_{\beta}^{j}$ and
using equations (4.1) and (4.2), we have

$\displaystyle 2b_{ij}\times B_{\alpha}^{i}$
$\displaystyle=\left(b_{i}B_{\alpha}^{i}\right)c_{j}+b_{j}c_{i}B_{\alpha}^{i}$
$\displaystyle 2b_{ij}B_{\alpha}^{i}$ $\displaystyle=0\times
c_{j}+b_{j}c_{i}B_{\alpha}^{i}$ $\displaystyle 2b_{ij}B_{\alpha}^{i}\times
B_{\beta}^{j}$
$\displaystyle=\left(b_{j}B_{\beta}^{j}\right)c_{i}B_{\alpha}^{i}$
$\displaystyle 2b_{ij}B_{\alpha}^{i}B_{\beta}^{j}$ $\displaystyle=0\times
c_{i}B_{\alpha}^{i}$ $\displaystyle 2b_{ij}B_{\alpha}^{i}B_{\beta}^{j}$
$\displaystyle=0$ $\displaystyle b_{ij}B_{\alpha}^{i}B_{\beta}^{j}$
$\displaystyle=0$

Again, multiplying equation 4.35 by $B_{\alpha}^{i}$ and $y^{j}$ and then
using Equation (4.1), we have

$\displaystyle 2b_{ij}\times B_{\alpha}^{i}y^{j}$
$\displaystyle=\left(b_{i}B_{\alpha}^{i}\right)c_{j}y^{j}+\left(b_{j}y^{j}\right)c_{i}B_{\alpha}^{i}$
$\displaystyle 2b_{ij}B_{\alpha}^{i}y^{j}$ $\displaystyle=0\times
c_{j}y^{j}+0\times c_{i}B_{\alpha}^{i}$ $\displaystyle
2b_{ij}B_{\alpha}^{i}y^{j}$ $\displaystyle=0$ $\displaystyle
b_{ij}B_{\alpha}^{i}y^{j}$ $\displaystyle=0$ $\displaystyle
b_{i0}B_{\alpha}^{i}$ $\displaystyle=0\quad\text{(contraction by $y^{j}$ is
taking place)}$

Again, consider equation (4.35)

$\displaystyle 2b_{ij}$ $\displaystyle=b_{i}c_{j}+b_{j}c_{i}$ $\displaystyle
2b_{ij}y^{j}$
$\displaystyle=b_{i}\left(c_{j}y^{j}\right)+\left(b_{j}y^{j}\right)c_{i}\quad\text{(multiplying
by $y^{j}$ both sides)}$ $\displaystyle 2b_{i0}$
$\displaystyle=b_{i}c_{0}+0\times c_{i}\quad\text{(contraction by $y^{j}$ is
taking place)}$ $\displaystyle 2b_{i0}$ $\displaystyle=b_{i}c_{0}$
$\displaystyle 2b_{i0}b^{i}$
$\displaystyle=\left(b_{i}b^{i}\right)c_{0}\quad\text{(multiplying by $b^{i}$
both sides)}$ $\displaystyle 2b_{i0}b^{i}$
$\displaystyle=b^{2}c_{0}\quad\text{ ($\because b^{2}=b_{i}b^{i}$)}$
$\displaystyle b_{i0}b^{i}$ $\displaystyle=\frac{b^{2}c_{0}}{2}$

Using this in equation 4.30 gives $H_{\alpha}=0.$ Now using 4.23 and 4.24 and
using $b_{00}=0$ and $b_{ij}B_{\alpha}^{i}B_{\beta}^{j}=0$, we get
$\lambda^{m}=0,$ $A_{j}^{i}B_{\beta}^{j}=0$ and
$B_{ij}B_{\alpha}^{i}B_{\beta}^{j}=\frac{1}{2\alpha}h_{\alpha\beta}$. Thus
using the equations 4.6, 4.7, 4.8, 4.12 and 4.20, we get

$\displaystyle
b_{r}D_{ij}^{r}B_{\alpha}^{i}B_{\beta}^{j}=-\frac{c_{0}b^{2}}{1+n(n+1)}h_{\alpha\beta}$
(4.36)

Thus using the relation $b_{i|j}=b_{ij}-b_{r}D_{ij}^{r}$ and equation 4.36,
equation 4.19 reduces to

$\displaystyle-\frac{c_{0}b^{2}}{1+n(n+1)}h_{\alpha\beta}+\sqrt{\frac{b^{2}}{1+n(n+1)}}H_{\alpha\beta}=0$
(4.37)

$\displaystyle H_{\alpha\beta}=\frac{c_{0}b}{\sqrt{1+n(n+1)}}h_{\alpha\beta}$

Thus we shown that the second fundamental tensor $H_{\alpha\beta}$ of
$M^{n-1}$ is proportional to its angular metric tensor $h_{\alpha\beta}$.

∎

######  Theorem 4.5.

Let $(M,F)$ be a Finsler manifold, where
$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n\in N$, is a
generalized square metric and $M^{n-1}$ be its associated hypersurface.Then
the hypersurface $M^{n-1}$ will be hyperplane of second kind if and only if

$\displaystyle b_{ij}$ $\displaystyle=eb_{i}b_{j}$

###### Proof.

We know from Lemma 3.8, hypersurface $M^{n-1}$ is a hyperplane of second kind
if $H_{\alpha}=0$ and $H_{\alpha\beta}=0$. Now we consider these two
sufficient conditions one by one.

  1. 1.

If $H_{\alpha\beta}=0$, then equation (4.37) becomes

$\displaystyle-\frac{c_{0}b^{2}}{1+n(n+1)}h_{\alpha\beta}+\sqrt{\frac{b^{2}}{1+n(n+1)}}\times
0=0$ $\displaystyle\implies-\frac{c_{0}b^{2}}{1+n(n+1)}h_{\alpha\beta}=0$
$\displaystyle\implies c_{0}=0$ $\displaystyle\implies c_{0}=c_{i}y^{i}=0$

$\implies$ there exist a function $e(x)$ such that $c_{i}(x)=e(x)b_{i}(x)$ and
this $c_{i}(x)=e(x)b_{i}(x)$ forces $c_{0}$ to vanish as follows:

$\displaystyle c_{0}$ $\displaystyle=c_{i}(x)y^{i}$
$\displaystyle=e(x)b_{i}(x)y^{i}$
$\displaystyle=e(x)\left(b_{i}(x)y^{i}\right)$ $\displaystyle=e(x)\times
0\quad\text{(as along the hypersurface $b_{i}y^{i}=0$)}$
$\displaystyle\therefore c_{0}$ $\displaystyle=0$

  2. 2.

Again, if $H_{\alpha}=0$, then Lemma 3.6 and Lemma 3.7 imply $H_{0}=0$.
We have already shown above $H_{0}=0$ is equivalent to

$\displaystyle 2b_{ij}=b_{i}c_{j}+b_{j}c_{i}$

Now we combine case 1 and case 2. For that, put the value of
$c_{i}(x)=e(x)b_{i}(x)$ obtained in case 1 to equation obtained in case 2, we
get

$\displaystyle 2b_{ij}$ $\displaystyle=b_{i}e(x)b_{j}(x)+b_{j}e(x)b_{i}(x)$
$\displaystyle=e\left(b_{i}b_{j}+b_{j}b_{i}\right)$ $\displaystyle=e\times
2b_{i}b_{j}$ $\displaystyle\therefore 2b_{ij}$ $\displaystyle=2eb_{i}b_{j}$
$\displaystyle\implies b_{ij}$ $\displaystyle=eb_{i}b_{j}$ (4.38)

Thus we shown that the hypersurface $M^{n-1}$ of the Finsler manifold $(M,F)$
will be hyperplane of second kind iff $b_{ij}=eb_{i}b_{j}$.

∎

######  Theorem 4.6.

Let $(M,F)$ be a Finsler manifold, where
$F(\alpha,\beta)=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n\in N$, is a
generalized square metric and $M^{n-1}$ be its associated hypersurface. Then
the hypersurface $M^{n-1}$ will not be hyperplane of third kind.

###### Proof.

We know from sufficient conditions of Lemma 3.9 a hypersurface becomes a
hyperplane of third kind if $H_{\alpha}=0,$ $H_{\alpha\beta}=0$ and
$M_{\alpha\beta}=0$. Now we consider these three sufficient conditions one by
one.

  1. 1.

If $H_{\alpha}=0$, then we get the condition
$2b_{ij}=b_{i}e(x)b_{j}(x)+b_{j}e(x)b_{i}(x)$, which has already been proved
above and is termed as the condition of hyperplane of first kind.

  2. 2.

If $H_{\alpha\beta}=0$, then we get the condition $b_{ij}=eb_{i}b_{j}$, which
has already been proved above and is termed as the condition of hyperplane of
second kind.

  3. 3.

Now put $M_{\alpha\beta}=0$ in Equation (4.15), we get

$\displaystyle
0=\frac{(n+1)}{2\alpha}\left(\sqrt{\frac{b^{2}}{1+n(n+1)}}\right)h_{\alpha\beta}$

which implies that no condition could be deduced to satisfy
$M_{\alpha\beta}=0$, i.e., it is impossible to find a condition under which a
hypersurface becomes a hyperplane of third kind, as term on the R.H.S. of the
above equation can never be zero.

Finally, we shown that hypersurface $M^{n-1}$ is not a hyperplane of third
kind. ∎

######  Corollary 4.7.

Let $(M,F)$ be a Finsler manifold, where $F$ may be any of the following
Finsler metrics obtained by generalized square metric
$F=\frac{(\alpha+\beta)^{n+1}}{\alpha^{n}}$, $n=1,2,3,......$

  1. 1.

$F=\frac{(\alpha+\beta)^{2}}{\alpha}$ (popularly known as square metric)

  2. 2.

$F=\frac{(\alpha+\beta)^{3}}{\alpha^{2}}$

  3. 3.

$F=\frac{(\alpha+\beta)^{4}}{\alpha^{3}}$, etc.,
Also let $M^{n-1}$ be the corresponding hypersurfaces of the given Finsler
manifold $(M,F)$. Then, in either case, show that the hypersurface is a
hyperplane of first kind, second kind and not of the third kind.

###### Proof.

By Theorems 4.4, 4.5 and 4.6 it can be easily deduce that the hypersurfaces
$M^{n-1}$ corresponding to different Finsler manifolds $(M,F)$ are a
hyperplane of first kind, second kind and not of the third kind. It is
remarkable that corresponding author has already been published a paper [13]
on part (1) of this corollary. ∎

######  Conclusion.

Now after all, one may ask why authors of the article is so interested to
carry forward the theory of hypersurface over Finsler space with generalized
square metric
$F(x,y)=\frac{[\alpha(x,y)+\beta(x,y)]^{n+1}}{[\alpha(x,y)]^{n}}$, where
$n=1,2,3,...$. My answer is very assertive that generalization of any theory
is always fascinating due to its nature to bring various special cases under
one umbrella. For example, the theorems that we have proved works for every
natural number $n\in N$.

## References

  * [1] V.K. Chaubey and Mishra  “Hypersurfaces of a Finsler space with a special ($\alpha$, $\beta$)-metric”  In _J. Contemp. Mathemat. Anal._ 52, 2017, pp. 1–7
  * [2] V.K. Chaubey and B.K. Tripathi  “Finslerian Hypersurfaces of a Finsler Spaces with ($\alpha$, $\beta$)-metric”  In _Journal of Dynamical Systems and Geometric Theories_ 12, 2014, pp. 19–27
  * [3] G. G.C. and Vinay Pandey  “On a hypersurface of a Finsler space with the special metric, $\alpha+\frac{\beta^{n+1}}{(\alpha-\beta)^{n}}$”  In _J.T.S._ 7, 2013, pp. 39–47
  * [4] Manish Kumar Gupta and Suman Sharma  “On Hypersurface of a Finsler space subjected to h-Matsumoto change”  In _https://arxiv.org/pdf/2205.03520.pdf._ 322.3-4, 2022, pp. 401–409
  * [5] H.Y. I.Y. and Y.D. Lee  “On a hypersurface of a special Finsler space with a metric $\alpha+\frac{\beta^{2}}{\alpha}$”  In _Korean J. Math. Sciences_ 8.1, 2001, pp. 93–101
  * [6] M. Kitayama  “On Finslerian hypersurfaces given by $\beta$ change”  In _Balkan J. of Geometry and It’s Applications_ 7, 2002, pp. 49–55
  * [7] V.K. Kropina  “On projective two dimensional Finsler space with a special metric”  In _Trudy Sem. Vektor. Tenzor. Anal._ 11, 1961, pp. 277–292
  * [8] M. Matsumoto  “A slope of a mountain is a Finsler surface with respect to a time measure”  In _J. Math. Kyoto Univ_ 29, 1989, pp. 17–25
  * [9] M. Matsumoto  “Foundations of Finsler Geometry and Special Finsler Spaces”  Kaiseisha Press, Otsu, 1986
  * [10] M. Matsumoto  “The induced and intrinsic Finsler connection of a hypersurface and Finslerian projective geometry”  In _J. Math. Kyoto Univ._ 25, 1985, pp. 107–144
  * [11] M. P.L.  “The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology”  Kluwer Academic Publisher, 1993
  * [12] PN Panday and Ganga Prasad Yadav  “Hypersurface of a Finsler space with Randers change of special ($\alpha$, $\beta$)-metric”  In _Bulletin of the Transilvania University of Brasov. Series III: Mathematics and Computer Science_ , 2018, pp. 195–206
  * [13] M Rafee, Avdhesh Kumar and GC Chaubey  “On the Hypersurface of a Finsler space with the square metric”  In _Int. J. Pure. Appl. Math_ 118, 2018, pp. 723–733
  * [14] G. Randers  “On an Asymmetrical Metric in the Four-Space of General Relativity”  In _Phys. Rev._ 2, 1941, pp. 195–199
  * [15] G. Shanker and Ravindra  “On the hypersurface of a second approximate Matsumoto metric $\alpha+\beta+\frac{\beta^{2}}{\alpha}+\frac{\beta^{2}}{\alpha^{2}}$”  In _International J. of Contemp. Math. Sciences_ 7.2, 2012, pp. 115–124
  * [16] G. Shanker and Vijeta singh  “On the hypersurface of a Finsler space with Randers change of generalized $(\alpha,\beta)-metric$”  In _International Journal of pure and applied Mathematics_ 105.2, 2015, pp. 223–234
  * [17] C. Shibata  “On invariant tensors of $\beta$-change of Finsler metrics”  In _J. Math. Kyoto Univ._ 24, 1984, pp. 163–188
  * [18] U.P. Singh and Bindu Kumari  “On a Matsumoto space”  In _International Journal of pure and applied Mathematics_ 32.4, 2001, pp. 521–531


# Multi-Modal Hybrid Architecture for Pedestrian Action Prediction

Amir Rasouli*, Tiffany Yau, Mohsen Rohani and Jun Luo Authors are with Noah’s
Ark Laboratory, Huawei, Markham, Canada.* Corresponding author
<EMAIL_ADDRESS>
###### Abstract

Pedestrian behavior prediction is one of the major challenges for intelligent
driving systems in urban environments. Pedestrians often exhibit a wide range
of behaviors and adequate interpretations of those depend on various sources
of information such as pedestrian appearance, states of other road users, the
environment layout, etc. To address this problem, we propose a novel multi-
modal prediction algorithm that incorporates different sources of information
captured from the environment to predict future crossing actions of
pedestrians. The proposed model benefits from a hybrid learning architecture
consisting of feedforward and recurrent networks for analyzing visual features
of the environment and dynamics of the scene. Using the existing 2D pedestrian
behavior benchmarks and a newly annotated 3D driving dataset, we show that our
proposed model achieves state-of-the-art performance in pedestrian crossing
prediction.

##  I Introduction

Road user behavior prediction is one of the fundamental challenges for
autonomous driving systems in urban environments. Predicting pedestrian
behavior, in particular, is of great importance as they are among the most
vulnerable road users and often exhibit a wide range of behaviors [1] that are
impacted by numerous environmental and social factors [2].

Pedestrian prediction in the context of driving can be done implicitly via
forecasting future trajectories or explicitly through predicting high-level
actions of pedestrians, e.g. crossing the road (see Figure 1). In either case,
behavior prediction requires a deep understanding of various contextual
information in order to achieve a high precision [3, 4, 5, 6, 7, 8]. In recent
years, many deep learning architectures have been proposed that exploit
various data modalities, such as visual features, pedestrian dynamics, pose,
ego-motion, etc. in order to predict future actions. These algorithms rely on
either feedforward methods for spatiotemporal reasoning over scene images or
recurrent-based architectures to combine dynamics and visual features in a
single framework (see [9] for an extensive review).

In this work, we propose a novel hybrid architecture that benefits from both
feedforward and recurrent architectures to generate a joint representation of
both visual features and scene dynamics. More specifically, our model uses
four modalities of data: semantic maps of the environment, local scenes of
pedestrians and their surroundings, and pedestrian and the ego-vehicle
dynamics. The visual and dynamics features are encoded using convolutional
layers and recurrent networks respectively. Their outputs are weighted using
two attention modules and combined to create a joint representation that is
used for action prediction.

Figure 1: Predicting pedestrian crossing in front of the ego-vehicle based on
various contextual information: local scene around the pedestrian, semantic
map of the environment and pedestrian and the ego-vehicle dynamics.

We evaluate the performance of the proposed algorithm on common 2D pedestrian
behavior benchmark datasets as well as a new 3D dataset created by adding 3D
bounding boxes and behavioral annotations to the existing autonomous driving
dataset, nuScenes [10]. We show that the proposed method achieves state-of-
the-art performance on multiple prediction metrics on these datasets.

##  II Related Works

###  II-A Behavior prediction

The topic of human behavior prediction is of great interest to robotics and
computer vision communities. Behavior prediction has been used in many
applications such as human-object [11, 12] and human-human interaction [13,
14], risk assessment [15, 16], anomaly detection [17], surveillance [18, 7],
sports forecasting [19, 20] and intelligent driving systems [21, 5].

###  II-B Behavior Prediction in Driving

In the context of intelligent driving, particularly in urban environments,
predicting the behavior of road users is vital for safe motion planning. In
this context, the dominant approach is to forecast future trajectories of
other road users [21, 22, 23, 24, 25]. Recent developments in the field,
however, suggest that predicting higher-level actions of road users can be
beneficial by providing an early risk assessment [3, 5, 26] or by improving
the accuracy of trajectory forecasting [4, 6, 7, 27]. These actions, for
example, can refer to various maneuvers performed by other vehicles, e.g.
changing lane, making a turn [27], or in the case of pedestrians, interaction
with other objects, e.g. opening a car door [4], or, more importantly,
crossing the road [6, 5, 26].

#### Pedestrian Crossing Prediction

In recent years, the topic of pedestrian crossing prediction has gained
momentum and many state-of-the-art algorithms have been proposed that take
advantage of various data modalities and learning architectures [9].

Crossing prediction algorithms use different architectures. Those based on
feedforward architectures [3, 28, 26, 29] often rely on unimodal data. For
instance, the method in [28] uses a series of 3D DenseNet blocks to reason
over image sequences of pedestrians. The authors of [26, 3] propose generative
encoder-decoder architectures using 3D convolutional layers. Here, the
algorithms first generate future scenes and then based on those images predict
crossing actions. The model in [29] relies on intermediate features, such as
pedestrian head orientation, motion state, and the state of other traffic
elements. Each of these components is detected using separate CNN-based object
classifiers the outputs of which are combined for predicting crossing action.

Alternatively, some algorithms rely on recurrent architectures and take
advantage of a combination of different modalities for prediction [30, 5, 8].
For instance, the authors of [8] use a multi-stream LSTM architecture. They
first encode visual features, optical flow images, and vehicle dynamics using
individual LSTMs the outputs of which are concatenated to generate a shared
representation. This representation is then fed into an embedding layer
followed by another LSTM. The output of the second LSTM is combined with the
shared representation for the final inference. The model proposed in [5] uses
a multi-layer recurrent architecture with five spatially stacked GRUs that
encode pedestrian appearance, pedestrian surrounding context, pedestrians’
poses and trajectories, and the ego-vehicle speed. The inputs are fed into the
network at different levels according to their complexity, e.g. visual
features at the bottom and speed at the top layers.

In this work, we use a hybrid approach that takes advantage of both
feedforward networks and recurrent architectures for a joint visuospatial and
dynamics reasoning over traffic scenes.

#### Pedestrian Behavior Datasets

Although there are many publicly available datasets for pedestrian trajectory
prediction [31, 32, 33, 34, 35], the choices for pedestrian action prediction,
particularly in the context of driving, are more limited. There are a number
of datasets that provide rich behavioral tags along with temporally coherent
spatial annotations that can be used for pedestrian action prediction,
including Joint Attention in Autonomous Driving (JAAD) [29], VIrtual
ENvironment for Action Analysis (VIENA2) [8], Pedestrian Intention Estimation
(PIE) [6], Trajectory Inference using Targeted Action priors Network (TITAN)
[4] and Stanford-TRI Intent Prediction (STIP) [30]. These datasets provide
image sequences of traffic scenes recorded using on-board cameras along with
2D bounding boxes for pedestrians and other traffic objects as well as
annotations for pedestrian behavior and ego-motion information. The drawback
of these datasets is that they do not contain 3D information such as bird’s
eye view map of the environment, 3D bounding boxes, etc. all of which are
important for real-world autonomous driving applications. In addition to 2D
benchmark datasets, we use a new dataset of 3D bounding boxes and behavioral
annotations that extends the nuScenes dataset.

##  III Method

###  III-A Problem statement

In this work, pedestrian action prediction is formulated as an optimization
problem $p(A_{i}^{t+m}|LS,M,PM,VM)$ where the goal is to estimate the
probability of crossing action $A_{i}^{t+m}\in\\{0,1\\}$ for some pedestrian
$i$, $1<i<n$ in some time $t+m$ in the future. Here, the prediction is based
on the observed local scene around the pedestrian
$LS=\\{ls^{1},ls^{2},...,ls^{t}\\}$, the semantic map of the environment
$M=\\{m^{1},m^{2},...,m^{t}\\}$, the pedestrian’s motion
$PM=\\{pm^{1},pm^{2},...,pm^{t}\\}$ and the ego-vehicle motion
$VM=\\{vm^{1},vm^{2},...,vm^{t}\\}$.

###  III-B Architecture

Figure 2: The architecture of the proposed model. The model processes four
different input modalities: semantic maps of the environment (2D or bird’s eye
view), local scene images of pedestrians and their motion (2D spatial or 3D
global coordinates + velocity), and ego-vehicle motion. Semantic maps and
images are encoded with two sets of Conv2D layers. The outputs of these layers
are combined and fed into another Conv2D layer followed by a dense embedding
layer to form visual representations which are weighted using Visual Attention
Module (VAM). The motion of pedestrians and the ego-vehicle is encoded using
two LSTMs, the outputs of which are concatenated and fed into Dynamic
Attention Module (DAM). The final joint representation is formed by
concatenating the outputs of the attention modules. The specifications of conv
layers for scene encoding are given as [number of filters, kernel size,
stride].

As highlighted in a survey of the past studies [2], there is a diverse set of
social and environmental factors that impact the way pedestrians make crossing
decision. To capture such contextual complexity, we propose a multi-modal
method that relies on a hybrid architecture for encoding visual and dynamics
information. Figure 2 illustrates the proposed method which can be divided
into three main parts:

  * •

Visual Encoding. This part of the model relies on 2D convolutional layers to
generate visual representations for semantic map of the environment and local
scene context of pedestrians.

  * •

Dynamics Encoding. This recurrent part of the model generates a joint
representation of scene dynamics, namely pedestrian and ego-vehicle motion.

  * •

Prediction. Using a joint representation of the visual and dynamics features,
the model estimates the probability of the crossing action in the future.

We discuss each of these components in more details in the subsections below.

###  III-C Visual Encoding

(a) Sequential (b) Atrous

(c) Multi-scale

Figure 3: Different strategies for processing map images. The specifications
of 2D conv layers are given as [number of filters, kernel size, stride]. In
b), rate refers to the rate of kernel dilation.

Visual information is crucial for robust prediction of pedestrian behavior as
it reflects the state of the pedestrians, e.g. pose, head orientation, motion,
gestures as well as their surrounding environment, e.g. the state of other
road users, road structure, signals, etc. For this purpose, we rely on two
different sources of visuospatial inputs: semantic maps and local scene images
around pedestrians.

Semantic map. Depending on the type of data, we either use semantic maps
generated from scene images or bird’s eye view maps of the environment and
traffic elements. The maps for each time-step are concatenated channel-wise.

When processing the maps, it is important to capture the interdependency
between different traffic components such as pedestrians, vehicles, road
structure, etc. To address this issue, we consider three strategies as shown
in Figure 3.

The first approach is a common sequential processing with successive
convolutional layers. This method captures local information in the image but
fails to represent broader context at higher resolutions. Furthermore,
consecutive downsampling can result in the loss of some of the finer visual
details. To address these drawbacks, we consider two alternatives: Atrous (or
dilated) convolution and multi-scale skip connection.

Atrous convolution is a special form of convolution in which the effective
field of view of a filter is increased by inserting gaps, parameterized as the
rate of dilation, between the elements of a given kernel as proposed in [36].
This allows the model to capture a broader spatial context without the need
for downsampling.

Multi-scale skip connections are often used to combine fine and coarse
features generated at different convolutional layers. Following this approach,
the output of the first conv layer and the upsampled (via deconvolution
operation) output of the last layer are fed into another conv layer for joint
processing. The specifications are shown in Figure 3.

Local scene. We use scene images to capture the changes in the appearance of
pedestrians and their local surroundings. At a given time step, we extract a
region around each pedestrian by scaling up the corresponding 2D spatial
bounding box and then adjust the dimensions of the scaled box so its width
matches its height. Similar to maps, we concatenate the images channel-wise
for a given observation sequence prior to processing by multiple conv layers.

In order to combine the visual features with different modalities, we employ a
late fusion technique in which we spatially concatenate the outputs of map and
local scene encodings and jointly process them with an additional
convolutional layer. The output of the fusion conv layer is flattened and fed
into a dense layer to reduce the dimension of the feature vector.

Visual Attention Module (VAM). Although visual features are important for
reasoning about future pedestrian action, it is important to emphasize those
features that are more relevant to the task. To achieve this goal, we weight
visual representation features by using an attention mechanism. We calculate
feature weight $\alpha_{i}$ for a given feature point $z_{i}$ as follows:

$\alpha_{i}=\frac{e^{f_{i}}}{\sum_{j}e^{f_{j}}},f_{i}=\sum_{k}\omega_{i,k}z_{k}+b_{i}.$

where $f_{i}$ is a single neuron in a fully-connected layer.

###  III-D Dynamics Encoding

When anticipating pedestrian action, dynamics features, such as pedestrian and
ego-motion, are of vital importance. For example, when a pedestrian is moving
towards the road, that can be an indicator of crossing intention. Likewise,
the ego-vehicle motion can directly impact whether a pedestrian makes a
crossing decision. If, for instance, the ego-vehicle is driving at a high
speed, the pedestrian will likely cross after the vehicle has passed. Given
the importance of dynamics information, we use pedestrian and the ego-vehicle
motion as described below.

Pedestrian motion. We use both pedestrian location and velocity for encoding
pedestrian motion. In the absence of 3D data, 2D bounding boxes, i.e. the
coordinates of the boxes $[x_{1},y_{1},x_{2},y_{2}]$ corresponding to top-left
and bottom-right corners of the boxes around pedestrians in image frames are
used. For the 3D dataset, we use global coordinates of pedestrians in 3D space
with respect to the map coordinates.

Velocities are obtained by calculating the displacement of the pedestrian
coordinates (either in 2D or 3D) with respect to the first coordinate at the
beginning of observation sequences. The combination of pedestrian locations
and velocities forms pedestrian motion information.

Ego-vehicle motion. We follow the same routine as for pedestrians and use the
ego-vehicle global coordinates and velocities. For 2D datasets, since ego-
location is not available, we only use the vehicle’s velocity.

Pedestrian and ego-motion features are encoded using two recurrent networks.
The hidden states $h^{1:t}$ of these networks are concatenated temporally to
form dynamics representations.

Dynamics Attention Module (DAM). Similar to the visual representations, we
employ an attention module to weight dynamics features. DAM is a 3D attention
mechanism inspired by [37] which receives as input a temporal data sequence
and generates a unified weighted representation computed as follows:

$DAM^{t}=tanh(W_{c}[c^{t}\oplus h^{t}])$
$c^{t}=\sum_{i\in[1:t]}\sigma({h^{t}}^{\prime}W_{a}h^{i})h^{i}$

$\sigma$ is a softmax function and $h^{i}$ is a hidden state of dynamics
representation for $i=1,2,...,t$. Here, $\oplus$ denotes the concatenation
operation.

###  III-E Prediction

The joint representation used for final prediction is formed by concatenating
the outputs of VAM and DAM. This representation is then fed into two
consecutive dense layers to make crossing predictions. For learning, we use
binary cross-entropy loss given by

$L=-\sum_{n}y_{n}\log{\hat{y}_{n}}+(1-y_{n})\log{(1-\hat{y}_{n})}.$

##  IV Empirical Evaluation

###  IV-A Datasets

For evaluations on 2D data, we choose two naturalistic pedestrian behavior
benchmarks, PIE [6], and JAAD [29] which are publicly available. The
alternatives, TITAN [4] and STIP [30] datasets are only available under
restrictive terms of use, and VIENA2 [8] only contains simulated samples.

Pedestrian Intention Estimation (PIE). PIE contains 6 hours of driving footage
recorded with an on-board camera. There are 1842 pedestrian samples with dense
2D bounding box annotations at 30Hz with behavioral tags along with motion
sensor recordings both from the vehicle and the camera. For the ego-vehicle
velocity, we use gyroscope readings from the camera. The data is split into
training and testing sets as recommended in [6].

Joint Attention in Autonomous Driving (JAAD). Similar to PIE, JAAD is a
dataset consisting of on-board video recordings with 346 clips and 2580
pedestrians annotated with 2D bounding boxes at 30Hz. Compared to PIE, JAAD
samples are less diversified, i.e. the majority of samples are pedestrians
walking on sidewalks and sequences are generally shorter. JAAD also lacks ego-
motion information, however, it provides driver’s actions, such as speeding
up, stopping, etc. which are used in place for ego-motion information. The
data is split into train/test sets following [38].

Pedestrian Prediction on nuScenes (PePScenes). Due to the lack of 3D driving
data with pedestrian behavior annotations, we created our own dataset by
adding annotations to the existing dataset nuScenes [10]. For behavioral
annotations, we added sample-wise and per-frame crossing labels to 719
pedestrian samples out of which 570 are not crossing. For behavioral
annotations, we selected the pedestrians that 1) appeared long enough in front
of the vehicle and 2) appeared to have the intention of crossing, e.g. moving
close to the curbside, looking at the traffic, etc.

In addition, we extended 2D/3D bounding box annotations on nuScenes from 2Hz
to 10Hz to make samples more suitable for pedestrian prediction. This is done
by interpolating boxes between two consecutive frames using pedestrian global
coordinates. The ratio of train/test data is 70/30.

Data preparation. We follow the same procedure as in [5] and clip sequences up
to the time of the crossing events, i.e. the moment the pedestrians start
crossing the road. In cases where no crossing occurs, the last frame is
selected instead. All models use $0.5$s observation length (or $5$ frames at
$10$Hz for all datasets) and sample sequences from each track between $1$ to
$2s$ to the event of crossing with $50\%$ overlap between each sample.
Overall, we get the following number of train/test samples: $3980/3185$ for
PIE, $3955/3110$ for JAAD and $2544/1146$ for PePScenes.

###  IV-B Implementation

For 2D datasets, we generate semantic maps using the algorithm in [36]
pretrained on Cityscapes [39] and use 14 main object classes appearing in
traffic scenes. For the PePScenes dataset, rasterized maps encoded as
3-channel images similar to [40] are used. The map is of size $30\times 30$
meters centered around the ego-vehicle.

To extract local scene images, we use $1.5x$ scaled versions of pedestrian 2D
bounding boxes in image frames. Unlike the 2D datasets, PePScenes contains
recordings from three front looking cameras that cover a wide-angle view in
front of the vehicle. To select a pedestrian of interest, we choose a camera
frame in which the pedestrian appears. If the pedestrian appears in two
adjacent cameras, we select the camera frame in which a larger portion of the
pedestrian is visible.

The details of convolutional layers for each module can be seen in Figures
2,3. For all recurrent networks, we use LSTM cells with 256 hidden units. We
set the dimension of the embedding dense layer for visual features to $512$
and the second to last layer to $256$.

###  IV-C Training

We trained the proposed model on all datasets end-to-end using RMSProp [41]
optimizer with batch size of $16$, learning rate of $0.00005$ for $50$ epochs
on the 2D datasets and $40$ on PePScenes. We applied $L2$ regularization of
$0.0001$ to LSTMs and the last dense layer. To compensate for data imbalance,
we set class weights based on the ratio of positive and negative samples.

###  IV-D Metrics

To evaluate the proposed model, common binary classification metrics as in [5]
are used including $accuracy$, Area Under the Curve ($AUC$), $F1$, and
$precision$. Using all these metrics, as opposed to only reporting on accuracy
[30], is important given the fact that the number of negative and positive
samples are imbalanced in all the datasets mentioned above. This means that
the models can favor one class over the others and still achieve very high
accuracy. AUC, F1, and precision show the balanced performance of the methods
by highlighting how on average each method identifies different classes and
distinguishes between them.

###  IV-E Models

We compare the performance of the proposed model to a series of baseline and
state-of-the-art crossing prediction models:
Trajectory-Based Forecaster (TF). As a baseline, we make crossing prediction
using only pedestrian trajectory information, i.e. pedestrian coordinates in
2D or 3D (as discussed in Sec. III-D), using an LSTM followed by a dense
layer.
Inception 3D (I3D) [42]. Given the similarities between action prediction and
recognition tasks, we use a state-of-the-art activity recognition model, I3D.
This is a feedforward model based on a series of 3D convolutional layers. As
input to this model, we use local scene images as described earlier.
Stacked with Fusion GRU (SF-GRU) [5]. This is a state-of-the-art pedestrian
crossing prediction algorithm based on a multi-level recurrent architecture in
which different modalities of data are infused and encoded gradually. This
model relies on five modalities of data, namely pedestrians’ appearances and
surrounding context, pedestrians’ poses and coordinates, and the ego-vehicle
speed. We also report on a variation of SF-GRU in which we combine all data
modalities and feed them to the bottom layer of a spatially Stacked GRUs
(S-GRU) similar to [43].
An alternative approach to stacked processing is to encode different
modalities of data individually and combine them prior to classification. We
follow the approach in [8] and process each input using a separate GRU without
the second GRU for joint processing. We refer to this model as Multi-Stream
GRU (MS-GRU).

All the poses are generated using the model in [44] pretrained on the MSCOCO
dataset [45].

###  IV-F Prediction on 2D Datasets

TABLE I: Performance of the proposed algorithm on PIE and JAAD datasets.

| PIE | JAAD
---|---|---
Models | Acc | AUC | F1 | Prec | Acc | AUC | F1 | Prec
TF | 0.75 | 0.73 | 0.61 | 0.55 | 0.76 | 0.72 | 0.54 | 0.40
I3D [42] | 0.63 | 0.58 | 0.42 | 0.37 | 0.79 | 0.71 | 0.49 | 0.42
MS-GRU [8] | 0.86 | 0.85 | 0.77 | 0.71 | 0.81 | 0.79 | 0.59 | 0.48
S-GRU [43] | 0.82 | 0.78 | 0.68 | 0.68 | 0.83 | 0.76 | 0.58 | 0.53
SF-GRU [5] | 0.87 | 0.85 | 0.78 | 0.74 | 0.83 | 0.79 | 0.59 | 0.50
| Sequential | 0.87 | 0.86 | 0.78 | 0.73 | 0.79 | 0.79 | 0.57 | 0.45
Ours | Atrous | 0.89 | 0.88 | 0.81 | 0.77 | 0.84 | 0.80 | 0.62 | 0.54
| Multi-scale | 0.88 | 0.86 | 0.79 | 0.75 | 0.77 | 0.79 | 0.56 | 0.42

We begin by evaluating the performance of the proposed models and state-of-
the-art algorithms discussed earlier on 2D datasets. As shown in Table I, the
proposed algorithm using atrous map processing achieves the best performance
on both datasets. On PIE, using multi-scale map encoding technique achieves
only small improvements while the atrous method improves more on all metrics,
especially on precision by $4\%$. On JAAD, only the atrous method offers
improvements on all metrics and the improvement gap is not as high as reported
on PIE. This difference is to some extent expected as JAAD is not as diverse
compared to PIE. For example, the majority of the samples in JAAD are
pedestrians that are walking alongside the vehicle on sidewalks. This is also
reflected in the changes in the performance of I3D. This method, despite using
only visual features achieves much better results on JAAD compared to PIE.
This means there is a high degree of visual similarity between samples.

Another possible explanation for smaller improvement on JAAD is the lack of
proper ego-motion information. JAAD only offers high-level actions of the
driver, which are not as informative as the actual dynamics of the vehicle.
The importance of the ego-vehicle dynamics is highlighted in Section IV-I.

Figure 4: Performance of the proposed model on 2D and 3D datasets. Red and
green boxes indicate future crossing or non-crossing actions respectively. The
results are divided into correct predictions on the left and incorrect ones on
the right.

###  IV-G Prediction on 3D Dataset

TABLE II: Performance of the proposed algorithm on PePScenes. Models | Acc | AUC | F1 | Prec
---|---|---|---|---
TF | 0.80 | 0.52 | 0.14 | 0.26
I3D [42] | 0.84 | 0.60 | 0.33 | 0.53
MS-GRU [8] | 0.85 | 0.66 | 0.46 | 0.59
S-GRU [43] | 0.86 | 0.69 | 0.51 | 0.64
SF-GRU [5] | 0.87 | 0.68 | 0.51 | 0.67
| Sequential | 0.85 | 0.63 | 0.39 | 0.65
Ours | Atrous | 0.88 | 0.71 | 0.56 | 0.77
| Multi-scale | 0.86 | 0.68 | 0.50 | 0.64

We repeat the same experiments as above using our newly annotated 3D dataset.
The results of this experiment are summarized in Table II. As in the 2D
experiment, the proposed model achieves state-of-the-art performance using
atrous encoding. The improvements, however, are more noticeable, in
particular, F1 and precision are increased by $5\%$ and $10\%$ respectively.
This is because bird’s eye view maps, first, contain finer details compared to
2D semantic maps, e.g. location of objects, road boundaries, etc., and second,
encode a larger area around the vehicle. As a result, avoiding downsampling
and using successively larger kernel sizes can capture information more
effectively.

###  IV-H Qualitative Analysis

Figure 4 shows the performance of the proposed model on 2D and 3D datasets. In
this figure, we can see that two cases are challenging to predict: people
standing on the curbside or road and have no crossing intention, e.g. the
person engaged in conversation or a construction worker operating next to the
road (top row). The pedestrian’s direction of motion can also be distracting.
For example, pedestrian might be moving towards the road or step on the road
but would not cross or might be standing at the intersection prior to
crossing.

###  IV-I Ablation

Earlier we discussed the importance of multi-modal processing and how
different sources of information help us capture different contextual elements
in traffic scenes. Here, we conduct an ablation study on the final proposed
model (with atrous encoding) using different input modalities. For this
experiment, we use both a 2D dataset (PIE) and a 3D dataset (PePScenes). We
selected PIE over JAAD because it has more diverse samples and it also
contains actual ego-vehicle dynamics information.

Table III shows the results of this experiment. On the 2D dataset, the method
achieves relatively high performance using only visual features. However, on
all metrics, using dynamics information is superior. This is due to the fact
that visual features in 2D space do not properly capture the dynamics of the
scene which are fundamental for action prediction. It should be noted that
despite such a discrepancy, visual features in conjunction with dynamics
information can help improve the results on all metrics, particularly on F1
and precision by $4\%$ and $8\%$ respectively.

TABLE III: Ablation study on different feature modalities.

| PIE | PePScenes
---|---|---
Input Modality | Acc | AUC | F1 | Prec | Acc | AUC | F1 | Prec
Local Scene | 0.67 | 0.54 | 0.29 | 0.37 | 0.83 | 0.57 | 0.27 | 0.46
Map + Local Scene | 0.75 | 0.65 | 0.49 | 0.59 | 0.84 | 0.73 | 0.55 | 0.53
Ped. Motion | 0.84 | 0.85 | 0.75 | 0.66 | 0.75 | 0.54 | 0.22 | 0.23
Ped. + Veh. Motion | 0.85 | 0.86 | 0.77 | 0.69 | 0.82 | 0.72 | 0.51 | 0.47
Visual + Dynamics | 0.89 | 0.88 | 0.81 | 0.77 | 0.88 | 0.71 | 0.56 | 0.77

The results on the 3D dataset show that similar to the 2D data, using only
images of pedestrians and their surroundings, the performance is relatively
poor. However, by adding the map information a significant improvement can be
achieved, especially on AUC, F1, and precision. This is primarily due to the
fact that the bird’s eye view map captures the changes in the dynamics of the
ego-vehicle and its surroundings much more effectively compared to 2D semantic
maps. In fact, when all metrics are considered, the performance of the model
is similar when using visual or dynamics features. Despite such similarity of
performance, there are still cases that are not learned properly using only
one type of feature. This is highlighted in the performance of the model
combining all features. Besides the AUC metric, the combined model performs
better on F1 and significantly better on accuracy by $4\%$ and precision by
$24\%$. Considering all metrics, the combined model is clearly superior to
others.

##  V Conclusion

In this work, we proposed a hybrid model for predicting pedestrian road-
crossing action. Our model benefits from both feedforward and recurrent
architectures for encoding different input modalities that capture both the
changes in the visual appearance and dynamics of the traffic scenes. Using
common 2D pedestrian behavior benchmark datasets and our newly annotated 3D
dataset, we showed that our proposed model achieves state-of-the-art
performance across all metrics.

Furthermore, by conducting an ablation study on the proposed model, we showed
how different sources of information can impact prediction accuracy. Our
findings suggest that, even though dynamics information is dominant in
predicting pedestrian behavior, visual features play a complementary role for
prediction and can result in improved performance when combined with dynamics
information.

## References

  * [1] A. Rasouli, I. Kotseruba, and J. K. Tsotsos, “Agreeing to cross: How drivers and pedestrians communicate,” in _Intelligent Vehicles Symposium (IV)_ , 2017\.
  * [2] A. Rasouli and J. K. Tsotsos, “Autonomous vehicles that interact with pedestrians: A survey of theory and practice,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 21, no. 3, pp. 900–918, 2019.
  * [3] M. Chaabane, A. Trabelsi, N. Blanchard, and R. Beveridge, “Looking ahead: Anticipating pedestrians crossing with future frames prediction,” in _WACV_ , 2020.
  * [4] S. Malla, B. Dariush, and C. Choi, “Titan: Future forecast using action priors,” in _CVPR_ , 2020.
  * [5] A. Rasouli, I. Kotseruba, and J. K. Tsotsos, “Pedestrian action anticipation using contextual feature fusion in stacked rnns,” in _BMVC_ , 2019.
  * [6] A. Rasouli, I. Kotseruba, T. Kunic, and J. K. Tsotsos, “Pie: A large-scale dataset and models for pedestrian intention estimation and trajectory prediction,” in _ICCV_ , 2019.
  * [7] J. Liang, L. Jiang, J. C. Niebles, A. G. Hauptmann, and L. Fei-Fei, “Peeking into the future: Predicting future person activities and locations in videos,” in _CVPR_ , 2019.
  * [8] M. S. Aliakbarian, F. S. Saleh, M. Salzmann, B. Fernando, L. Petersson, and L. Andersson, “VIENA: A driving anticipation dataset,” in _ACCV_ , C. V. Jawahar, H. Li, G. Mori, and K. Schindler, Eds., 2019.
  * [9] A. Rasouli, “Deep learning for vision-based prediction: A survey,” _arXiv:2007.00095_ , 2020.
  * [10] H. Caesar, V. Bankiti, A. H. Lang, S. Vora, V. E. Liong, Q. Xu, A. Krishnan, Y. Pan, G. Baldan, and O. Beijbom, “nuscenes: A multimodal dataset for autonomous driving,” in _CVPR_ , 2020.
  * [11] M. Liu, S. Tang, Y. Li, and J. Rehg, “Forecasting human object interaction: Joint prediction of motor attention and actions in first person video,” in _ECCV_ , 2020.
  * [12] A. Piergiovanni, A. Angelova, A. Toshev, and M. S. Ryoo, “Adversarial generative grammars for human activity prediction,” in _ECCV_ , 2020.
  * [13] H. Joo, T. Simon, M. Cikara, and Y. Sheikh, “Towards social artificial intelligence: Nonverbal social signal prediction in a triadic interaction,” in _CVPR_ , 2019.
  * [14] T. Yao, M. Wang, B. Ni, H. Wei, and X. Yang, “Multiple granularity group interaction prediction,” in _CVPR_ , 2018.
  * [15] M. Strickland, G. Fainekos, and H. B. Amor, “Deep predictive models for collision risk assessment in autonomous driving,” in _ICRA_ , 2018.
  * [16] K.-H. Zeng, S.-H. Chou, F.-H. Chan, J. Carlos Niebles, and M. Sun, “Agent-centric risk assessment: Accident anticipation and risky region localization,” in _CVPR_ , 2017.
  * [17] D. Epstein, B. Chen, and C. Vondrick, “Oops! predicting unintentional action in video,” in _CVPR_ , 2020.
  * [18] Y. Ma, X. Zhu, X. Cheng, R. Yang, J. Liu, and D. Manocha, “Autotrajectory: Label-free trajectory extraction and prediction from videos using dynamic points,” in _ECCV_ , 2020.
  * [19] M. Qi, J. Qin, Y. Wu, and Y. Yang, “Imitative non-autoregressive modeling for trajectory forecasting and imputation,” in _CVPR_ , 2020.
  * [20] P. Felsen, P. Agrawal, and J. Malik, “What will happen next? forecasting player moves in sports videos,” in _ICCV_ , 2017.
  * [21] L. Fang, Q. Jiang, J. Shi, and B. Zhou, “Tpnet: Trajectory proposal network for motion prediction,” in _CVPR_ , 2020.
  * [22] M. Liang, B. Yang, W. Zeng, Y. Chen, R. Hu, S. Casas, and R. Urtasun, “Pnpnet: End-to-end perception and prediction with tracking in the loop,” in _CVPR_ , 2020.
  * [23] T. Phan-Minh, E. C. Grigore, F. A. Boulton, O. Beijbom, and E. M. Wolff, “Covernet: Multimodal behavior prediction using trajectory sets,” in _CVPR_ , 2020.
  * [24] R. Chandra, U. Bhattacharya, A. Bera, and D. Manocha, “Traphic: Trajectory prediction in dense and heterogeneous traffic using weighted interactions,” in _CVPR_ , 2019.
  * [25] N. Rhinehart, K. M. Kitani, and P. Vernaza, “R2p2: A reparameterized pushforward policy for diverse, precise generative path forecasting,” in _ECCV_ , 2018.
  * [26] P. Gujjar and R. Vaughan, “Classifying pedestrian actions in advance using predicted video of urban driving scenes,” in _ICRA_ , 2019.
  * [27] S. Casas, W. Luo, and R. Urtasun, “Intentnet: Learning to predict intention from raw sensor data,” in _CORL_ , 2018.
  * [28] K. Saleh, M. Hossny, and S. Nahavandi, “Real-time intent prediction of pedestrians for autonomous ground vehicles via spatio-temporal densenet,” in _ICRA_ , 2019.
  * [29] A. Rasouli, I. Kotseruba, and J. K. Tsotsos, “Are they going to cross? a benchmark dataset and baseline for pedestrian crosswalk behavior,” in _ICCVW_ , 2017.
  * [30] B. Liu, E. Adeli, Z. Cao, K.-H. Lee, A. Shenoi, A. Gaidon, and J. C. Niebles, “Spatiotemporal relationship reasoning for pedestrian intent prediction,” _IEEE Robotics and Automation Letters_ , vol. 5, no. 2, pp. 3485–3492, 2020\.
  * [31] J. Liang, L. Jiang, K. Murphy, T. Yu, and A. Hauptmann, “The garden of forking paths: Towards multi-future trajectory prediction,” in _CVPR_ , 2020.
  * [32] P. Sun, H. Kretzschmar, X. Dotiwalla, A. Chouard, V. Patnaik, P. Tsui, J. Guo, Y. Zhou, Y. Chai, B. Caine, V. Vasudevan, W. Han, J. Ngiam, H. Zhao, A. Timofeev, S. Ettinger, M. Krivokon, A. Gao, A. Joshi, Y. Zhang, J. Shlens, Z. Chen, and D. Anguelov, “Scalability in perception for autonomous driving: Waymo open dataset,” in _CVPR_ , 2020.
  * [33] W. Liu, W. Luo, D. Lian, and S. Gao, “Future frame prediction for anomaly detection – a new baseline,” in _CVPR_ , 2018.
  * [34] A. Robicquet, A. Sadeghian, A. Alahi, and S. Savarese, “Learning social etiquette: Human trajectory understanding in crowded scenes,” in _ECCV_ , 2016.
  * [35] A. Lerner, Y. Chrysanthou, and D. Lischinski, “Crowds by example,” _Computer graphics forum_ , vol. 26, no. 3, pp. 655–664, 2007.
  * [36] L.-C. Chen, G. Papandreou, F. Schroff, and H. Adam, “Rethinking atrous convolution for semantic image segmentation,” _arXiv:1706.05587_ , 2017.
  * [37] M.-T. Luong, H. Pham, and C. D. Manning, “Effective approaches to attention-based neural machine translation,” _arXiv:1508.04025_ , 2015.
  * [38] A. Rasouli, I. Kotseruba, and J. K. Tsotsos, “It’s not all about size: On the role of data properties in pedestrian detection,” in _ECCVW_ , 2018.
  * [39] M. Cordts, M. Omran, S. Ramos, T. Rehfeld, M. Enzweiler, R. Benenson, U. Franke, S. Roth, and B. Schiele, “The cityscapes dataset for semantic urban scene understanding,” in _CVPR_ , 2016.
  * [40] H. Cui, V. Radosavljevic, F.-C. Chou, T.-H. Lin, T. Nguyen, T.-K. Huang, J. Schneider, and N. Djuric, “Multimodal trajectory predictions for autonomous driving using deep convolutional networks,” in _ICRA_ , 2019.
  * [41] T. Tieleman and G. Hinton, “Lecture 6.5-rmsprop, coursera: Neural networks for machine learning,” Tech. Rep., 2012.
  * [42] J. Carreira and A. Zisserman, “Quo vadis, action recognition? a new model and the kinetics dataset,” in _CVPR_ , 2017.
  * [43] J. Yue-Hei Ng, M. Hausknecht, S. Vijayanarasimhan, O. Vinyals, R. Monga, and G. Toderici, “Beyond short snippets: Deep networks for video classification,” in _CVPR_ , 2015.
  * [44] Z. Cao, T. Simon, S.-E. Wei, and Y. Sheikh, “Realtime multi-person 2d pose estimation using part affinity fields,” in _CVPR_ , 2017.
  * [45] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, and C. L. Zitnick, “Microsoft COCO: Common objects in context,” in _ECCV_ , 2014.