CR-Lightlike Submanifolds of a Golden Semi-

Riemannian Manifold

Mobin Ahmad

Department of Mathematics and Statistics

Faculty of Science, Integral University,

Lucknow-226026, India

E-mail<EMAIL_ADDRESS>

Mohammad Aamir Qayyoom

Department of Mathematics and Statistics

Faculty of Science, Integral University,

Lucknow-226026, India

E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we define and study CR-lightlike submanifolds of a golden semi-
Riemannian manifold. We investigate some properties of geodesic CR-
submanifolds of a golden semi-Riemannian manifold. Moreover, we obtain many
interesting results for totally geodesic and totally umbilical CR-submanifolds
on a golden Riemannian manifold. A non-trivial example of CR-lightlike
submanifold of a golden semi-Riemannian manifold is also constructed.

Mathematics Subject Classification (2010): 53C15, 53C40, 53C20.

Keywords and Phrases: golden structure, semi-Riemannian manifold, CR-
lightlike submanifolds, totally umbilical, geodesic.

##  1 Introduction

The theory of submanifolds of a manifold is one of the most interesting topics
in differential geometry. According to the behaviour of the tangent bundle of
a submanifold, we have three classes of submanifolds: holomorphic
submanifolds, totally real submanifolds and CR-submanifolds has been
introduced by the author in [16]. Let $\overline{M}$ be an almost Hermitian
manifold and let $J$ be the almost complex structure of $\overline{M}.$ A real
submanifold $M$ of $\overline{M}$ is called a CR submanifold if there exists a
differential distribution $D$ on $M$ satisfying

$(i)J(D_{x})=D_{x}$

and

$(ii)J(D_{x}^{\bot})\subset T_{x}M^{\bot}$

for each $x\in M,$ where $D^{\bot}$ is the complementary orthogonal
distribution to $D$ and $T_{x}M^{\bot}$ is normal space to $M$ at $x.$
Holomorphic submanifold and totally real submanifolds are particular cases of
CR-submanifolds.
Later, the CR-submanifolds have been extensively studied by several geometers
( see in [4], [6], [15], [12], [13] [17]). Also, Some properties of CR-
submanifold have been investigated in various studies (see [1], [2], [3], [5],
[6], [7], [10], [19], [20], [31], [35], [37]).
Bejancu and Duggal [21] and Kupeli [32] developed the general theory of
lightlike submanifolds of a semi-Riemannian manifold. In [14], the authors
constructed the principal vector bundles to lightlike submanifolds which has a
non-trivial intersection with the tangent bundle of lightlike submanifolds in
a semi-Riemann manifold and obtained Gauss-Weingarten formulae as well as
other properties of this submanifold. Duggal and Bejancu [22] studied
lightlike hypersurfaces of indefinite Kaehler manifolds. Duggal and Sahin [23]
studied lightlike submanifolds of indefinite Sasakian manifolds. Acet et. al
[11] studied lightlike submanifolds of a para-Sasakian manifold. Sahin and
Gunes [33] studied the geodesic CR-lightlike submanifolds in a Kaehler
manifold. They also studied the integrability of distributions in CR-lightlike
submanifolds in [34]. Gonga et. al [29] studied totally umbilical CR-lightlike
submanifolds of indefinite Kaehler manifolds.

On the other hand, Crasmareanu and Hretcanu [18] constructed the golden
structure on a differentiable manifold $(\overline{M},g)$ as a particular case
of polynomial structure [28]. Gezer et. al investigated the integrability
conditions of golden Riemannian structure. The golden structure was also
studied in various studies (see [24], [25], [26], [27], [36]). Hretcanu C.E.
[30] studied submanifolds in Riemannian manifold with golden structure. The
authors studied some properties of submanifoldsof Riemannian manifolds with
golden structure in [8]. They also studied hypersurfaces of a metallic
Riemannian manifold in [9].
The growing importance of lightlike submanifolds in mathematical physics, in
particular, their use in relativity and many more, motivated the authors to
study lightlike submanifolds extensively. In this paper, we study CR-lightlike
submanifold of a golden semi-Riemannian manifold. The paper is organized as
follows:
In section 2, we define a golden structure manifold and CR-lightlike
submanifolds of a golden Riemannian manifold. In section 3, we establish
several properties of geodesic CR-lightlike submanifolds on a golden
Riemannian manifold. In section 4, we investigate some properties of totally
umbilical CR-lightlike submanifolds. In the last section, we give a non-
trivial example of CR-submanifolds of a golden semi-Riemannian manifold.

##  2 Definition and preliminaries

Let $(\overline{M},{g})$ is a $(m+n)-$dimensional semi-Riemannian manifold,
$m,n\geq 1$ and ${g}$ be a semi-Riemannian metric on $\overline{M}.$ We denote
by $q$ the constant index of ${g}$ and we suppose that $\overline{M}$ is not
Riemannian manifold.
Let $M$ be a lightlike submanifold of dimension $m$ of $\overline{M}.$ In this
case there exists a smooth distribution on $M,$ named a radical distribution
$Rad(TM)$ such that $Rad(TM)=TM\cap TM^{\bot},$ $\forall p\in M.$ If the rank
of $RadTM$ is $r\geq 0,$ $M$ is called an $r-$lightlike submanifold of
$\overline{M}.$ Then there arise four cases:
I: $0<r<min\\{m,n\\};$ the submanifold is called an $r-$lightlike submanifold.
II: $1<r=n<m;$ The submanifold is called a coisotropic submanifold.
III: $1<r=m<n;$ The submanifold is called a isotropic submanifold.
IV: $1<r=n=m;$ The submanifold is called totally lightlike submanifold.
Let $M$ be an $r-$lightlike submanifold of $\overline{M}.$ We consider the
complementary distribution $S(TM)$ of $Rad(TM)$ on $TM$ which is called a
screen distribution. Then we have the direct orthogonal sum

$TM=RadTM\bot S(TM).$

As, $S(TM)$ is a non-degenerate vector subbundle of $T\overline{M}|_{M},$ we
put

$T\overline{M}|_{M}=S(TM)\bot S(TM)^{\bot},$

where $S(TM)^{\bot}$ is the complementary orthogonal vector subbundle of
$S(TM)$ in $T\overline{M}|_{M}.$
$S(TM),S(TM^{\bot})$ are non-degenerate, we have the following orthogonal
direct decomposition

$S(TM)^{\bot}=S(TM^{\bot})\bot S(TM^{\bot})^{\bot}.$

###### Theorem 2.1

[21] Let $(M,g,S(TM),S(TM)^{\bot})$ be an $r-$lightlike submanifold of a semi-
Riemannian manifold $(\overline{M},{g}).$ Then there exists a complementary
vector bundle $ltr(TM)$ called a lightlike transversal bundle of $Rad(TM)$ in
$S(TM^{\bot})^{\bot}$ and the basis of $\Gamma(ltr(TM)|_{V})$ consists of
smooth sections $\\{N_{1},.....,N_{r}\\}$ of $S(TM^{\bot})^{\bot}|_{V}$ such
that

${g}(N_{i},\xi_{j})=\delta_{ij},\;{g}(N_{i},N_{j})=0,\;i,j=0,1,2,,,,r,$

where $\\{\xi_{1},.....,\xi_{r}\\}$ is a basis of $\Gamma(RadTM)|_{U}.$

We consider the vector bundle

$tr(TM)=ltr(TM)\bot S(TM^{\bot}).$

Thus,

$T\overline{M}=TM\oplus tr(TM)=S(TM)\bot S(TM^{\bot})\bot(Rad(TM)\oplus
ltr(TM).$

Let $\overline{\nabla}$ be the Levi-Civita connection on $\overline{M},$ we
have

$\overline{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y),\;\forall X,Y\in\Gamma(TM)$ (2.1)

and

$\overline{\nabla}_{X}V=-A_{V}X+\nabla_{X}^{\bot}V,\;\forall
X\in\Gamma(TM)\;and\;V\in\Gamma(tr(TM)),$ (2.2)

where $\\{\nabla_{X}Y,A_{V}X\\}$ and $\\{h(X,Y),\nabla_{X}^{\bot}V\\}$ belongs
to $\Gamma(TM)$ and $\Gamma(tr(TM)),$ respectively.
Using projection $L:tr(TM)\rightarrow ltr(TM),$ and $S:tr(TM)\rightarrow
S(TM^{\bot}),$ we have

$\overline{\nabla}_{X}Y=\nabla_{X}Y+h^{l}(X,Y)+h^{s}(X,Y),$ (2.3)
$\overline{\nabla}_{X}N=-A_{N}X+\nabla_{X}^{l}N+D^{s}(X,N),$ (2.4)

and

$\overline{\nabla}_{X}W=-A_{W}X+\nabla_{X}^{s}+D^{l}(X,W)$ (2.5)

for any $X,Y\in\Gamma(TM),\;N\in\Gamma(ltr(TM))$ and
$W\in\Gamma(S(TM^{\bot})),$
where
$h^{l}(X,Y)=Lh(X,Y),\;h^{s}(X,Y)=Sh(X,Y),\;\nabla_{X}^{l}N,\;D^{l}(X,W)\in\Gamma(ltr(TM)),\;\nabla_{X}^{s}\;D^{s}(X,N)\in\Gamma(S(TM^{\bot}))$
and $\nabla_{X}Y,\;A_{N}X,\;A_{W}X\in\Gamma(TM).$
$P$ denotes the projection morphism of $TM$ to the screen distribution, we
consider the distribution

$\nabla_{X}PY=\nabla_{X}^{*}PY+h^{*}(X,PY),$
$\nabla_{X}\xi=-A_{\xi}^{*}X+\nabla_{X}^{*t}\xi,$ (2.6)

where $X,Y\in\Gamma(TM),\;\xi\in\Gamma(Rad(TM)).$
Then, we have the following equations

${g}(h^{l}(X,PY),\xi)=g(A_{\xi}^{*}X,PY),$ ${g}(h^{*}(X,PY),N)=g(A_{N}X,PY),$
(2.7) $g(A_{\xi}^{*}PX,PY)=g(PX,A_{\xi^{*}}PY),$ $A^{*}_{\xi}\xi=0.$

Consider that $\overline{\nabla}$ is a metric connection, we have

$(\nabla_{X}{g})(Y,Z)={g}(h^{l}(X,Y),Z)+{g}(h^{l}(X,YZ),Y).$ (2.8)

Using the properties of linear connection, we have

$(\nabla_{X}h^{l})(Y,Z)=\nabla_{X}^{l}(h^{l}(Y,Z))-h^{l}(\overline{\nabla}_{X}Y,Z)-h^{l}(Y,\overline{\nabla}_{X}Z),$
(2.9)
$(\nabla_{X}h^{s})(Y,Z)=\nabla_{X}^{s}(h^{s}(Y,Z))-h^{s}(\overline{\nabla}_{X}Y,Z)-h^{s}(Y,\overline{\nabla}_{X}Z).$
(2.10)

###### Definition 2.2

Let $(\overline{M},{J},{g})$ be a real m-dimensional golden semi-Riemannian
manifold and $M$ be a real n-dimensional submanifold of $\overline{M},$ that
is

${J}^{2}={J}+I,$ (2.11)

where ${J}$ is a non-null tensor of type $(1,1)$ and metric ${g}$ is
${J}-$compatible, $i.e$

${g}({J}X,Y)={g}(X,{J}X)$ (2.12)

then, $\overline{M}$ is called a locally golden manifold if $J$ is parallel
with respect to $\overline{\nabla}$, i.e. $\overline{\nabla}_{X}J=0$ $\forall
X,Y\in\Gamma(T\overline{M}).$

###### Definition 2.3

A submanifold $M$ of a golden semi-Riemannian manifold $\overline{M}$ is said
to be a CR-lightlike submanifold if the following two conditions are
fulfilled:
(i) ${J}(Rad(TM))$ is a distribution on $M$ such that

$Rad(TM)\cap{J}Rad(TM)=\\{0\\}.$

(ii) There exist vector bundles
$S(TM),\;S(TM^{\bot}),\;ltr(TM),\;D_{0}\;and\;D^{\prime}$ over $M$ such that

$S(TM)=\\{{J}(RadTM)\oplus D^{\prime}\\}\bot
D_{0},\;{J}D_{0}=D_{0},\;{J}D^{\prime}=L_{1}\bot L_{2},$

where $D_{0}$ is a non-degenerate distribution on $M$ and $L_{1},$ $L_{2},$
are vector bundles of $ltr(TM)$ and $S(TM^{\bot})$ respectively.

From the definition of CR-lightlike submanifold, we have

$TM=D\oplus D^{\prime},$

where $D=Rad(TM)\bot{J}Rad(TM)\bot D_{0}.$
We denote by $S$ and $Q$ the projection on $D$ and $D^{\prime},$ respectively.
Then we have

${J}X=fX+wX$

for any $X,Y\in\Gamma(TM),$ where $fX={J}SX$ and $wX={J}QX.$
On the other hand, we have

${J}V=BV+CV$

for any $V\in\Gamma(tr(TM)),$ where $BV\in\Gamma(TM)$ and
$CV\in\Gamma(tr(TM)).$ Unless $M_{1}$ and $M_{2}$ are supposed to as
${J}L_{1}$ and ${J}L_{2},$ respectively.

###### Lemma 2.4

Let $M$ be a CR-lightlike submanifold of golden semi-Riemannian manifold and
screen distribution be totally geodesic. Then $\nabla_{X}Y\in\Gamma(S(TM)),$
where $X,Y\in\Gamma(S(TM)).$

Proof  For any $X,Y\in\Gamma(S(TM)),$

$\displaystyle{g}(\nabla_{X}Y,N)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y-h(X,Y),N)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y,N)-{g}(h(X,Y),N)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y,N)$ $\displaystyle=$
$\displaystyle-{g}(Y,\overline{\nabla}_{X}N).$

Using (2.4), we have

$\displaystyle{g}(\nabla_{X}Y,N)$ $\displaystyle=$
$\displaystyle-{g}(Y,-A_{N}X+\nabla_{X}^{\bot}N)$ $\displaystyle=$
$\displaystyle{g}(Y,A_{N}X)-{g}(Y,\nabla_{X}^{\bot}N)$ $\displaystyle=$
$\displaystyle{g}(Y,A_{N}X).$

Using (2.7), we get

${g}(\nabla_{X}Y,N)={g}(h^{*}(X,Y),N).$

Since screen distribution is totally geodesic, then $h^{*}(X,Y)=0,$

${g}(\overline{\nabla}_{X}Y,N)=0.$

Using Theorem 2.1, we have

$\nabla_{X}Y\in\Gamma(S(TM)),$

where $X,Y\in\Gamma(S(TM)).$

###### Lemma 2.5

Let $M$ be a CR-lightlike submanifold a locally golden semi-Riemannian
manifold $\overline{M}.$ Then $\nabla_{X}JX=J\nabla_{X}X$ for any
$X\in\Gamma(D_{0}).$

Proof  Let $X,Y\in\Gamma(D_{0})$, using (2.1), we have

$\displaystyle{g}(\nabla_{X}JX,Y)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}JX-h(X,JX),Y)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}JX,Y)-{g}(h(X,JX),Y)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}JX,Y)$ $\displaystyle=$
$\displaystyle{g}((\overline{\nabla}_{X}J)X+J(\overline{\nabla}_{X}X),Y).$

Since $\overline{\nabla}_{X}J=0,$

$\displaystyle{g}(\nabla_{X}JX,Y)$ $\displaystyle=$
$\displaystyle{g}(J(\overline{\nabla}_{X}X),Y)$ $\displaystyle=$
$\displaystyle{g}(J(\overline{\nabla}_{X}X),Y)$ $\displaystyle=$
$\displaystyle{g}((\overline{\nabla}_{X}X),JY)$ $\displaystyle=$
$\displaystyle{g}(\nabla_{X}X,JY)-{g}(h(X,X),JY)$ $\displaystyle=$
$\displaystyle{g}(\nabla_{X}X,JY)$ $\displaystyle{g}(\nabla_{X}JX,Y)$
$\displaystyle=$ $\displaystyle{g}(J(\nabla_{X}X),Y),$
$\displaystyle{g}(\nabla_{X}JX-J(\nabla_{X}X),Y)$ $\displaystyle=$
$\displaystyle 0.$

Non-degeneracy of $D_{0},$ implies

$\nabla_{X}JX=J(\nabla_{X}X),$

where $X\in\Gamma(D_{0}).$

##  3  Geodesic CR-lightlike submanifolds

###### Definition 3.1

A CR-lightlike submanifold of a golden semi-Riemannian manifold is called
mixed geodesic CR-lightlike submanifold if its second fundamental form $h$
satisfies

$h(X,U)=0,$

where $X\in\Gamma(D)$ and $U\in\Gamma(D^{\prime}).$

###### Definition 3.2

A CR-lightlike submanifold of a golden semi-Riemannian manifold is called
$D-$geodesic CR-lightlike submanifold if its second fundamental form $h$
satisfies

$h(X,Y)=0,$

where $X,Y\in\Gamma(D).$

###### Definition 3.3

A CR-lightlike submanifold of a golden semi-Riemannian manifold is called
$D^{\prime}-$geodesic CR-lightlike submanifold if its second fundamental form
$h$ satisfies

$h(U,V)=0,$

where $U,V\in\Gamma(D^{\prime}).$

###### Theorem 3.4

Let $\overline{M}$ be a golden semi-Riemannian manifold and $M$ be a CR-
lightlike submanifold of $\overline{M}.$ Then, M is totally geodesic if and
only if

$(L_{\xi}{g})(X,Y)=0$

and

$(L_{W}{g})(X,Y)=0$

for any
$X,Y\in\Gamma(TM),\;\xi\in\Gamma(Rad(TM))\;and\;W\in\Gamma(S(TM^{\bot})).$

Proof Since M is totally geodesic, then

$h(X,Y)=0$

for any $X,Y\in\Gamma(TM).$
By definition of lightlike submanifolds, $h(X,Y)=0$ if and only if

${g}(h(X,Y),\xi)=0$

and

${g}(h(X,Y),W)=0.$

$\displaystyle{g}(h(X,Y),\xi)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y-\nabla_{X}Y,\xi)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y,\xi)-{g}(\nabla_{X}Y,\xi)$
$\displaystyle=$ $\displaystyle{g}({\nabla}_{X}Y,\xi)$ $\displaystyle=$
$\displaystyle\overline{\nabla}_{X}{g}(Y,\xi)-{g}(Y,\overline{\nabla}_{X}\xi)$
$\displaystyle=$ $\displaystyle-{g}(Y,\overline{\nabla}_{X}\xi)$
$\displaystyle=$ $\displaystyle-{g}(Y,[X,\xi]+\overline{\nabla}_{\xi}X$
$\displaystyle=$ $\displaystyle-{g}(Y,[X,\xi])-g(Y,\overline{\nabla}_{\xi}X)$
$\displaystyle=$
$\displaystyle-{g}(Y,[X,\xi])+{g}(\overline{\nabla}_{\xi}Y,X)$
$\displaystyle=$
$\displaystyle-{g}(Y,[X,\xi])+{g}([\xi,Y],X)+{g}(\overline{\nabla}_{Y}\xi,X)$
$\displaystyle=$
$\displaystyle-{g}(Y,[X,\xi])+{g}(X,[\xi,Y])+{g}(\overline{\nabla}_{Y}\xi,X)$
$\displaystyle=$
$\displaystyle-(L_{\xi}{g})(X,Y)+{g}(\overline{\nabla}_{Y}\xi,X)$
$\displaystyle=$
$\displaystyle-(L_{\xi}{g})(X,Y)-{g}(\xi,\overline{\nabla}_{Y}X)$
$\displaystyle=$
$\displaystyle-(L_{\xi}{g})(X,Y)-{g}(\xi,{\nabla}_{Y}X+h(X,Y))$
$\displaystyle=$
$\displaystyle-(L_{\xi}{g})(X,Y)-{g}(\xi,{\nabla}_{Y}X)-{g}(\xi,h(X,Y)))$
$\displaystyle=$ $\displaystyle-(L_{\xi}{g})(X,Y)-{g}(\xi,h(X,Y)))$
$\displaystyle 2{g}(h(X,Y)$ $\displaystyle=$
$\displaystyle-(L_{\xi}{g})(X,Y).$

Since ${g}(h(X,Y),\xi)=0,$ we have

$(L_{\xi}{g})(X,Y)=0.$

Similarly,

$\displaystyle\ {g}(h(X,Y),W)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y-\nabla_{X}Y,W)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y,W)$ $\displaystyle=$
$\displaystyle-{g}(Y,[X,W]+\overline{\nabla}_{W}X)$ $\displaystyle=$
$\displaystyle-{g}(Y,[X,W])-{g}(Y,\overline{\nabla}_{W}X)$ $\displaystyle=$
$\displaystyle-{g}(Y,[X,W])+{g}(\overline{\nabla}_{W}Y,X)$ $\displaystyle=$
$\displaystyle-{g}(Y,[X,W])+{g}([W,Y],X)+{g}(\overline{\nabla}_{Y}W,X)$
$\displaystyle=$
$\displaystyle-{g}(Y,[X,W])+{g}(X,[W,Y])+{g}(\overline{\nabla}_{Y}W,X)$
$\displaystyle=$ $\displaystyle-(L_{W}{g})(X,Y)+{g}(\overline{\nabla}_{Y}W,X)$
$\displaystyle=$ $\displaystyle-(L_{W}{g})(X,Y)-{g}(W,\overline{\nabla}_{Y}X)$
$\displaystyle=$ $\displaystyle-(L_{W}{g})(X,Y)-{g}(W,{\nabla}_{Y}X+h(X,Y))$
$\displaystyle=$
$\displaystyle-(L_{W}{g})(X,Y)-{g}(W,{\nabla}_{Y}X)-{g}(W,h(X,Y)))$
$\displaystyle=$ $\displaystyle-(L_{W}{g})(X,Y)-{g}(W,h(X,Y))$ $\displaystyle
2{g}(h(X,Y),W)$ $\displaystyle=$ $\displaystyle-(L_{W}{g})(X,Y).$

Since ${g}(h(X,Y),W)=0,$ we get

$(L_{W}{g})(X,Y)=0$

for any $W\in\Gamma(S(TM^{\bot})).$

###### Lemma 3.5

Let $M$ be a CR-lightlike submanifold of a golden semi-Riemannian manifold
$\overline{M}.$ Then

${g}(h(X,Y),W)={g}(A_{W}X,Y)$

for any
$X\in\Gamma(D),\;Y\in\Gamma(D^{\prime})\;and\;W\in\Gamma(S(TM^{\bot})).$

Proof Using (2.1), we have

$\displaystyle{g}(h(X,Y),W)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y-\nabla_{X}Y,W)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y,W)-{g}(\nabla_{X}Y,W)$
$\displaystyle=$ $\displaystyle{g}(\overline{\nabla}_{X}Y,W)$ $\displaystyle=$
$\displaystyle{g}(Y,\overline{\nabla}_{X}W).$

From (2.5) it follows that

$\displaystyle{g}(h(X,Y),W)$ $\displaystyle=$
$\displaystyle-{g}(Y,-A_{W}X+\nabla_{X}^{s}W+D^{s}(X,W))$ $\displaystyle=$
$\displaystyle{g}(Y,A_{W}X)-{g}(Y,\nabla_{X}^{s}W)-{g}(Y,D^{s}(X,W))$
$\displaystyle{g}(h(X,Y),W)$ $\displaystyle=$ $\displaystyle{g}(Y,A_{W}X),$

where $X\in\Gamma(D),\;Y\in\Gamma(D^{\prime})\;and\;W\in\Gamma(S(TM^{\bot})).$

###### Theorem 3.6

Let $M$ be a CR-lightlike submanifold of a golden semi-Riemannian manifold
$\overline{M}.$ Then $M$ is mixed geodesic if and only if

$A^{*}_{\xi}X\in\Gamma(D_{0}\bot{J}L_{1})$

and

$A_{W}X\in\Gamma(D_{0}\bot Rad(TM)\bot{J}L_{1})$

for any
$X\in\Gamma(D),\;\xi\in\Gamma(Rad(TM))\;and\;W\in\Gamma(S(TM^{\bot})).$

Proof For any
$X\in\Gamma(D),\;Y\in\Gamma(D^{\prime})\;and\;W\in\Gamma(S(TM^{\bot})),$ we
have
From (2.1),

$\displaystyle{g}(h(X,Y),\xi)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y-\nabla_{X}Y,\xi)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y,\xi)-{g}(\nabla_{X}Y,\xi)$
$\displaystyle=$ $\displaystyle{g}(\overline{\nabla}_{X}Y,\xi)$
$\displaystyle=$ $\displaystyle-{g}(Y,\overline{\nabla}_{X}\xi).$

Again using (2.1), we get

$\displaystyle{}{g}(h(X,Y),\xi)$ $\displaystyle=$
$\displaystyle-{g}(Y,\nabla_{X}\xi+h(X,\xi))$ $\displaystyle=$
$\displaystyle-{g}(Y,\nabla_{X}\xi)-{g}(Y,h(X,\xi))$ $\displaystyle=$
$\displaystyle-{g}(Y,\nabla_{X}\xi).$

Using (2.6), we have

$\displaystyle{}{g}(h(X,Y),\xi)$ $\displaystyle=$
$\displaystyle-{g}(Y,-A_{\xi}^{*}X+\nabla^{*t}_{X}\xi)$ $\displaystyle=$
$\displaystyle{g}(Y,A_{\xi}^{*}X)+{g}(Y,\nabla^{*t}_{X}\xi)$
$\displaystyle{g}(h(X,Y),\xi)$ $\displaystyle=$
$\displaystyle{g}(Y,A_{\xi}^{*}X)$ $\displaystyle{g}(Y,A_{\xi}^{*}X)$
$\displaystyle=$ $\displaystyle 0.$

Since CR-lightlike submanifold $M$ is mixed geodesic. Then

${g}(h(X,Y),\xi)=0$ $\Rightarrow{g}(Y,A_{\xi}^{*}X)=0$ $\Rightarrow
A_{\xi}^{*}X\in\Gamma(D_{0}\bot{J}L_{1}),$

where $X\in\Gamma(D),Y\in\Gamma(D^{\prime}).$
From (2.1), we get

$\displaystyle{g}(h(X,Y),W)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y-\nabla_{X}Y,W)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}Y,W)-{g}(\nabla_{X}Y,W)$
$\displaystyle=$ $\displaystyle{g}(\overline{\nabla}_{X}Y,W)$ $\displaystyle=$
$\displaystyle-{g}(Y,\overline{\nabla}_{X}W).$

From (2.5), we get

$\displaystyle{}{g}(h(X,Y),W)$ $\displaystyle=$
$\displaystyle-{g}(Y,-A_{W}X+\nabla_{X}^{s}W+D^{l}(X,W))$ $\displaystyle=$
$\displaystyle{g}(Y,A_{W}X)-{g}(Y,\nabla_{X}^{s}W)-{g}(Y,D^{l}(X,W))$
$\displaystyle{g}(h(X,Y),W)$ $\displaystyle=$ $\displaystyle{g}(Y,A_{W}X).$

Since, $M$ is mixed geodesic, then ${g}(h(X,Y),W)=0$

$\Rightarrow{g}(Y,A_{W}X)=0.$ $A_{W}X\in\Gamma(D_{0}\bot
Rad(TM)\bot{J}L_{1}).$

###### Theorem 3.7

Let $M$ be a CR-lightlike submanifold of a semi-Riemannian manifold
$\overline{M}.$ Then $M$ is $D^{\prime}-$geodesic if and only if $A_{W}Z$ and
$A_{\xi}^{*}Z$ have no component in $M_{2}\bot{J}Rad(TM)$ for any
$Z\in\Gamma(D^{\prime}),\;\xi\in\Gamma(Rad(TM))\;and\;W\in\Gamma(S(TM^{\bot})).$

Proof From (2.1) we have

$\displaystyle{g}(h(Z,V),W)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{Z}V-\nabla_{X}Y,W)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{Z}V,W)-{g}(\nabla_{X}Y,W)$
$\displaystyle=$ $\displaystyle-\overline{g}(\nabla_{X}Y,W),$

where $Z,V\in\Gamma(D^{\prime}).$
Using (2.5), we have

$\displaystyle{g}(h(Z,V),W)$ $\displaystyle=$
$\displaystyle-{g}(V,-A_{W}Z+\nabla_{Z}^{s}+D^{l}(Z,W))$ $\displaystyle=$
$\displaystyle{g}(V,A_{W}Z)-{g}(V,\nabla_{Z}^{s})-{g}(V,D^{l}(Z,W))$
$\displaystyle{g}(h(Z,V),W)$ $\displaystyle=$ $\displaystyle{g}(V,A_{W}Z).$
(3.1)

Since $M$ is $D^{\prime}-$geodesic, then ${g}(h(Z,V),W)=0.$
From (3.1), we get

${g}(V,A_{W}Z)=0.$

Now,

$\displaystyle{g}(h(Z,V),\xi)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{Z}V-\nabla_{Z}V,\xi)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{Z}V,\xi)-{g}(\nabla_{Z}V,\xi)$
$\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{Z}V,\xi)=-{g}(V,\overline{\nabla}_{Z}\xi).$

From (2.6), we have

$\displaystyle{}{g}(h(Z,V),\xi)$ $\displaystyle=$
$\displaystyle-{g}(V,-A_{\xi}^{*}Z+\nabla_{Z}^{*t}\xi)$ $\displaystyle=$
$\displaystyle{g}(V,A_{\xi}^{*}Z)-{g}(V,\nabla_{Z}^{*t}\xi)$
$\displaystyle{g}(h(Z,V),\xi)$ $\displaystyle=$
$\displaystyle{g}(V,A_{\xi}^{*}Z)$ $\displaystyle{g}(h(Z,V),\xi)$
$\displaystyle=$ $\displaystyle{g}(A_{\xi}^{*}V,Z).$

Since $M$ is $D^{\prime}-$ geodesic, then

${g}(h(Z,V),\xi)=0$ $\Rightarrow{g}(A_{\xi}^{*}V,Z)=0.$

Thus, $A_{W}Z$ and $A^{*}_{\xi}Z$ have no component in $M_{2}\bot JRad(TM).$

###### Lemma 3.8

Let $M$ be a CR-lightlike submanifold of a golden semi-Riemannian manifold
$\overline{M}.$ If the distribution $D$ is integrable, then the following
conditions hold:
(i)
$J{g}(D^{l}({J}X,W),Y)-{g}(D^{l}(X,W),{J}Y)={g}(A_{W}{J}X,Y)-{g}(A_{W}X,{J}Y),$
(ii) ${g}(D^{l}({J}X),\xi)={g}(A_{W}X,{J}\xi),$
(iii) ${g}(D^{l}(X,W),\xi)={g}(A_{W}{J}X,J\xi)-{g}(A_{W}X,J\xi),$
where $X,Y\in\Gamma(TM),\;\xi\in\Gamma(Rad(TM))$ and
$W\in\Gamma(S(TM^{\bot})).$

Proof From equation (2.5), we have

$\displaystyle{g}(D^{l}({J}X,W),Y)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{JX}W+A_{W}{J}X-\nabla_{JX}^{s}W,Y)$
$\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{JX}W,Y)+{g}(A_{W}{J}X,Y)-{g}(\nabla_{JX}^{s}W,Y)$
$\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{JX}W,Y)+{g}(A_{W}{J}X,Y)$
$\displaystyle=$
$\displaystyle-{g}(W,\overline{\nabla}_{JX}Y)+{g}(A_{W}{J}X,Y).$

Using (2.1), we have

$\displaystyle{}{g}(D^{l}({J}X,W),Y)$ $\displaystyle=$
$\displaystyle-{g}(W,{\nabla}_{JX}Y+h({J}X,Y))+{g}(A_{W}{J}X,Y)$
$\displaystyle=$
$\displaystyle-{g}(W,{\nabla}_{JX}Y)-{g}(W,h({J}X,Y))+{g}(A_{W}{J}X,Y)$
$\displaystyle=$ $\displaystyle-{g}(W,h({J}X,Y))+{g}(A_{W}{J}X,Y)$
$\displaystyle=$ $\displaystyle-{g}(W,h(X,{J}Y))+{g}(A_{W}{J}X,Y).$

Again using (2.1), we have

$\displaystyle{}{g}(D^{l}({J}X,W),Y)$ $\displaystyle=$
$\displaystyle-{g}(W,\overline{\nabla}_{X}{J}Y-\nabla_{X}{J}Y)+{g}(A_{W}{J}X,Y)$
$\displaystyle=$
$\displaystyle-{g}(W,\overline{\nabla}_{X}{J}Y)+{g}(W,\nabla_{X}{J}Y)+{g}(A_{W}{J}X,Y)$
$\displaystyle=$
$\displaystyle-{g}(W,\overline{\nabla}_{X}{J}Y)+{g}(A_{W}{J}X,Y)$
$\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{X}W,{J}Y)+{g}(A_{W}{J}X,Y).$

Using (2.5), we get

$\displaystyle{}{g}(D^{l}({J}X,W),Y)$ $\displaystyle=$
$\displaystyle{g}(-A_{W}X+\nabla_{X}^{s}W+D^{l}(X,W),{J}Y)+$ $\displaystyle=$
$\displaystyle{g}(A_{W}{J}X,Y)-{g}(A_{W}X,{J}Y)+{g}(\nabla_{X}^{s}W,{J}Y)+$
$\displaystyle{g}(D^{l}(X,W),{J}Y)+{g}(A_{W}{J}X,Y)$ $\displaystyle=$
$\displaystyle-{g}(A_{W}X,{J}Y)+{g}(A_{W}{J}X,Y)$
$\displaystyle{}{g}(D^{l}({J}X,W),Y)-{g}(D^{l}(X,W),{J}Y)$ $\displaystyle=$
$\displaystyle{g}(A_{W}{J}X,Y)-{g}(A_{W}X,{J}Y).$

(ii) Using (2.5), we have

$\displaystyle{g}(D^{l}({J}X,W),\xi)$ $\displaystyle=$
$\displaystyle{g}(A_{W}{J}X-\nabla_{{J}X}^{s}W+{\nabla}_{{J}X}W,\xi)$
$\displaystyle=$
$\displaystyle{g}(A_{W}{J}X,\xi)-{g}(\nabla_{{J}X}^{s}W,\xi)+\overline{g}({\nabla}_{{J}X}W,\xi)$
$\displaystyle=$
$\displaystyle{g}(A_{W}{J}X,\xi)+{g}(\overline{\nabla}_{{J}X}W,\xi)$
$\displaystyle=$
$\displaystyle{g}(A_{W}{J}X,\xi)-{g}(W,\overline{\nabla}_{{J}X}\xi).$

Using (2.6), we get

$\displaystyle{}{g}(D^{l}({J}X,W),\xi)$ $\displaystyle=$
$\displaystyle{g}(A_{W}{J}X,\xi)+{g}(W,A^{*}_{\xi}{J}X)-{g}(W,\nabla^{*t}_{{J}X},\xi)$
$\displaystyle{g}(D^{l}({J}X),\xi)$ $\displaystyle=$
$\displaystyle{g}(A_{W}{J}X,\xi)$ $\displaystyle{g}(D^{l}({J}X),\xi)$
$\displaystyle=$ $\displaystyle{g}(A_{W}X,{J}\xi).$

(iii) Replacing $Y$ by $J\xi$ in (i), we have

$\displaystyle
J{g}(D^{l}({J}X,W),J\xi)-{g}(D^{l}(X,W),{J^{2}}\xi)={g}(A_{W}{J}X,J\xi)-{g}(A_{W}X,{J^{2}}\xi).$

Using (2.11), we get

$\displaystyle J{g}(D^{l}({J}X,W),J\xi)-{g}(D^{l}(X,W),(J+I)\xi)$
$\displaystyle=$ $\displaystyle{g}(A_{W}{J}X,J\xi)-k,{g}(A_{W}X,(J+I)\xi)$
$\displaystyle
J{g}(D^{l}({J}X,W),J\xi)-{g}(D^{l}(X,W),J\xi)-{g}(D^{l}(X,W),\xi)$
$\displaystyle=$ $\displaystyle{g}(A_{W}{J}X,J\xi)-{g}(A_{W}X,J\xi)-$
$\displaystyle{g}(A_{W}X,\xi).$ $\displaystyle{g}(D^{l}(X,W),\xi)$
$\displaystyle=$ $\displaystyle{g}(A_{W}{J}X,J\xi)-{g}(A_{W}X,J\xi).$

##  4 Totally Umbilical CR-lightlike submanifolds.

###### Definition 4.1

A lightlike submanifold $(M,g)$ of a semi-Riemannian manifold
$(\overline{M},{g})$ is said to be totally umbilical in $\overline{M}$ if
there is a smooth transversal vector field $H\in\Gamma(tr(TM))$ on $M,$ called
the transversal curvature vector field of $M,$ such that

$h(X,Y)=H{g}(X,Y),$

where $X,Y\in\Gamma(TM).$
Using (2.3), $M$ is a totally umbilical if and only if there exists smooth
vector fields $H^{l}\in\Gamma(ltr(TM))$ and
$H^{s}(X,Y)\in\Gamma(S(TM^{\bot})),$ such that

$h^{l}(X,Y)=H^{l}{g}(X,Y),\;\;h^{s}(X,Y)=H^{s}{g}(X,Y).$

###### Theorem 4.2

Let $M$ be a totally umbilical CR-lightlike submanifold of a locally golden
manifold $\overline{M}$ and screen distribution be totally geodesic. Then

$A_{JZ}W=A_{JW}Z,\;\forall\;W,Z\in\Gamma D^{\prime}.$

Proof  Since $\overline{M}$ be a golden semi-Riemannian manifold, therefore

$J\overline{\nabla}_{Z}W=\overline{\nabla}_{Z}JW.$

Using (2.1) and (2.2), we have

$\displaystyle J(\nabla_{Z}W+h(Z,W))$ $\displaystyle=$ $\displaystyle-
A_{JW}Z+\nabla_{Z}^{t}JW$ $\displaystyle J(\nabla_{Z}W)+J(h(Z,W))$
$\displaystyle=$ $\displaystyle-A_{JW}Z+\nabla_{Z}^{t}JW.$ (4.1)

Interchange $Z$ and $W$ in the above equation, we have

$J(\nabla_{W}Z)+J(h(W,Z))=-A_{JZ}W+\nabla_{W}^{t}JZ.$ (4.2)

Substract equations (4.1) and (4.2), we get

$\displaystyle J(\nabla_{Z}W)-J(\nabla_{W}Z)+J(h(Z,W))-J(h(W,Z))$
$\displaystyle=$ $\displaystyle A_{JZ}W-A_{JW}Z+\nabla_{Z}^{t}JW$
$\displaystyle-\nabla_{W}^{t}JZ$ $\displaystyle J(\nabla_{Z}W)-J(\nabla_{W}Z)$
$\displaystyle=$ $\displaystyle A_{JZ}W-A_{JW}Z+\nabla_{Z}^{t}JW$
$\displaystyle-\nabla_{W}^{t}JZ$ $\displaystyle
J(\nabla_{Z}W-\nabla_{W}Z)-\nabla_{Z}^{t}JW+\nabla_{W}^{t}JZ$ $\displaystyle=$
$\displaystyle A_{JZ}W-A_{JW}Z.$ (4.3)

Taking inner product with $X\in\Gamma(D_{0})$ in (4.3), we have

$\displaystyle{g}(J\nabla_{Z}W,X)-{g}(J(\nabla_{W}Z,X)-{g}(\nabla_{W}^{t}JZ,X)-{g}(\nabla_{W}^{t}JZ,X)$
$\displaystyle=$ $\displaystyle{g}(A_{JZ}W,X)$ $\displaystyle-{g}(A_{JW}Z,X)$
$\displaystyle{g}(J(\nabla_{W}Z,X)-{g}(J(\nabla_{W}Z,X)$ $\displaystyle=$
$\displaystyle{g}(A_{JZ}W,X)$ $\displaystyle-{g}(A_{JW}Z,X).$

Using (2.12), we get

$\displaystyle{g}(\nabla_{W}Z,JX)-{g}(\nabla_{W}Z,JX)$ $\displaystyle=$
$\displaystyle{g}(A_{JZ}W,X)-{g}(A_{JW}Z,X)$
$\displaystyle{g}(A_{JZ}W-A_{JW}Z,X)$ $\displaystyle=$
$\displaystyle{g}(\nabla_{W}Z,JX)-{g}(\nabla_{W}Z,JX).$ (4.4)

Now,

$\displaystyle{g}(\nabla_{W}Z,JX)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{W}Z-h(W,Z),JX)$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{W}Z,JX)-{g}(h(W,Z),JX),$ $\displaystyle=$
$\displaystyle{g}(\overline{\nabla}_{W}Z,JX)$ $\displaystyle=$
$\displaystyle-{g}(Z,{\nabla}_{W}JX)$ $\displaystyle{g}(\nabla_{W}Z,JX)$
$\displaystyle=$
$\displaystyle-{g}(Z,(\overline{\nabla}_{W}J)X-J(\overline{\nabla}_{W}X)).$

Since $J$ is parallel to $\overline{\nabla},$ i.e. $\overline{\nabla}_{X}J=0$

${g}(\nabla_{W}Z,JX)=-J(\overline{\nabla}_{W}X)).$

Using (2.3), we have

$\displaystyle{g}(\nabla_{W}Z,JX)$ $\displaystyle=$
$\displaystyle-{g}(JZ,\nabla_{W}X+h^{s}(W,X)+h^{l}(W,X))$
$\displaystyle{g}(\nabla_{W}Z,JX)$ $\displaystyle=$
$\displaystyle-{g}(JZ,\nabla_{W}X)-{g}(JZ,h^{s}(W,X))-{g}(JZ,h^{l}(W,X)).$
(4.5)

Since $M$ is a totally umbilical CR-lightlike submanifold and screen
distribution is totally geodesic, then

$h^{s}(W,X)=H^{s}{g}(W,X)=0$

and

$h^{l}(W,X)=H^{l}{g}(W,X)=0,$

where $W\in\Gamma(D^{\prime})$ and $X\in\Gamma(D_{0}).$
From (4.5), we have

${g}(\nabla_{W}Z,JX)=-{g}(JZ,\nabla_{W}X).$

From Lemma 2.4, we get

${g}(\nabla_{W}Z,JX)=0.$

Similarly,

${g}(\nabla_{Z}W,JX)=0$

Using (4.4), we have

$g(A_{JZ}W-A_{JW}Z,X)=0.$

Since $D_{0}$ is non-degenerate, then

$A_{JZ}W-A_{JW}Z=0$ $\Rightarrow A_{JZ}W=A_{JW}Z.$

###### Definition 4.3

For a CR-lightlike submanifold $M,$ a plane $X\wedge Z$ with
$X\in\Gamma(D_{0})$ and $Z\in\Gamma(D^{\prime})$ is called a CR-lightlike
section. The sectional curvature $\overline{K}(\pi)$ of a CR-lightlike section
$\pi$ is called CR-lightlike sectional curvature.

###### Theorem 4.4

Let $M$ be a totally umbilical CR-lightlike submanifold $M$ of a golden
manifold $\overline{M}.$ Then, the CR-lightlike sectional curvature of $M$
vanishes, that is $\overline{K}(\pi)=0$ for all CR-lightlike section $\pi.$

Proof  Since, $M$ is a totally umbilical CR-lightlike submanifold of
$\overline{M}$ then from (2.9) and (2.10), we have

$\displaystyle(\nabla_{X}h^{l})(Y,Z)=g(Y,Z)\nabla^{l}_{X}H^{l}-H^{l}\\{(\nabla_{X}g)(Y,Z)\\},$
(4.6)
$\displaystyle(\nabla_{X}h^{s})(Y,Z)=g(Y,Z)\nabla^{s}_{X}H^{s}-H^{s}\\{(\nabla_{X}g)(Y,Z)\\}.$
(4.7)

For a CR-lightlike section $\pi=X\wedge
Z,\;X\in\Gamma(D_{0}),Z\in\Gamma(D^{\prime}).$
From (2.8), we have $(\nabla_{X}g)(Y,Z)=0.$ Therefore, from (4.6) and (4.7),
we get

$\displaystyle(\nabla_{X}h^{l})(Y,Z)=g(Y,Z)\nabla^{l}_{X}H^{l},$ (4.8)
$\displaystyle(\nabla_{X}h^{s})(Y,Z)=g(Y,Z)\nabla^{s}_{X}H^{s}.$ (4.9)

Now, from (4.8) and (4.9), we get

$\displaystyle\\{\overline{R}(X,Y)Z\\}^{tr}$ $\displaystyle=$ $\displaystyle
g(Y,Z)\nabla^{l}_{X}H^{l}-g(X,Z)\nabla^{l}_{Y}H^{l}+g(Y,Z)D^{l}(X,H^{s})$
(4.10)
$\displaystyle-g(X,Z)D^{l}(Y,H^{s})+g(Y,Z)\nabla^{s}_{X}H^{s}-g(X,Z)\nabla^{s}_{Y}H^{s}$
$\displaystyle+g(Y,Z)D^{s}(X,H^{l})-g(X,Z)D^{s}(Y,H^{l}).$

For any $V\in\Gamma(tr(TM))$, from equation (4.10) we get

$\displaystyle\overline{R}(X,Y,Z,V)$ $\displaystyle=$ $\displaystyle
g(Y,Z){g}(\nabla^{l}_{X}H^{l},V)-g(X,Z){g}(\nabla^{l}_{Y}H^{l},V)+$
$\displaystyle g(Y,Z){g}(D^{l}(X,H^{s}),V)-g(X,Z){g}(D^{l}(Y,H^{s}),V)+$
$\displaystyle
g(Y,Z){g}(\nabla^{s}_{X}H^{s},V)-g(X,Z){g}(\nabla^{s}_{Y}H^{s},V)$
$\displaystyle+g(Y,Z){g}(D^{s}(X,H^{l}),V)-g(X,Z){g}(D^{s}(Y,H^{l},V).$
$\displaystyle\overline{R}(X,Z,X,Z)$ $\displaystyle=$ $\displaystyle
g(Z,X){g}(\nabla^{l}_{X}H^{l},Z)-g(X,X){g}(\nabla^{l}_{Z}H^{l},Z)$
$\displaystyle+g(Z,X){g}(D^{l}(X,H^{s}),Z)-g(X,X){g}(D^{l}(Z,H^{s}),Z)+$
$\displaystyle
g(Z,X){g}(\nabla^{s}_{X}H^{s},Z)-g(X,X){g}(\nabla^{s}_{Z}H^{s},Z)$
$\displaystyle+g(Z,X){g}(D^{s}(X,H^{l}),Z)-g(X,X){g}(D^{s}(Z,H^{l},Z).$
$\displaystyle{R}(X,Z,JX,JZ)$ $\displaystyle=$ $\displaystyle
g(Z,JX){g}(\nabla^{l}_{X}H^{l},JZ)-g(X,JX){g}(\nabla^{l}_{Z}H^{l},JZ)+$
$\displaystyle g(Z,JX){g}(D^{l}(X,H^{s}),JZ)-g(X,JX){g}(D^{l}(Z,H^{s}),JZ)+$
$\displaystyle
g(Z,JX){g}(\nabla^{s}_{X}H^{s},JZ)-g(X,JX){g}(\nabla^{s}_{Z}H^{s},JZ)+$
$\displaystyle g(Z,JX){g}(D^{s}(X,H^{l}),JZ)-g(X,JX){g}(D^{s}(Z,H^{l},JZ).$

For any unit vectors $X\in\Gamma(D)$ and $Z\in\Gamma(D^{\prime}),$ we have

$\overline{R}(X,Z,JX,JZ)=\overline{R}(X,Z,X,Z)=0.$

Since,

$K(\gamma)=K_{N}(X\wedge Y)=g(\overline{R}(X,Y)Y,X),$

where

$\overline{R}(X,Z,X,Z)=g(\overline{R}(X,Z)X,Z)$

or,

$\overline{R}(X,Z,JX,JZ)=g(\overline{R}(X,Z)JX,JZ)$

i.e.

$\overline{K}(\pi)=0$

for all CR-sections $\pi.$

##  5 Example

Example 5.1  We consider a semi-Riemannian manifold $R_{2}^{6}$ of signature
$(-,-,+,+,+,+)$ with respect to the canonical basis $(\frac{\partial}{\partial
x_{1}},\frac{\partial}{\partial x_{2}},\frac{\partial}{\partial
x_{3}},\frac{\partial}{\partial x_{4}},\frac{\partial}{\partial
x_{5}},\frac{\partial}{\partial x_{6}})$ and a submanifold $M$ of codimension
$2$ in $R_{2}^{6}$ given by equations

$x_{5}=x_{1}cos\alpha-x_{2}sin\alpha-x_{3}x_{4}tan\alpha$
$x_{6}=x_{1}sin\alpha-x_{2}cos\alpha-x_{3}x_{4},$

where $\alpha\in R-\\{\frac{\pi}{2}+k\pi;\ k\in z\\}.$ The structure on
$R^{6}_{2}$ is defined by

$J(\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial
x_{2}},\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{4}},\frac{\partial}{\partial x_{5}},\frac{\partial}{\partial
x_{6}})=(\overline{\phi}\ \frac{\partial}{\partial
x_{1}},\overline{\phi}\frac{\partial}{\partial
x_{2}},\phi\frac{\partial}{\partial x_{3}},\phi\frac{\partial}{\partial
x_{4}},\phi\frac{\partial}{\partial x_{5}},\phi\frac{\partial}{\partial
x_{6}}).$

Now,

$J^{2}(\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial
x_{2}},\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{4}},\frac{\partial}{\partial x_{5}},\frac{\partial}{\partial
x_{6}})=((\overline{\phi}+1)\ \frac{\partial}{\partial
x_{1}},(\overline{\phi}+1)\frac{\partial}{\partial
x_{2}},(\phi+1)\frac{\partial}{\partial
x_{3}},(\phi+1)\frac{\partial}{\partial x_{4}},$
$(\phi+1)\frac{\partial}{\partial x_{5}},(\phi+1)\frac{\partial}{\partial
x_{6}})$ $J^{2}(\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial
x_{2}},\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{4}},\frac{\partial}{\partial x_{5}},\frac{\partial}{\partial
x_{6}})=J(\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial
x_{2}},\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{4}},\frac{\partial}{\partial x_{5}},\frac{\partial}{\partial x_{6}})\ +\
(\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial x_{2}},$
$\frac{\partial}{\partial x_{3}},\frac{\partial}{\partial
x_{4}},\frac{\partial}{\partial x_{5}},\frac{\partial}{\partial x_{6}}).$
$J^{2}=J+I.$

It follows that $(R_{2}^{6},J)$ is a golden semi-Reimannian manifold.

The tangent bundle $TM$ is spanned by

$X_{0}=-sin\alpha\ \frac{\partial}{\partial x_{5}}-cos\alpha\
\frac{\partial}{\partial x_{6}}-\phi\ \frac{\partial}{\partial x_{2}},$
$X_{1}=-\phi\ sin\alpha\ \frac{\partial}{\partial x_{5}}-\phi\ cos\alpha\
\frac{\partial}{\partial x_{6}}+\ \frac{\partial}{\partial x_{2}},$
$X_{2}=\frac{\partial}{\partial x_{5}}-\overline{\phi}\ sin\alpha\
\frac{\partial}{\partial x_{2}}+\frac{\partial}{\partial x_{1}},$
$X_{3}=-\overline{\phi}\ cos\alpha\ \frac{\partial}{\partial
x_{2}}+\frac{\partial}{\partial x_{4}}+i\frac{\partial}{\partial x_{6}}.$

Thus, $M$ is a 1-lightlike submanifold of $R_{2}^{6}$ with
$RadTM=Span\\{X_{0}\\}$. Using golden structure of $R_{2}^{6},$ we obtain that
$X_{1}=J(X_{0}).$ Thus $J(RadTM)$ is a distribution on $M.$ Hence $M$ is a CR-
lightlike submanifold.

Acknowledgement. All authors would like to thank Integral University, Lucknow,
India for providing financial support under the seed money project
IUL/IIRC/SMP/2021/010 to the present work (manuscript number
IU/R$\&$D/2022-MCN0001708).

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# Jacob’s ladders, existence of almost linear increments of the Hardy-
Littlewood integral and new types of multiplicative laws

Jan Moser  Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
<EMAIL_ADDRESS>
###### Abstract.

In this paper we prove that there is a continuum set of increments with some
minimal structure for the Hardy - Littlewood integral. The result implies a
number of new properties of the Hardy - Littlewood integral.

###### Key words and phrases:

Riemann zeta-function

##  1\. Introduction

### 1.1.

In this paper we shall prove some new results of completely new type for
certain class of increments of the classical Hardy - Littlewood integral111See
[3].

(1.1) $J(T)=\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t.$

We use for this purpose the theory of Jacob’s ladders introduced in our papers
[7] – [10]. Namely it is true: for every sufficiently big $T>0$ and for every
fixed $k\in\mathbb{N}$ there is a sequence

(1.2) $\\{\overset{r}{T}(T)\\}_{r=1}^{k}:\
T<\overset{1}{T}(T)<\overset{2}{T}(T)<\dots<\overset{k}{T}(T)$

that the following formula

(1.3)
$\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim(1-c)\overset{r-1}{T},\ T\to\infty$

holds true, where $c$ is the Euler’s constant.

Since (see (1.1))

(1.4)
$\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=J(\overset{r}{T})-J(\overset{r-1}{T})=\Delta
J(\overset{r-1}{T},\overset{r}{T}),\ r=1,\dots,k$

then the set of integrals in (1.3) represents the set of correspondings
increments of the Hardy-Littlewood integral (1.1).

######  Remark 1.

Consequently, we have constructed vector function of increments of the Hardy -
Littlewood integral

(1.5)
$\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\ r=1,\dots,k,\ T\in(T_{0},\infty),$

with positive and sufficiently big $T_{0}$ that fulfils the following
property:

(1.6)
$\lim_{T\to\infty}\frac{1}{\overset{r-1}{T}}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=1-c,\ r=1,\dots,k.$

### 1.2.

Now, we shall present some strange formulas in the theory of the Riemann zeta-
functions following from (1.3).

  * (A)

Multiplicative formulas of the type:

(1.7)
$\begin{split}&\int_{T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T}}^{[T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim\\\
&\frac{1}{(1-c)^{k-1}}\prod_{r=1}^{k}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\ T\to\infty,\end{split}$

where we use the following notation

(1.8)
$[T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T}]^{1}=\overset{1}{\wideparen{T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T}}}.$

  * (B)

Additive formula of the type:

(1.9)
$\int_{T+\overset{1}{T}+\cdots+\overset{k-1}{T}}^{[T+\overset{1}{T}+\cdots+\overset{k-1}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim\sum_{r=1}^{k}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\ T\to\infty.$

  * (C)

And mixed formulas, for example:

(1.10)
$\begin{split}&\int_{T+\overset{1}{T}\overset{2}{T}}^{[T+\overset{1}{T}\overset{2}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim\\\
&\int_{T}^{\overset{1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t+\\\
&\frac{1}{1-c}\int_{\overset{1}{T}}^{[\overset{1}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\times\int_{\overset{2}{T}}^{[\overset{2}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\ T\to\infty.\end{split}$

######  Remark 2.

The formula (1.7) can be rewritten in the form

(1.11)
$\begin{split}&\int\int_{\square_{k}}\dots\int\prod_{r=1}^{k}\left|\zeta\left(\frac{1}{2}+i(x_{r-1}+\overset{r-1}{T})\right)\right|^{2}{\rm
d}x_{0}{\rm d}x_{1}\dots\rm dx_{k-1}\sim\\\
&\int_{T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T}}^{[T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\ T\to\infty,\end{split}$

where

(1.12) $\square_{k}:\ 0\leq x_{r-1}\leq\overset{r}{T}-\overset{r-1}{T},\
r=1,\dots,k,\ \square_{k}\subset\mathbb{R}^{k}.$

######  Remark 3.

Our formula (1.7) establishes a nonlinear asymptotic dependence of each of the
integrals in the set

(1.13)
$\left\\{\int_{T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T}}^{[T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\
\int_{T}^{\overset{1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\dots,\int_{\overset{k-1}{T}}^{\overset{k}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\right\\}$

on remaining elements. On the other hand, our formula (1.9) establishes a
linear dependence between the integrals in the set

(1.14)
$\left\\{\int_{T+\overset{1}{T}+\cdots+\overset{k-1}{T}}^{[T+\overset{1}{T}+\cdots+\overset{k-1}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\
\int_{T}^{\overset{1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\dots,\int_{\overset{k-1}{T}}^{\overset{k}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\right\\}.$

######  Remark 4.

Main distances of the segments of integration in (1.7) are given by:

(1.15)
$\begin{split}&\rho_{k}\\{[T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T},[T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T}]^{1}];[\overset{k-1}{T},\overset{k}{T}]\\}=\\\
&T\times\overset{1}{T}\times\cdots\times\overset{k-1}{T}-\overset{k}{T}>T^{k}-(1+\epsilon)T>0.9\times
T^{k}.\end{split}$

Comp. (3.3) with $0<\epsilon$ being sufficiently small. Therefore they are big
with big $k$. For example

(1.16) $\rho_{257}>0.9\times T^{257}.$

### 1.3.

Next, we have obtained also a nonlinear formula that generates corresponding
Hardy - Littlewood integral by means of the increments, (1.3) and (1.4).
Namely the following equality holds true:

(1.17)
$\begin{split}&\int_{0}^{\overset{r-1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim\\\
&\frac{1}{1-c}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\times\ln\left\\{\frac{e^{\Lambda_{1}}}{1-c}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\right\\},\\\ &r=1,\dots,k,\ T\to\infty,\end{split}$

where

(1.18) $\Lambda_{1}=2c-1-\ln 2\pi.$

######  Remark 5.

Our formula (1.17) expresses the following property: _smaller_ integral
generates _bigger_ one.

######  Remark 6.

As a consequence, our result (1.3) and its implications labelled (1.7) –
(1.11) and (1.17)222And also some others, we shall present later. express new
kind of phenomena connected with the Hardy - Littlewood integral (1.1) on
distances

$(1-c)\pi(T),$

where, as usually, the symbol $\pi(T)$ stands for the prime-counting function.

##  2\. Brief history of the Hardy - Littlewood integral (1918)

### 2.1.

The origin of the Hardy - Littlewood integral lies in the asymptotic formula

(2.1) $\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim T\ln T,\ T\to\infty,$

see [3], pp. 122, 151 – 156. Littlewood formula [5] ($0<\delta$ being
sufficiently small):

(2.2) $\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=T\ln T-(1+\ln 2\pi)T+\mathcal{O}(T^{\frac{3}{4}+\delta})$

was the next step. And finally, Ingham proved the following formula [6]:

(2.3) $\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=T\ln T-(1+\ln 2\pi-2c)T+\mathcal{O}(T^{\frac{1}{2}}\ln T),$

that contains complete set of main terms.

######  Remark 7.

We see that the first Ingham’s contribution to the formula (2.3) is in
addition of the term $-2cT$ that was missing in the previous Littlewood
formula (2.2). The second contribution of Ingham is quite essential
improvement of the error term in Littlewood’s formula, namely by $33.3\%$.

######  Remark 8.

Let us remind also some of other considerable refinements

(2.4) $\mathcal{O}(T^{\frac{5}{12}+\delta}),\ 16.6\%;\
\mathcal{O}(T^{\frac{1}{3}+\delta}),\ 20\%$

of the error term in (2.3) due to Titschmarsh [11] and Balasubramanian [1],
correspondingly.

We can include previous results into the following statement.

######  Formula1.

(2.5) $\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=T\ln T-(1+\ln 2\pi-2c)T+R(T).$

We will refer to (2.5) as to Hardy-Littlewood-Ingham formula with the error
term $R(T)$.

### 2.2.

Further, let us remind the fundamental Good’s $\Omega$-theorem, [2]:

(2.6) $R(T)=\Omega(T^{\frac{1}{4}}),\ T\to\infty.$

######  Remark 9.

It is possible to write down

(2.7) $R(T)=\mathcal{O}(T^{a+\delta}),\ \frac{1}{4}\leq a\leq\frac{1}{3},$

for the error term. In this form $a$ denotes the least valid exponent for
which the estimate (2.7) holds true and $a=\frac{1}{4}$ represents the Good’s
$\Omega$-theorem and $a=\frac{1}{3}$ represents the main Balasubramanian’s
exponent.

######  Remark 10.

It is fully sufficient to use the exponent $\frac{1}{3}$ despite there is also
the second Balasubramanian’s exponent that is equal to $\frac{27}{82}$, that
is an improvement by $1.18\%$, see [1].

##  3\. Main result as the next step in the evolution of the Hardy -
Littlewood formula (1918)

### 3.1.

We use the following notions from our papers [7] – [10]:

  * (a)

The Jacob’s ladder $\varphi_{1}(T)$,

  * (b)

direct iterations of Jacob’s ladder:

$\begin{split}&\varphi_{1}^{0}(T)=T,\ \varphi_{1}^{1}(T)=\varphi_{1}(T),\
\varphi_{1}^{2}(T)=\varphi_{1}(\varphi_{1}(T)),\ \dots,\\\
&\varphi_{1}^{r}(T)=\varphi_{1}(\varphi_{1}^{r-1}(T)),\
r=1,\dots,k,\end{split}$

  * (c)

reverse iterations of Jacob’s ladder (by means of the inverse function
$\varphi_{1}^{-1}(T)$):

(3.1) $\begin{split}&\varphi_{1}^{-1}(T)=\overset{1}{T},\
\varphi_{1}^{-2}(T)=\varphi_{1}^{-1}(\overset{1}{T})=\overset{2}{T},\dots,\\\
&\varphi_{1}^{-r}(T)=\varphi_{1}^{-1}(\overset{r-1}{T})=\overset{r}{T},\
r=1,\dots,k,\end{split}$

that is, for example,

(3.2) $\varphi_{1}(\overset{r}{T})=\overset{r-1}{T}$

for every fixed $k\in\mathbb{N}$ and every sufficiently big $T>0$.

Next, we use also the following properties:333See [9]

(3.3)
$\begin{split}&\overset{r}{T}-\overset{r-1}{T}\sim(1-c)\pi(\overset{r}{T}),\
\pi(\overset{r}{T})\sim\frac{\overset{r}{T}}{\ln\overset{r}{T}},\
r=1,\dots,k,\\\
&\overset{0}{T}=T<\overset{1}{T}(T)<\overset{2}{T}(T)<\dots<\overset{k}{T}(T),\\\
&T\sim\overset{1}{T}\sim\overset{2}{T}\sim\dots\sim\overset{k}{T},\
T\to\infty.\end{split}$

######  Remark 11.

The asymptotic behaviour of the points

$\\{T,\overset{1}{T},\dots,\overset{k}{T}\\}$

is as follows: As $T\to\infty$ these points recede unboundedly each from other
and all together are receding to infinity. Hence, the set of these points
behaves at $T\to\infty$ as one-dimensional Friedmann-Hubble expanding
Universe.

### 3.2.

Here we formulate our main result.

######  Theorem 1.

For every fixed $k\in\mathbb{N}$ and every sufficiently big $T>0$ we have:

(3.4)
$\begin{split}&\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=(1-c)\overset{r-1}{T}+\mathcal{O}(T^{a+\delta}),\\\ &1-c\approx 0.42,\
r=1,\dots,k,\ T\to\infty\end{split}$

where, by (3.1):

(3.5) $\overset{r}{T}=\overset{r}{T}(T)=\varphi_{1}^{-r}(T).$

######  Remark 12.

The existence of linear main terms in (3.4), linear with respect to variables
(3.2), is a new phenomenon in the theory of the Riemann’s
$\zeta\left(\frac{1}{2}+it\right)$-function. For example, it is true444Remark
9, $a=\frac{1}{3}$. that

(3.6)
$\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=(1-c)\overset{r-1}{T}+\mathcal{O}(T^{\frac{1}{3}+\delta}),\ T\to\infty.$

######  Remark 13.

Let us notice explicitly, that our formula (3.4) inherits every true estimate
for its error term from HLI formula (2.5).555Comp. (2.7) and Remark 8.

##  4\. The first class of consequences of (3.4)

### 4.1.

At the first place the following statement is obtained by (3.4).

######  Corollary 1.

(4.1)
$\begin{split}&\frac{1}{\overset{r-1}{T}}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=1-c+\mathcal{O}(T^{-1+a+\delta}),\\\ &r=1,\dots,k,\
T\to\infty,\end{split}$

where (see Remark 9)

(4.2) $\mathcal{O}(T^{-1+a+\delta})=\mathcal{O}(T^{-\frac{2}{3}+\delta}).$

######  Remark 14.

The formula (4.1) expresses certain asymptotic conservation law controlling
chaotic oscillations of the Riemann’s
$\zeta\left(\frac{1}{2}+it\right)$-function.

Let us remind in connection with chaotic oscillations of the Riemann’s zeta-
function on the critical line the following:

  * (A)

Spectral form of the Riemann-Siegel formula666Comp. [10], subsection 1.4.

(4.3)
$\begin{split}&Z(t)=\sum_{n\leq\tau(x)}\frac{2}{\sqrt{n}}\cos\\{t\omega_{n}(x)+\psi(x)\\}+Q(x),\\\
&\tau(x)=\sqrt{\frac{x}{2\pi}},\ Q(x)=\mathcal{O}(x^{-\frac{1}{4}}),\
t\in[x,x+V],\\\ &V=(0,x^{\frac{1}{4}}],x\in[T,\overset{k}{T}],\\\
&|Z(t)|=\left|\zeta\left(\frac{1}{2}+it\right)\right|,\end{split}$

where the functions

$\frac{2}{\sqrt{n}}\cos\\{t\omega_{n}(x)+\psi(x)\\}$

are Riemann’s oscillators with:

    * (a)

Amplitude

$\frac{2}{\sqrt{n}},$

    * (b)

Incoherent local phase constant

$\psi(x)=-\frac{x}{2}-\frac{\pi}{8},$

    * (c)

Nonsynchronized local times

$t\in[x,x+V],$

    * (d)

Local spectrum of cyclic frequencies

$\\{\omega_{n}(x)\\}_{n\leq\tau(x)},\ \omega_{n}(x)=\ln\frac{\tau(x)}{n}.$

  * (B)

The graph of the function $|\zeta\left(\frac{1}{2}+it\right)|$ after
transformation

$Z(t)\to|Z(t)|=\left|\zeta\left(\frac{1}{2}+it\right)\right|,$

in the neighbourhood of the first Lehmer’s pair of zeroes of $Z(t)$, see [4],
pp. 296, 297, for example.

### 4.2.

Next, let us remind the Euler’s $c$-constant definition

$c=\lim_{n\to\infty}\left\\{1+\frac{1}{2}+\dots+\frac{1}{n}-\ln
n\right\\}\approx 0.5772157$

and, for example, Dirichlet’s formula for the same constant

(4.4) $c=\int_{0}^{\infty}\left\\{\frac{1}{1+t}-e^{-t}\right\\}{\rm d}t.$

Now we obtain the following result in this direction.

######  Corollary 2.

(4.5)
$c=1-\lim_{T\to\infty}\frac{1}{\overset{r-1}{T}}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,$

or

(4.6)
$\bar{c}=\lim_{T\to\infty}\frac{1}{\overset{r-1}{T}}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t$

for _the complementary_ Euler’s constant $\bar{c}=1-c$.

######  Remark 15.

Our formula (4.5) gives new expression for the Euler’s $c$ and this expression
is generated by the Riemann’s $\zeta\left(\frac{1}{2}+it\right)$-function
together with the reverse iterations of the Jacob’s ladder, see (3.1).

### 4.3.

Since

(4.7) $\frac{1}{\overset{r}{T}-\overset{r-1}{T}}=\mathcal{O}\left(\frac{\ln
T}{T}\right),$

see (3.3), then we have the following.

######  Corollary 3.

(4.8)
$\begin{split}&\frac{1}{\overset{r}{T}-\overset{r-1}{T}}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\\\
&(1-c)\frac{\overset{r-1}{T}}{\overset{r}{T}-\overset{r-1}{T}}+\mathcal{O}(T^{-1+a+\delta}\ln
T),\ r=1,\dots,k,\ T\to\infty.\end{split}$

######  Remark 16.

Another new phenomenon is stated here. Namely existence of rational
expressions for the main members of the mean values in variables (3.2).

### 4.4.

Since, see (3.3)

(4.9) $\overset{r}{T}-\overset{r-1}{T}\geq(1-o(1))(1-c)\frac{T}{\ln T},\
T\to\infty,$

and (see (3.4) and (3.6))

(4.10) $T^{a+\delta}<T^{\frac{1}{3}+\delta},$

then we have the following asymptotic formula.

######  Corollary 4.

(4.11)
$\begin{split}&\int_{\overset{r}{T}}^{\overset{r+1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t-\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\\\ &(1-c)(\overset{r}{T}-\overset{r-1}{T})+\mathcal{O}(T^{a+\delta}),\
r=1,\dots,k,\ T\to\infty.\end{split}$

##  5\. The second class of consequences of (3.4)

### 5.1.

Since (see (3.3) and (3.6))

(5.1)
$\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=(1-c)\overset{r-1}{T}\left\\{1+\mathcal{O}(T^{-\frac{2}{3}+\delta})\right\\},$

then

(5.2)
$\prod_{r=1}^{k}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\left\\{1+\mathcal{O}(T^{-\frac{2}{3}+\delta}\right\\}\times\prod_{r=1}^{k}(1-c)\overset{r-1}{T}.$

Next, we have by (5.1) with $r=1$

(5.3)
$\int_{T}^{\overset{1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=(1-c)T\left\\{1+\mathcal{O}(T^{-\frac{2}{3}+\delta})\right\\}$

and we put herein the following777See (3.3).

(5.4) $\begin{split}&T\to
T\times\overset{1}{T}\times\dots\times\overset{k-1}{T}\sim(T)^{k},\\\
&\overset{1}{T}\to[T\times\overset{1}{T}\times\dots\times\overset{k-1}{T}]^{1},\end{split}$

with the result

(5.5)
$\begin{split}&\int_{T\times\overset{1}{T}\times\dots\times\overset{k-1}{T}}^{[T\times\overset{1}{T}\times\dots\times\overset{k-1}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\\\
&\left\\{1+\mathcal{O}(T^{-\frac{2}{3}+\delta})\right\\}\frac{1}{(1-c)^{k-1}}\prod_{r=1}^{k}(1-c)\overset{r-1}{T}.\end{split}$

And consequently, see (5.2) and (5.5), we obtain the following.

######  Corollary 5.

(5.6)
$\int_{T\times\overset{1}{T}\times\dots\times\overset{k-1}{T}}^{[T\times\overset{1}{T}\times\dots\times\overset{k-1}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim\frac{1}{(1-c)^{k-1}}\prod_{r=1}^{k}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\ T\to\infty.$

######  Remark 17.

We can obtain also the formula:

(5.7)
$\int_{(T)^{k}}^{[(T)^{k}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim\frac{1}{(1-c)^{k-1}}\left\\{\int_{T}^{\overset{1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\right\\}^{k},\ T\to\infty,$

where, of course, $\overset{1}{T}=[T]^{1}$. Since

$T<\overset{1}{T}<\dots<\overset{k-1}{T}$

the formula (5.7) does not follow directly from (5.6).

######  Remark 18.

Since888See (3.4) with $r=1$.

(5.8)
$\int_{T+\dots+\overset{k-1}{T}}^{[T+\dots+\overset{k-1}{T}]^{1}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\sum_{r=1}^{k}(1-c)\overset{r-1}{T}+\mathcal{O}(T^{a+\delta}),$

then (1.9) follows directly. Mixed formulas of type (1.10) can be obtained by
combining corresponding types (1.7) and (1.9).

##  6\. Nonlinear phenomena in which Hardy - Littlewood integral is generated
by the increment (1.4)

The increments of the Hardy - Littlewood integral are defined, of course, as
differences of integrals, see (1.4). Now, we can provide, in opposite
direction, a nonlinear formula that generates the Hardy - Littlewood integral
by means of the increments.

### 6.1.

######  Theorem 2.

The following formula

(6.1)
$\begin{split}&\int_{0}^{\overset{r-1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\\\
&\frac{1}{1-c}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\times\left\\{\frac{e^{\Lambda_{1}}}{1-c}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\right\\}+\\\ &\mathcal{O}(T^{a+\delta}\ln T),\ r=1,\dots,k,\
T\to\infty\end{split}$

holds true, where

(6.2) $\Lambda_{1}=2c-1-\ln 2\pi.$

######  Remark 19.

Nonlinear transformation of the increments (for segments
$[\overset{r-1}{T},\overset{r}{T}]$) onto the corresponding Hardy-Littlewood
integral (on adjacent segments $[0,\overset{r}{T}]$)

(6.3)
$\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\xrightarrow{\mathcal{T}_{1}}\int_{0}^{\overset{r-1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\ r=1,\dots,k$

is defined by the formula (6.1).

### 6.2.

######  Theorem 3.

The following formula

(6.4)
$\begin{split}&\int_{0}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\\\
&\frac{1}{1-c}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\times\left\\{\frac{e^{\Lambda_{2}}}{1-c}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\right\\}+\\\ &\mathcal{O}(T^{a+\delta}\ln T),\ r=1,\dots,k,\
T\to\infty\end{split}$

holds true, where

(6.5) $\Lambda_{2}=c-\ln 2\pi.$

######  Remark 20.

Nonlinear transformation of the increments (for segments
$[\overset{r-1}{T},\overset{r}{T}]$) onto the corresponding Hardy-Littlewood
integral (on adjacent segments $[0,\overset{r}{T}]$)

(6.6)
$\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\xrightarrow{\mathcal{T}_{2}}\int_{0}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t,\ r=1,\dots,k$

is defined by the formula (6.4). Transformation $\mathcal{T}_{2}$ is then a
continuation of the corresponding increment onto the whole segment
$[0,\overset{r}{T}]$.

##  7\. Next steps in the evolution of the Hardy - Littlewood formula and
proofs of Theorems

### 7.1.

Let us remind that we have introduced the Jacob’s ladder

$\varphi_{1}(T)=\frac{1}{2}\varphi(T)$

in our paper [7] (comp. also [8]), where the function $\varphi(T)$ is an
arbitrary solution of the noblinear integral equation (also introduced in [7])

$\int_{0}^{\mu[x(T)]}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}e^{-\frac{2}{x(T)}t}{\rm
d}t=\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm d}t,$

where each admissible function $\mu(y)$ generates a solution

$y=\varphi(T;\mu)=\varphi(T);\ \mu(y)\geq 7y\ln y.$

We call the function $\varphi_{1}(T)$ as Jacob’s ladder since analogy with the
Jacob’s dream in Chumash, Bereishis, 28:12.

### 7.2.

Next, 83 years after HLI formula (2.5), we have prove the following new
result, see [7].

######  Formula2.

The Hardy-Littlewood integral has, in addition to previously known HLI
expressions possessing an unbounded error terms at $T\to\infty$, the following
infinite number of almost exact representations

(7.1)
$\begin{split}&\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\varphi_{1}(T)\ln\\{\varphi_{1}(T)\\}+\\\ &(c-\ln
2\pi)\varphi_{1}(T)+c_{0}+\mathcal{O}\left(\frac{\ln T}{T}\right),\end{split}$

where $c_{0}$ is a constant from the Titschmarsh-Kober-Atkinson formula.

### 7.3.

Further, we have obtained, see [7], (6.2), the following.

######  Formula3.

(7.2) $T-\varphi_{1}(T)\sim(1-c)\pi(T);\ \pi(T)\sim\frac{T}{\ln T},\
T\to\infty,$

where the Jacob’s ladder can be viewed as the complementary function to the
function $(1-c)\pi(T)$ in the sense

(7.3) $\varphi_{1}(T)+(1-c)\pi(T)\sim T,\ T\to\infty.$

######  Remark 21.

We have used the following change

(7.4) $\frac{T}{\ln T}\to\pi(T),\ T\to\infty$

in the first asymptotic formula (7.2), see also [7], (6.2). This is however
completely correct in the asymptotic region by the prime-number law. At the
same time, we can give also a motivation for the change (7.4). Namely,
following999Comp. [12], p. 2.

$\zeta(s)=\exp\left\\{s\int_{2}^{\infty}\frac{\pi(x)}{x(x^{s}-1){\rm
d}x}\right\\},\ \operatorname{Re}(s)>1$

we can get that the function $\pi(x)$ is one element of the set of five
functions generating the function $\zeta(s)$.

######  Remark 22.

Our formula (7.1) represents the further step in the evolution of the Hardy -
Littlewood integral (1918) for the hypertranscendental Riemann’s function
$|\zeta\left(\frac{1}{2}+it\right)|^{2}$.

### 7.4.

Now, we can make a final step toward the proof of Theorem 1.

###### Proof.

By making use of the substitution

$T\to\overset{r-1}{T}$

in (2.5) and

$T\to\overset{r}{T}$

in (7.1)101010See also (3.2). we obtain the following couple of formulas.

(7.5)
$\int_{0}^{\overset{r-1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\overset{r-1}{T}\ln\overset{r-1}{T}-(1+\ln
2\pi-2c)\overset{r-1}{T}+\mathcal{O}(T^{a+\delta}),$ (7.6)
$\int_{0}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\overset{r-1}{T}\ln\overset{r-1}{T}-(c-\ln
2\pi)\overset{r-1}{T}+c_{0}+\mathcal{O}\left(\frac{\ln T}{T}\right),$

$r=1,\dots,k,\ T\to\infty$, correspondingly. If we subtract (7.5) from (7.6)
we obtain instantly (3.4). ∎

######  Remark 23.

Our formula (3.4) represents a kind of interaction between (7.5) and (7.6)
generated by the Jacob’s ladder by means of the property111111See (3.2).

$\overset{r}{T}=\varphi_{1}^{-r}(T),\ T\to\infty.$

######  Remark 24.

In fact, we have used the interaction between the HLI formula and our formula
(7.1) for the first time in our paper [7]. Namely, the formula (7.2)
represents the result of this interaction, see [7], (6.2). This last formula
generates all the necessary properties of the set of iterations of the Jacob’s
ladder, see [9].

### 7.5.

Now we give the proof of Theorem 2.

###### Proof.

We get from (7.5) by (3.3) and (6.2) that

(7.7)
$\begin{split}&\int_{0}^{\overset{r-1}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\overset{r-1}{T}\ln\left\\{e^{\Lambda_{1}}\overset{r-1}{T}\right\\}+\mathcal{O}(T^{a+\delta})=\\\
&\overset{r-1}{T}\ln\left\\{e^{\Lambda_{1}}\overset{r-1}{T}\right\\}\left\\{1+\mathcal{O}\left(\frac{T^{-1+a+\delta}}{\ln
T}\right)\right\\}.\end{split}$

Next, see (3.4),

(7.8)
$\begin{split}&\overset{r-1}{T}=\frac{1}{1-c}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t+\mathcal{O}(T^{a+\delta})=\\\
&\frac{1}{1-c}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\times\left\\{1+\mathcal{O}(T^{-1+a+\delta})\right\\},\end{split}$

and by (7.8)

(7.9)
$\begin{split}&\ln(e^{\Lambda_{1}}\overset{r-1}{T})=\ln\left\\{\frac{e^{\Lambda_{1}}}{1-c}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\right\\}+\mathcal{O}(T^{-1+a+\delta})=\\\
&\ln\left\\{\frac{e^{\Lambda_{1}}}{1-c}\int_{\overset{r-1}{T}}^{\overset{r}{T}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\right\\}\times\left\\{1+\mathcal{O}\left(\frac{T^{-1+a+\delta}}{\ln
T}\right)\right\\}.\end{split}$

Now, (6.1) follows from (7.7) by (7.8) and (7.9). ∎

######  Remark 25.

We can obtain (6.4) by a similar way.

I would like to thank Michal Demetrian for his moral support of my study of
Jacob’s ladders.

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