# Lagrangian fields, Calabi functions, and local symplectic groupoids

Alexander Karabegov  Department of Mathematics, Abilene Christian University,
ACU Box 28012, Abilene, TX 79699-8012<EMAIL_ADDRESS>
###### Abstract.

A Lagrangian field on a symplectic manifold $M$ is a family
$\Lambda=\\{\Lambda_{x}|x\in M\\}$ of pointed Lagrangian submanifolds of $M$.
This notion is a generalization of a real Lagrangian polarization for which
each $\Lambda_{x}$ is the leaf containing $x$. Two Lagrangian fields $\Lambda$
and $\tilde{\Lambda}$ are called transversal if $\Lambda_{x}$ intersects
$\tilde{\Lambda}_{x}$ transversally at $x$ for every $x$. Two transversal
Lagrangian fields determine an almost para-Kähler structure on $M$. We
construct a local symplectic groupoid on a neighborhood of the zero section of
$T^{\ast}M$ from two transversal Lagrangian fields on $M$. The Lagrangian
manifold of $n$-cycles of this groupoid in $(T^{\ast}M)^{n}$ has a generating
function whose germ around the diagonal of $M^{n}$ is given by the $n$-point
cyclic Calabi function of a closed (1,1)-form on a neighborhood of the
diagonal of $M^{2}$ obtained from the symplectic form on $M$.

###### Key words and phrases:

local symplectic groupoids, Calabi functions, Lagrangian fields

###### 2010 Mathematics Subject Classification:

22A22, 53D17, 53D12

##  1\. Introduction

A symplectic groupoid over a Poisson manifold $M$ is a geometric object which
is a heuristic semiclassical counterpart of an algebra of quantum observables
on $M$. Symplectic groupoids were first introduced by Karasev [9], Weinstein
[11], and Zakrzewski [12]. Symplectic groupoids can be treated as a starting
point for quantization (see [3] and [6]).

In this paper we construct local symplectic groupoids over almost para-Kähler
manifolds. These groupoids are based on an additional geometric structure, a
Lagrangian field, which is a generalization of a Lagrangian polarization. A
Lagrangian field on a symplectic manifold $M$ is a family
$\Lambda=\\{\Lambda_{x}|x\in M\\}$ of pointed Lagrangian submanifolds of $M$
(with $x\in\Lambda_{x}$). It induces a Lagrangian distribution
$\\{T_{x}\Lambda_{x},x\in M\\}$ on $M$ which is not necessarily integrable.

Two Lagrangian fields $\Lambda$ and $\tilde{\Lambda}$ are called transversal
if $\Lambda_{x}$ intersects $\tilde{\Lambda}_{x}$ transversally at $x$ for
every $x\in M$. The corresponding transversal Lagrangian distributions
determine an almost para-Kähler structure on $M$.

We construct a local symplectic groupoid on a neighborhood of the zero section
of $T^{\ast}M$ from two transversal Lagrangian fields on $M$. Such groupoids
have a rich geometric structure. We expect that they can be used for
quantization of almost para-Kähler manifolds.

##  2\. General facts on groupoids

In this section we give basic facts on groupoids and fix various conventions
that will be used in the rest of the paper.

A groupoid $\Sigma\rightrightarrows M$ is a small category with the set of
objects $M$ and the set of morphisms $\Sigma$, where all morphisms are
invertible. The injective unit mapping $\epsilon:M\to\Sigma$ maps an object
$x$ to the identity morphism $\mathrm{id}_{x}:x\to x$. The set $\epsilon(M)$
of the identity morphisms in $\Sigma$ is called the set of units of the
groupoid. The source and target mappings $s,t:\Sigma\to M$ map a morphism
$\alpha:x\to y$ to $s(\alpha)=x$ and $t(\alpha)=y$. The involutive inverse
mapping $i:\Sigma\to\Sigma$ maps $\alpha\in\Sigma$ to $i(\alpha)=\alpha^{-1}$.
We will write the composition of $\alpha:y\to z$ and $\beta:x\to y$ as
$\alpha\beta:x\to z$. Two morphisms $\alpha$ and $\beta$ are thus composable
if $s(\alpha)=t(\beta)$. An element
$(\alpha_{1},\ldots,\alpha_{n})\in\Sigma^{n}$ is an $n$-cycle in $\Sigma$ if
the composition of the elements $\alpha_{1},\ldots,\alpha_{n}\in\Sigma$ is
defined and equals an identity morphism,

$\alpha_{1}\alpha_{2}\ldots\alpha_{n}=\mathrm{id}_{x},$

where $x=s(\alpha_{n})=t(\alpha_{1})$. We denote the set of $n$-cycles in
$\Sigma$ by $\Sigma_{n}$. So, the graph of $i$ is $\Sigma_{2}$.

The set of objects of every groupoid in this paper is denoted by $M$. We
denote the diagonal of the set $M^{n}$ by $M_{n}$, which should not lead to
confusion 111This notation agrees with that for the set of $n$-cycles if $M$
is treated as the identity groupoid where every morphism is an identity..

Example The pair groupoid of a set $M$ has the set of objects $M$ and the set
of morphisms $M^{2}$ with $\epsilon(x)=(x,x),i(x,y)=(y,x),s(x,y)=y$, and
$t(x,y)=x$, so that the unique morphism $\alpha:y\to x$ is $\alpha=(x,y)$. The
set of units is the diagonal ${M_{2}}\subset M^{2}$. The composition of
$(x,y)$ and $(y,z)$ is $(x,y)(y,z)=(x,z)$. An $n$-cycle is

$\\{\alpha_{1},\alpha_{2},\ldots,\alpha_{n}\\}=\\{(x_{1},x_{2}),(x_{2},x_{3}),\ldots,(x_{n},x_{1})\\},$

where $\alpha_{i}:x_{i+1}\to x_{i}$ for $1\leq i\leq n-1$ and
$\alpha_{n}:x_{1}\to x_{n}$.

A groupoid $\Sigma\rightrightarrows M$ is a Lie groupoid if $M$ and $\Sigma$
are manifolds, all structure mappings are smooth, the unit mapping is an
embedding, and the source and target mappings are surjective submersions. A
local Lie groupoid has the same properties as a Lie groupoid, but in order for
two elements to be composable they should also be close to each other (see [5]
and [1]). A neighborhood of the unit space of a Lie groupoid is a local Lie
groupoid.

A symplectic groupoid $\Sigma\rightrightarrows M$ is a Lie groupoid where
$\Sigma$ is a symplectic manifold, the space of units $\epsilon(M)$ is
Lagrangian, $i$ is an anti-symplectomorphism, and the set $\Sigma_{3}$ of
3-cycles is a Lagrangian submanifold of $\Sigma^{3}$ 222In the conventional
definition of a symplectic groupoid an equivalent condition is used: the
product space
$\\{(\alpha,\beta,\gamma)|s(\alpha)=t(\beta),\gamma=\alpha\beta\\}$ is a
Lagrangian submanifold of $\Sigma\times\Sigma\times\bar{\Sigma}$, where
$\bar{\Sigma}$ is a copy of $\Sigma$ with the opposite symplectic structure..
It can be shown that then the space $\Sigma_{n}$ of $n$-cycles is a Lagrangian
submanifold of $\Sigma^{n}$ for all $n\geq 2$. There exists a unique Poisson
structure on $M$ such that $s$ is a Poisson morphism and $t$ is an anti-
Poisson morphism.

If $M=(M,\omega)$ is a symplectic manifold and $\bar{M}=(M,-\omega)$ is a copy
of $M$ with the opposite symplectic structure, then the pair groupoid
$\bar{M}\times M\rightrightarrows M$ with the product symplectic structure on
$\bar{M}\times M$ is a symplectic groupoid over $M$. Its diagonal unit space
${M_{2}}$ is Lagrangian.

If $M$ is compact, by Weinstein’s neighborhood theorem there exists a
neighborhood $V$ of the zero section $Z_{T^{\ast}M}$ in $T^{\ast}M$ and a
mapping $f:V\to\bar{M}\times M$ which identifies $Z_{T^{\ast}M}$ with the
diagonal ${M_{2}}$ of $\bar{M}\times M$ as copies of $M$ (i.e., if $z_{x}$ is
the unique element in $Z_{T^{\ast}M}\cap T^{\ast}_{x}M$, then $f(z_{x})=(x,x)$
for all $x\in M$) and is a symplectomorphism of $V$ onto $f(V)$. Then
$V\rightrightarrows M$ is a local symplectic groupoid over $M$ with
$s=\pi_{2}\circ f$ and $t=\pi_{1}\circ f$, where $\pi_{1}$ and $\pi_{2}$ are
the projections of $\bar{M}\times M$ onto the respective factors.

Given a symplectic manifold $M$ equipped with two transversal Lagrangian
fields $\Lambda$ and $\tilde{\Lambda}$, we construct in this paper a local
symplectic groupoid $V\rightrightarrows M$, where $V$ is a neighborhood of
$Z_{T^{\ast}M}$ in $T^{\ast}M$ (the compactness of $M$ is not required). For
this groupoid, $s(T^{\ast}_{x}M\cap V)\subset\Lambda_{x}$ and
$t(T^{\ast}_{x}M\cap V)\subset\tilde{\Lambda}_{x}$ for all $x\in M$. We also
show that the Lagrangian manifold of $n$-cycles of this groupoid has a
generating function whose germ around the diagonal $M_{n}$ of $M^{n}$ is given
by the $n$-point cyclic Calabi function of a closed (1,1)-form on a
neighborhood of the diagonal of $M^{2}$ obtained from the symplectic form on
$M$.

Acknowledgments I am very grateful to Th.Th.Voronov for an important
discussion on Lagrangian fields and to Alan Weinstein for valuable
suggestions.

##  3\. A local symplectic groupoid generated by two transversal Lagrangian
fields

Below we give a technical definition of a Lagrangian field on a symplectic
manifold $(M,\omega)$ of dimension $2m$.

######  Definition 3.1.

A Lagrangian field $\Lambda=(E,\lambda)$ on $M$ is a vector bundle $\pi:E\to
M$ of rank $m$ equipped with a surjective submersion $\lambda:E\to M$ whose
restriction to each fiber $E_{x},x\in M$, is an embedding onto a Lagrangian
submanifold $\Lambda_{x}:=\lambda(E_{x})$ of $M$ and such that $\lambda=\pi$
on the zero section $Z_{E}$ of $E$.

If $z_{x}$ is the unique element of $Z_{E}\cap E_{x}$, we have
$\lambda(z_{x})=\pi(z_{x})=x$, whence $x\in\Lambda_{x}$ for all $x\in M$.

######  Lemma 3.1.

There exists a unique horizontal 1-form $\phi$ on the total space of $E$ which
vanishes on $Z_{E}$ and satisfies the equation

(1) $\lambda^{\ast}\omega-\pi^{\ast}\omega=d\phi.$

###### Proof.

Let $F:[0,1]\times E\to M$ be the homotopy $F(t,z)=\lambda(t\cdot z)$ between
the mappings $\pi$ and $\lambda$, where $z\mapsto t\cdot z$ is the scalar
multiplication by $t$ on the fibers of $E$. Denote by $i_{t}:E\to[0,1]\times
E$ the inclusion mapping $i_{t}(z)=(t,z)$. We will show that the form

$\phi:=\int_{0}^{1}i^{\ast}_{t}\iota_{\frac{\partial}{\partial
t}}F^{\ast}\omega\,dt,$

where $\iota_{\frac{\partial}{\partial t}}$ is an insertion operator,
satisfies the conditions of the lemma. Equality (1) holds by the homotopy
formula. For any $x\in M$, the mapping $F$ maps $[0,1]\times E_{x}$ onto the
Lagrangian manifold $\Lambda_{x}$, whence

$F^{\ast}\omega|_{[0,1]\times E_{x}}=0.$

If $z\in Z_{E}$, then $F$ maps $[0,1]\times\\{z\\}$ to the point $\pi(z)$. It
implies that the form $i^{\ast}_{t}\iota_{\frac{\partial}{\partial
t}}F^{\ast}\omega$ on $E$ is horizontal and its restriction to $Z_{E}$
vanishes. Hence, the same holds true for the 1-form $\phi$.

Let $\psi$ be a closed horizontal 1-form on $E$ vanishing on $Z_{E}$. If
$\eta$ is a vertical vector field on $E$, then $\iota_{\eta}\psi=0$.
Therefore, $\mathcal{L}_{\eta}\psi=0$, where $\mathcal{L}_{\eta}$ is the Lie
derivative with respect to $\eta$. It follows that $\psi$ is the pullback of a
1-form on $M$ by $\pi$. Since the restriction of $\psi$ to $Z_{E}$ vanishes,
we see that $\psi=0$, which implies the uniqueness of $\phi$. Thus, the 1-form
$\phi$ satisfies all conditions of the lemma. ∎

Let $\Lambda=(E,\lambda)$ and $\tilde{\Lambda}=(\tilde{E},\tilde{\lambda})$ be
two Lagrangian fields on $M$. We denote the bundle projection for $\tilde{E}$
by $\tilde{\pi}$, the zero section by $Z_{\tilde{E}}$, the unique element of
$Z_{\tilde{E}}\cap\tilde{E}_{x}$ by $\tilde{z}_{x}$, and set
$\tilde{\Lambda}_{x}:=\tilde{\lambda}(\tilde{E}_{x})$ for each $x\in M$.
According to Lemma 3.1, there exists a unique horizontal 1-form $\tilde{\phi}$
on $\tilde{E}$ vanishing on $Z_{\tilde{E}}$ and satisfying

(2) $\tilde{\lambda}^{\ast}\omega-\tilde{\pi}^{\ast}\omega=d\tilde{\phi}.$

Consider the vector bundle

$\hat{E}:=E\oplus\tilde{E}=\\{(z,\tilde{z})\in
E\times\tilde{E}|\pi(z)=\tilde{\pi}(\tilde{z})\\}.$

The bundle projection $\hat{\pi}:\hat{E}\to M$ is such that
$\hat{\pi}(z,\tilde{z})=\pi(z)=\tilde{\pi}(\tilde{z})$. We introduce
projections $\tau:\hat{E}\to E$ and $\tilde{\tau}:\hat{E}\to\tilde{E}$ given
by $\tau(z,\tilde{z})=z$ and $\tilde{\tau}(z,\tilde{z})=\tilde{z}$. Then
$\hat{\pi}=\pi\tau=\tilde{\pi}\tilde{\tau}$. We get the following pullbacks of
(1) and (2) by $\tau$ and $\tilde{\tau}$, respectively:

$\tau^{\ast}\lambda^{\ast}\omega-\hat{\pi}^{\ast}\omega=d\tau^{\ast}\phi\mbox{
and
}\tilde{\tau}^{\ast}\tilde{\lambda}^{\ast}\omega-\hat{\pi}^{\ast}\omega=d\tilde{\tau}^{\ast}\tilde{\phi}.$

Subtracting these equations, we get that

(3)
$\tau^{\ast}\lambda^{\ast}\omega-\tilde{\tau}^{\ast}\tilde{\lambda}^{\ast}\omega=-d(\tilde{\tau}^{\ast}\tilde{\phi}-\tau^{\ast}\phi).$

The 1-form $\tilde{\tau}^{\ast}\tilde{\phi}-\tau^{\ast}\phi$ is horizontal on
$\hat{E}$ and vanishes on the zero section $Z_{\hat{E}}$. It determines a
fiberwise nonlinear mapping

$\chi:\hat{E}\to T^{\ast}M$

which maps a point $(z,\tilde{z})\in\hat{E}_{x}$ to the unique element
$\chi(z,\tilde{z})\in T^{\ast}_{x}M$ whose pullback by $\hat{\pi}$ is

$(d\hat{\pi}_{(z,\tilde{z})})^{\ast}\chi(z,\tilde{z})=(\tilde{\tau}^{\ast}\tilde{\phi}-\tau^{\ast}\phi)(z,\tilde{z})\in
T^{\ast}_{(z,\tilde{z})}\hat{E}.$

The mapping $\chi$ identifies the zero sections $Z_{\hat{E}}$ and
$Z_{T^{\ast}M}$ as copies of $M$.

Let $\theta$ be the canonical 1-form on $T^{\ast}M$. If $\\{x^{i}\\}$ are
local coordinates on $M$ and $\\{\xi_{i}\\}$ are the dual fiber coordinates on
$T^{\ast}M$, then $\theta=\xi_{i}dx^{i}$, where we use summation over repeated
lower and upper indices. We have

$\chi^{\ast}\theta=\tilde{\tau}^{\ast}\tilde{\phi}-\tau^{\ast}\phi.$

Thus, the left-hand side of (3) is equal to the pullback of the canonical
symplectic form $-d\theta$ on $T^{\ast}M$ by the mapping $\chi$.

######  Definition 3.2.

Lagrangian fields $\Lambda$ and $\tilde{\Lambda}$ on $M$ are called
transversal if $\Lambda_{x}$ transversally intersects $\tilde{\Lambda}_{x}$ at
$x$ for all $x\in M$.

######  Lemma 3.2.

Let $\Lambda$ and $\tilde{\Lambda}$ be two transversal Lagrangian fields on
$M$.

  1. (i)

There exists a neighborhood $W_{1}$ of $Z_{\hat{E}}$ in $\hat{E}$ such that
$\chi|_{W_{1}}$ is a diffeomorphism of $W_{1}$ onto a neighborhood of
$Z_{T^{\ast}M}$ in $T^{\ast}M$.

  2. (ii)

There exists a neighborhood $W_{2}$ of $Z_{\hat{E}}$ in $\hat{E}$ such that
the mapping

$\tilde{\lambda}\tilde{\tau}\times\lambda\tau|_{W_{2}}:(z,\tilde{z})\mapsto(\tilde{\lambda}(\tilde{z}),\lambda(z))$

is a diffeomorphism of $W_{2}$ onto a neighborhood of the diagonal $M_{2}$ in
$M^{2}$.

###### Proof.

Let ${x_{0}}$ be an arbitrary point in $M$. Since $\lambda$ and
$\tilde{\lambda}$ identify $Z_{E}$ and $Z_{\tilde{E}}$, respectively, with
$M$, there exists a coordinate chart $(\hat{U},\\{x^{i}\\})$ on $M$ around
${x_{0}}$, trivializations $U\times\mathbb{R}^{m}$ of $E$ and $\tilde{E}$ over
a neighborhood $U$ of ${x_{0}}$ in $\hat{U}$, and an open ball $B$ in
$\mathbb{R}^{m}$ centered at the origin such that $\lambda(U\times
B)\subset\hat{U}$ and $\tilde{\lambda}(U\times B)\subset\hat{U}$. We denote
the coordinates on $\mathbb{R}^{m}$ by $\\{v^{\alpha}\\}$ and identify
$U\times\mathbb{R}^{m}$ with coordinate charts on $E$ and $\tilde{E}$. Then
locally a point in $E$ or $\tilde{E}$ can be written as $(x,v)$. The
transversality of the Lagrangian fields $\Lambda$ and $\tilde{\Lambda}$ means
that the $2m\times 2m$-matrix

(4) $\begin{bmatrix}-\frac{\partial\lambda^{i}}{\partial
v^{\alpha}}(x,0)&\frac{\partial{\tilde{\lambda}}^{i}}{\partial
v^{\beta}}(x,0)\end{bmatrix}$

is nondegenerate for all $x\in U$. These trivializations of $E$ and
$\tilde{E}$ induce a trivialization
$U\times\mathbb{R}^{m}\times\mathbb{R}^{m}$ of $\hat{E}$ over $U$. A point
$(z,\tilde{z})\in\hat{E}$ with $z=(x,v)$ and $\tilde{z}=(x,w)$ will have the
coordinates $(x,v,w)$.

To prove (i), consider the neighborhood $U\times B\times B$ in $\hat{E}$ and
let $\\{\xi_{i}\\}$ be the fiber coordinates on $T^{\ast}U$ dual to
$\\{x^{i}\\}$. If $\phi=\phi_{i}(x,v)dx^{i}$ on $U\times\mathbb{R}^{m}\approx
E|_{U}$ and $\tilde{\phi}=\tilde{\phi}_{i}(x,v)dx^{i}$ on
$U\times\mathbb{R}^{m}\approx\tilde{E}|_{U}$, the mapping $\chi$ will be
written in coordinates as

$x^{i}=x^{i}\quad\xi_{j}={\tilde{\phi}}_{j}(x,w)-\phi_{j}(x,v).$

The Jacobian of this mapping is

$\begin{bmatrix}\delta^{i}_{k}&0&0\\\ \frac{\partial\tilde{\phi}_{j}}{\partial
x^{k}}(x,w)-\frac{\partial\phi_{j}}{\partial
x^{k}}(x,v)&-\frac{\partial\phi_{j}}{\partial
v^{\alpha}}(x,v)&\frac{\partial{\tilde{\phi}}_{j}}{\partial
v^{\beta}}(x,w)\end{bmatrix}.$

Since $\phi_{i}(x,0)=0$ and $\tilde{\phi}_{i}(x,0)=0$, at the point $(x,0,0)$
the Jacobian is equal to

(5) $\begin{bmatrix}\delta^{i}_{k}&0&0\\\ 0&-\frac{\partial\phi_{j}}{\partial
v^{\alpha}}(x,0)&\frac{\partial{\tilde{\phi}}_{j}}{\partial
v^{\beta}}(x,0)\end{bmatrix}.$

We will show that the matrix

(6) $\begin{bmatrix}-\frac{\partial\phi_{j}}{\partial
v^{\alpha}}(x,0)&\frac{\partial{\tilde{\phi}}_{j}}{\partial
v^{\beta}}(x,0)\end{bmatrix}$

is nondegenerate. In local coordinates we write

$\omega=\frac{1}{2}\omega_{ij}(x)dx^{i}\wedge dx^{j}$

and get from (1) that

$\frac{\partial\phi_{k}}{\partial
v^{\alpha}}=\omega_{ij}(\lambda)\frac{\partial\lambda^{i}}{\partial
v^{\alpha}}\frac{\partial\lambda^{j}}{\partial x^{k}}.$

It follows from the equality $\lambda(x,0)=x$ that

$\frac{\partial\phi_{k}}{\partial
v^{\alpha}}(x,0)=\frac{\partial\lambda^{i}}{\partial
v^{\alpha}}(x,0)\omega_{ik}(x).$

Similarly,

$\frac{\partial\tilde{\phi}_{k}}{\partial
v^{\alpha}}(x,0)=\frac{\partial\tilde{\lambda}^{i}}{\partial
v^{\alpha}}(x,0)\omega_{ik}(x).$

Therefore, we have the following matrix product representation of (6),

$\begin{bmatrix}-\frac{\partial\phi_{k}}{\partial
v^{\alpha}}(x,0)&\frac{\partial\tilde{\phi}_{k}}{\partial
v^{\beta}}(x,0)\end{bmatrix}=\begin{bmatrix}-\frac{\partial\lambda^{i}}{\partial
v^{\alpha}}(x,0)&\frac{\partial\tilde{\lambda}^{i}}{\partial
v^{\beta}}(x,0)\end{bmatrix}\cdot\begin{bmatrix}\omega_{ik}(x)\end{bmatrix}.$

Since the matrix (4) is nondegenerate, we see that (6) is also nondegenerate,
which implies that the Jacobian (5) is nondegenerate as well. Now part (i) of
the lemma follows from the fact that the mapping $\chi$ identifies
$Z_{\hat{E}}$ with $Z_{T^{\ast}M}$ as copies of $M$ and from the generalized
inverse function theorem.

To prove part (ii), consider a neighborhood $U\times B\times B$ as in part (i)
and the mapping

(7) $(x,v,w)\mapsto(\tilde{\lambda}(x,w),\lambda(x,v))$

from $U\times B\times B$ to $M\times M$. It maps $(x,0,0)$ to $(x,x)$. Its
Jacobian at $(x,v,w)$ is

$\begin{bmatrix}\frac{\partial\tilde{\lambda}^{i}}{\partial
x^{k}}(x,w)&0&\frac{\partial\tilde{\lambda}^{i}}{\partial v^{\beta}}(x,w)\\\
\frac{\partial\lambda^{j}}{\partial
x^{k}}(x,v)&\frac{\partial\lambda^{j}}{\partial
v^{\alpha}}(x,v)&0\end{bmatrix}.$

At the point $(x,0,0)$ the Jacobian is equal to

$\begin{bmatrix}\delta^{i}_{k}&0&\frac{\partial\tilde{\lambda}^{i}}{\partial
v^{\beta}}(x,0)\\\ \delta^{j}_{k}&\frac{\partial\lambda^{j}}{\partial
v^{\alpha}}(x,0)&0\end{bmatrix}.$

It is nondegenerate because (4) is nondegenerate. Part (ii) also follows from
the generalized inverse function theorem. ∎

Using the notations from Lemma 3.2, we set

$W:=W_{1}\cap W_{2}\subset\hat{E}\mbox{ and }V:=\chi(W)\subset T^{\ast}M.$

There exists a unique mapping $f:V\to M^{2}$ which is a diffeomorphism of $V$
onto $f(V)=(\tilde{\lambda}\tilde{\tau}\times\lambda\tau)(W)$ satisfying

(8) $f\circ\chi|_{W}=\tilde{\lambda}\tilde{\tau}\times\lambda\tau|_{W}.$

The neighborhood $V$ contains $Z_{T^{\ast}M}$ and $f(V)$ contains the diagonal
$M_{2}$ of $M^{2}$. The mapping $f$ identifies $Z_{T^{\ast}M}$ with $M_{2}$ as
copies of $M$. Recall that $\pi_{1}$ and $\pi_{2}$ are the projections of
$M^{2}$ onto the respective factors.

######  Theorem 3.1.

The mapping $f$ determines a local symplectic groupoid $V\rightrightarrows M$
with $s=\pi_{2}\circ f$ and $t=\pi_{1}\circ f$ such that

(9) $s(T^{\ast}_{x}M\cap V)\subset\Lambda_{x}\mbox{ and }t(T^{\ast}_{x}M\cap
V)\subset\tilde{\Lambda}_{x}$

for all $x\in M$.

###### Proof.

Eqn (3) implies that

$f^{\ast}(\pi^{\ast}_{2}\omega-\pi^{\ast}_{1}\omega)|_{V}=-d\theta|_{V},$

which means that $V$ is a local symplectic groupoid over $M$ with $s$ and $t$
as in the statement of the lemma. The inclusions (9) hold true because $f$
maps $T^{\ast}_{x}M\cap V$ to the set

$\\{(\tilde{\lambda}(\tilde{z}),\lambda(z))|z\in
E_{x},\tilde{z}\in\tilde{E}_{x}\\}$

and for $z\in E_{x}$ and $\tilde{z}\in\tilde{E}_{x}$ one has
$\lambda(z)\in\Lambda_{x}$ and
$\tilde{\lambda}(\tilde{z})\in\tilde{\Lambda}_{x}$. ∎

It follows from (8) that

(10) $s\circ\chi|_{W}=\lambda\tau|_{W}\mbox{ and
}t\circ\chi|_{W}=\tilde{\lambda}\tilde{\tau}|_{W}.$

##  4\. The intersection mapping $\gamma$

In this section we will show that for two transversal Lagrangian fields
$\Lambda=(E,\lambda)$ and $\tilde{\Lambda}=(\tilde{E},\tilde{\lambda})$ on $M$
there exists a neighborhood $V_{\Gamma}$ of the diagonal $M_{2}$ in $M^{2}$
and a mapping $\gamma:V_{\Gamma}\to M$ such that
$\gamma(x,y)\in\Lambda_{x}\cap\tilde{\Lambda}_{y}$ for all $(x,y)\in
V_{\Gamma}$. We will also show that $\Omega:=\gamma^{\ast}\omega$ is a closed
(1,1)-form with respect to the product structure on $V_{\Gamma}$.

Consider the surjective submersion
$(z,\tilde{z})\mapsto(\lambda(z),\tilde{\lambda}(\tilde{z}))$ of
$E\times\tilde{E}$ onto $M\times M$. The inverse image of the diagonal $M_{2}$
of $M^{2}$ is a $4m$-dimensional submanifold

$\Gamma:=\\{(z,\tilde{z})\in
E\times\tilde{E}|\lambda(z)=\tilde{\lambda}(\tilde{z})\\}$

of $E\times\tilde{E}$. Because $\lambda(z)=\pi(z)$ for $z\in Z_{E}$ and
$\tilde{\lambda}(\tilde{z})=\tilde{\pi}(\tilde{z})$ for $\tilde{z}\in
Z_{\tilde{E}}$, the zero section $Z_{\hat{E}}=\\{(z,\tilde{z})\in Z_{E}\times
Z_{\tilde{E}}|\pi(z)=\tilde{\pi}(z)\\}$ of $\hat{E}$ is also a submanifold of
$\Gamma$.

######  Lemma 4.1.

If $\Lambda$ and $\tilde{\Lambda}$ are transversal Lagrangian fields on $M$,
there exists a neighborhood $W_{\Gamma}$ of $Z_{\hat{E}}$ in $\Gamma$ and a
neighborhood $V_{\Gamma}$ of the diagonal $M_{2}$ in $M^{2}$ such that the
mapping

$\pi\times\tilde{\pi}\big{|}_{W_{\Gamma}}:(z,\tilde{z})\mapsto(\pi(z),\tilde{\pi}(\tilde{z}))$

is a diffeomorphism of $W_{\Gamma}$ onto $V_{\Gamma}$.

###### Proof.

Let ${x_{0}}$ be an arbitrary point in $M$. As in the proof of Lemma 3.2, we
fix a coordinate chart $(\hat{U},\\{x^{i}\\})$ on $M$ around ${x_{0}}$, a
neighborhood $U$ of ${x_{0}}$ in $\hat{U}$, trivializations
$U\times\mathbb{R}^{m}$ of $E$ and $\tilde{E}$ over $U$, and an open ball $B$
in $\mathbb{R}^{m}$ centered at the origin such that $\lambda(U\times
B)\subset\hat{U}$ and $\tilde{\lambda}(U\times B)\subset\hat{U}$. In the
calculations below we use local coordinates $(x,v)$ on
$U\times\mathbb{R}^{m}\approx E|_{U}$ and $(y,w)$ on
$U\times\mathbb{R}^{m}\approx\tilde{E}|_{U}$. Consider the function

$F(x,v,y,w):=\lambda(x,v)-\tilde{\lambda}(y,w)$

from $U\times B\times U\times B$ to $\mathbb{R}^{2m}$. Then $F(x,0,x,0)=0$,
because $\lambda(x,0)=x$ and $\tilde{\lambda}(x,0)=x$. Since $\Lambda$ and
$\tilde{\Lambda}$ are transversal, the matrix

$\begin{bmatrix}\frac{\partial F^{i}}{\partial
v^{\alpha}}(x,0,x,0)&\frac{\partial F^{i}}{\partial
w^{\beta}}(x,0,x,0)\end{bmatrix}=\begin{bmatrix}\frac{\partial\lambda^{i}}{\partial
v^{\alpha}}(x,0)&-\frac{\partial\tilde{\lambda}^{i}}{\partial
w^{\beta}}(x,0)\end{bmatrix}$

is nondegenerate for all $x\in U$. By the implicit function theorem, there
exists a neighborhood $U_{0}$ of ${x_{0}}$ in $U$ and unique mappings $g(x,y)$
and $\tilde{g}(x,y)$ from $U_{0}^{2}$ to $B$ such that
$g({x_{0}},{x_{0}})=0,\tilde{g}({x_{0}},{x_{0}})=0$, and

$F(x,g(x,y),y,\tilde{g}(x,y))=\lambda(x,g(x,y))-\tilde{\lambda}(y,\tilde{g}(x,y))=0$

for all $(x,y)\in U_{0}^{2}$. Then the mapping

$G(x,y):=(x,g(x,y),y,\tilde{g}(x,y))$

maps $U_{0}^{2}$ to $\Gamma$. In particular,
$G({x_{0}},{x_{0}})=({x_{0}},0,{x_{0}},0)$. It is a diffeomorphism of
$U_{0}^{2}$ onto $G(U_{0}^{2})$ whose inverse is
$\pi\times\tilde{\pi}\big{|}_{G(U_{0}^{2})}$. Thus the mapping

$\pi\times\tilde{\pi}\big{|}_{\Gamma}:\Gamma\to M^{2}$

is a submersion at all points of $Z_{\hat{E}}$. It also identifies
$Z_{\hat{E}}$ with $M_{2}$ as copies of $M$. The lemma follows from the
generalized inverse function theorem. ∎

Lemma 4.1 implies that there exists a unique mapping $\gamma:V_{\Gamma}\to M$
such that

(11)
$\gamma(\pi(z),\tilde{\pi}(\tilde{z}))=\lambda(z)=\tilde{\lambda}(\tilde{z})$

for all $(z,\tilde{z})\in W_{\Gamma}$. The main property of this mapping is
that

(12) $\gamma(x,y)\in\Lambda_{x}\cap\tilde{\Lambda}_{y}$

for all $(x,y)\in V_{\Gamma}$. We will show that $\gamma(x,x)=x$ for all $x\in
M$. We have $(z_{x},\tilde{z}_{x})\in Z_{\hat{E}}\subset W_{\Gamma}$ and

$\gamma(x,x)=\gamma(\pi(z_{x}),\tilde{\pi}(\tilde{z}_{x}))=\lambda(z_{x})=\pi(z_{x})=x.$

Let $\epsilon:M\to M^{2}$ be the diagonal inclusion, $\epsilon(x)=(x,x)$ for
$x\in M$. We have shown that

(13) $\gamma\epsilon=\mathrm{id}_{M}.$

We introduce a closed 2-form on $V_{\Gamma}$,

$\Omega:=\gamma^{\ast}\omega.$

It follows from (13) that $\epsilon^{\ast}\Omega=\omega$ and therefore

(14) $\gamma^{\ast}\epsilon^{\ast}\Omega=\Omega.$

######  Lemma 4.2.

The form $\Omega$ is a $(1,1)$-form with respect to the product structure on
$M^{2}$.

###### Proof.

The tangent space $T_{(x,y)}V_{\Gamma}$ at any point $(x,y)\in V_{\Gamma}$ is
the direct sum of the subspaces of vectors of type $(1,0)$ and $(0,1)$ with
respect to the product structure on $M^{2}$,

$T_{(x,y)}V_{\Gamma}=T^{(1,0)}_{(x,y)}V_{\Gamma}\oplus
T^{(0,1)}_{(x,y)}V_{\Gamma}.$

Any tangent vector $u\in T^{(1,0)}_{(x,y)}V_{\Gamma}$ is the tangent vector at
the initial point $(x(0),y)=(x,y)$ of some path $t\mapsto(x(t),y)$ in
$V_{\Gamma}$. It follows from (12) that $\gamma$ maps this path to a path in
the Lagrangian manifold $\tilde{\Lambda}_{y}$ with the initial point
$\gamma(x,y)$. Therefore,

$d\gamma_{(x,y)}(u)\in T_{\gamma(x,y)}\tilde{\Lambda}_{y}.$

Similarly, if $u\in T^{(0,1)}_{(x,y)}V_{\Gamma}$, then

$d\gamma_{(x,y)}(u)\in T_{\gamma(x,y)}\Lambda_{x}.$

Thus, if tangent vectors $u$ and $u^{\prime}$ in $T_{(x,y)}V_{\Gamma}$ are
both of type $(1,0)$ or both of type $(0,1)$, then $\Omega(u,u^{\prime})=0$.
It implies that $\Omega$ is of type $(1,1)$ with respect to the product
structure on $M^{2}$. ∎

##  5\. Calabi functions

In [2] Calabi introduced a diastasic function $D_{X}$ of a real-analytic
Kähler form $\omega_{X}$ on a Kähler manifold $X$. A local real-analytic
potential $\Phi_{X}(z,\bar{z})$ of $\omega_{X}$ is defined up to a summand
which is the real part of an analytic function. If $\Phi(z,\bar{w})$ is a
complex analytic continuation of $\Phi(z,\bar{z})$, the diastatic function is
locally defined as follows:

$D_{X}=\Phi_{X}(z,\bar{w})+\Phi_{X}(w,\bar{z})-\Phi_{X}(z,\bar{z})-\Phi_{X}(w,\bar{w}).$

The function $D_{X}$ does not depend on the choice of the local potential
$\Phi_{X}$ and is defined on a neighborhood of the diagonal of $X^{2}$. One
can similarly define multivariate functions corresponding to a closed
(1,1)-form on a product space.

Let $X$ and $Y$ be two manifolds, $W$ be an open subset of $X\times Y$, and
$\Omega$ be a closed 2-form on $W$ of type (1,1) with respect to the product
structure on $W$. Denote by $X_{k}$ and $Y_{l}$ the diagonals of $X^{k}$ and
$Y^{l}$, respectively, by $\pi_{ij}:X^{k}\times Y^{l}\to X\times Y$ the
projection that maps $(x_{1},\ldots,x_{k},y_{1},\ldots,y_{l})\in X^{k}\times
Y^{l}$ to $(x_{i},y_{j})$, and by $i:X\times Y\mapsto X^{k}\times Y^{l}$ the
diagonal inclusion that maps $(x,y)\in X\times Y$ to
$(x,\ldots,x,y,\ldots,y)\in X_{k}\times Y_{l}$. The set

$\hat{W}:=\bigcap\limits_{i,j}\pi_{ij}^{-1}(W)$

is an open subset of $X^{k}\times Y^{l}$ containing $i(W)$. Let $C_{ij}$ be a
$k\times l$-matrix such that $\sum_{i}C_{ij}=0$ for all $j$ and
$\sum_{j}C_{ij}=0$ for all $i$. The form

$\hat{\Omega}:=\sum\limits_{i,j}C_{ij}\pi_{ij}^{\ast}\Omega$

is a closed 2-form on $\hat{W}$ of type (1,1) with respect to the product
structure on $X^{k}\times Y^{l}$. Let $d=d_{x}+d_{y}$ be the corresponding
decomposition of the exterior derivative operator on $\hat{W}$.

######  Definition 5.1.

Let $W_{0}$ be a neighborhood of $i(W)$ in $\hat{W}$. We say that a function
$C$ on $W_{0}$ is a Calabi function associated with the form $\Omega$ and the
matrix $C_{ij}$ if $\hat{\Omega}=d_{x}d_{y}C$, $C|_{(X_{k}\times Y^{l})\cap
W_{0}}=0$, and $C|_{(X^{k}\times Y_{l})\cap W_{0}}=0$.

The condition $\hat{\Omega}=d_{x}d_{y}C$ means that $C$ is a potential of the
closed (1,1)-form $\hat{\Omega}$. We will show that the Calabi functions
associated with the form $\Omega$ and the matrix $C_{ij}$ exist. We fix
Riemannian metrics on $X$ and $Y$ and for every point $(a,b)\in W$ choose
geodesically convex neighborhoods $U_{a,b}$ of $a$ in $X$ and $V_{a,b}$ of $b$
in $Y$ such that $U_{a,b}\times V_{a,b}\subset W$. There exists a potential
$\Phi_{a,b}(x,y)$ of the form $\Omega$ on $U_{a,b}\times V_{a,b}$. It is
determined up to a summand $f(x)+g(y)$. We define a Calabi function $C_{a,b}$
on $U^{k}_{a,b}\times V^{l}_{a,b}$ by the formula

$C_{a,b}(x_{1},\ldots,x_{k},y_{1},\ldots,y_{l}):=\sum\limits_{ij}C_{ij}\Phi_{a,b}(x_{i},y_{j}).$

It does not depend on the choice of the potential $\Phi_{a,b}$. Moreover, for
another point $(c,d)\in W$ the functions $C_{a,b}$ and $C_{c,d}$ agree on

$(U^{k}_{a,b}\times V^{l}_{a,b})\cap(U^{k}_{c,d}\times
V^{l}_{c,d})=(U_{a,b}\cap U_{c,d})^{k}\times(V_{a,b}\cap V_{c,d})^{l}.$

Therefore, the functions $C_{a,b}$ determine a Calabi function $C$ on

$W_{0}:=\bigcup\limits_{(a,b)\in W}U^{k}_{a,b}\times V^{l}_{a,b}.$

It is easy to check that the Calabi functions associated with the form
$\Omega$ and the matrix $C_{ij}$ define the same germ around $i(W)$.

In the rest of the paper we will use a modification of these Calabi functions.
Assume that $X=Y$ and $W$ is an open subset of $X^{2}$ containing the diagonal
$X_{2}$. Let $C_{n}$ be a Calabi function associated with a closed (1,1)-form
$\Omega$ on $W$ and an $n\times n$-matrix $C_{ij}$. We introduce a modified
Calabi function

$\mathcal{C}_{n}(x_{1},\ldots,x_{n}):=C_{n}(x_{1},\ldots,x_{n},x_{1},\ldots,x_{n})$

which is defined on a neighborhood of the diagonal $X_{n}$ in $X^{n}$.

##  6\. A generating function of the Lagrangian manifold of $n$-cycles

Let $\Omega$ be the closed (1,1)-form on $V_{\Gamma}\subset M^{2}$ introduced
in Section 6 and $C_{ij}$ be an $n\times n$-matrix such that

$C_{ij}=\begin{cases}1&\mbox{ if }j=i+1\mbox{ for }1\leq i\leq n-1\mbox{ or
}(i,j)=(n,1);\\\ -1&\mbox{ if }i=j;\\\ 0&\mbox{ otherwise.}\end{cases}$

Then there exists a neighborhood $V_{0}$ of the diagonal $M_{n}$ in $M^{n}$
and a Calabi function $\mathcal{C}_{n}$ on $V_{0}$ associated with the form
$\Omega$ and the matrix $C_{ij}$. For any point ${x_{0}}\in M$ there exists a
contractible neighborhood $U$ of ${x_{0}}$ in $M$ such that $U^{2}\subset
V_{\Gamma}$, $U^{n}\subset V_{0}$, and

(15)
$\displaystyle\mathcal{C}_{n}(x_{1},\ldots,x_{n})=\Phi(x_{1},x_{2})-\Phi(x_{2},x_{2})+\Phi(x_{2},x_{3})-$
$\displaystyle\Phi(x_{3},x_{3})+\ldots-\Phi(x_{n},x_{n})+\Phi(x_{n},x_{1})-\Phi(x_{1},x_{1})$

for any potential $\Phi(x,y)$ of the form $\Omega$ on $U^{2}$. We call the
function $\mathcal{C}_{n}$ the cyclic $n$-point Calabi function of the form
$\Omega$. Now we will show that the germ of the generating function of the
Lagrangian manifold of $n$-cycles of the local symplectic groupoid constructed
in Theorem 3.1 around the diagonal $M_{n}$ of $M^{n}$ is given by the function
$\mathcal{C}_{n}$.

If $F(x_{1},\ldots,x_{n})$ is a function of $n$ variables and
$(U,\\{x^{i}\\})$ is a coordinate chart on $M$, we will use the following
notations for the partial derivatives of the function $F$ on $U^{n}$:

$\frac{\partial F}{\partial x_{k}^{i}}=\partial^{k}_{i}F.$

In particular, $\Omega=(\partial^{1}_{i}\partial^{2}_{j}\Phi)(x,y)dx^{i}\wedge
dy^{j}$ and

$\frac{\partial}{\partial
x^{i}}(\Phi(x,x))=(\partial^{1}_{i}\Phi)(x,x)+(\partial^{2}_{i}\Phi)(x,x).$

Since $\epsilon^{\ast}\Omega=\omega$, we see that locally

(16) $\omega=(\partial^{1}_{i}\partial^{2}_{j}\Phi)(x,x)dx^{i}\wedge dx^{j}.$

Shrinking $U$ around ${x_{0}}$, if necessary, we can find an open ball $B$ in
$\mathbb{R}^{m}$ centered at the origin and functions $g,\tilde{g}:U^{2}\to B$
such that

(17) $\gamma(x,y)=\lambda(x,g(x,y))=\tilde{\lambda}(y,\tilde{g}(x,y))$

for all $(x,y)\in U^{2}$ as in the proof of Lemma 4.1.

We will use the coordinates $(x^{i},u^{i},y^{i})$ on $U^{3}$ and the dual
coordinates $(u^{i},\xi_{i})$ on $T^{\ast}U$. Let $\kappa:U^{3}\to\hat{E}$ be
the mapping given by

$\kappa:(x,u,y)\mapsto(u,g(u,y),\tilde{g}(x,u)).$

Denote by $\pi_{i},i=1,2,3$, the projections of $U^{3}$ onto the respective
factors and set $\pi_{12}(x,u,y):=(x,u)$ and $\pi_{23}(x,u,y):=(u,y)$. Recall
that the mappings $\tau:\hat{E}\to E$ and $\tilde{\tau}:\hat{E}\to\tilde{E}$
are given by $\tau(z,\tilde{z})=z$ and $\tilde{\tau}(z,\tilde{z})=\tilde{z}$.
We write the point $(u,g(u,y),\tilde{g}(x,u))$ of $\hat{E}$ as $(z,\tilde{z})$
with $z=(u,g(u,y))$ and $\tilde{z}=(u,\tilde{g}(x,u))$. We see from (17) that

(18) $\lambda\tau\kappa=\gamma\pi_{23}\mbox{ and
}\tilde{\lambda}\tilde{\tau}\kappa=\gamma\pi_{12}.$

In the calculations below we will use the commutative diagrams

(19)
$\textstyle{U^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\scriptstyle{\gamma\pi_{23}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi|_{W}}$$\scriptstyle{\lambda\tau}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{M}$
$\textstyle{U^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\scriptstyle{\gamma\pi_{12}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi|_{W}}$$\scriptstyle{\tilde{\lambda}\tilde{\tau}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\textstyle{M}$

whose commutativity follows from Theorem 3.1, (10), and (18). Pulling back Eqn
(1) by $\tau\kappa$, we get that

(20)
$\displaystyle\pi_{23}^{\ast}\gamma^{\ast}\omega-\pi_{2}^{\ast}\omega=d\kappa^{\ast}\tau^{\ast}\phi$

on $U^{2}$. A simple calculation shows that the equality

(21) $\displaystyle(\partial^{1}_{i}\partial^{2}_{j}\Phi)(u,y)du^{i}\wedge
dy^{j}-(\partial^{1}_{i}\partial^{2}_{j}\Phi)(u,u)du^{i}\wedge du^{j}=$
$\displaystyle
d(((\partial^{1}_{i}\Phi)(u,u)-(\partial^{1}_{i}\Phi)(u,y))du^{i})$

holds on $U^{3}$. The left-hand sides of (20) and (21) agree. The 1-forms
$\kappa^{\ast}\tau^{\ast}\phi$ and
$((\partial^{1}_{i}\Phi)(u,u)-(\partial^{1}_{i}\Phi)(u,y))du^{i}$ on $U^{3}$
are horizontal with respect to the projection $\pi_{2}$, vanish on the
diagonal of $U^{3}$, and have equal differentials. Therefore, they coincide:

(22)
$\kappa^{\ast}\tau^{\ast}\phi=((\partial^{1}_{i}\Phi)(u,u)-(\partial^{1}_{i}\Phi)(u,y))du^{i}.$

Similarly, pulling back Eqn (2) by $\tilde{\tau}\kappa$, we get that

(23)
$\displaystyle\pi_{12}^{\ast}\gamma^{\ast}\omega-\pi_{2}^{\ast}\omega=d\kappa^{\ast}\tilde{\tau}^{\ast}\tilde{\phi}$

on $U^{3}$. We see from Eqn (23) and the equality

$\displaystyle(\partial^{1}_{i}\partial^{2}_{j}\Phi)(x,u)dx^{i}\wedge
du^{j}-(\partial^{1}_{i}\partial^{2}_{j}\Phi)(u,u)du^{i}\wedge du^{j}=$
$\displaystyle
d(((\partial^{2}_{i}\Phi)(x,u)-(\partial^{2}_{i}\Phi)(u,u))du^{i})$

that

(24)
$\kappa^{\ast}\tilde{\tau}^{\ast}\tilde{\phi}=((\partial^{2}_{i}\Phi)(x,u)-(\partial^{2}_{i}\Phi)(u,u))du^{i}$

on $U^{3}$. We introduce the Calabi function

(25) $T(x,u,y)=\Phi(x,u)-\Phi(u,u)+\Phi(u,y)-\Phi(x,y)$

on $U^{3}$ and obtain from (22) and (24) that

$\displaystyle\frac{\partial T(x,u,y)}{\partial
u^{i}}du^{i}=((\partial^{2}_{i}\Phi)(x,u)-(\partial^{2}_{i}\Phi)(u,u))du^{i}\hskip
28.45274pt$
$\displaystyle-((\partial^{1}_{i}\Phi)(u,u)-(\partial^{1}_{i}\Phi)(u,y))du^{i}=\kappa^{\ast}(\tilde{\tau}^{\ast}\tilde{\phi}-\tau^{\ast}\phi).$

It follows that the mapping $\chi\kappa:U^{3}\to V$ maps $(x,u,y)$ to

$(u^{i},\xi_{i})=\left(u^{i},\frac{\partial T(x,u,y)}{\partial u^{i}}\right).$

We see from the diagrams (19) that

(26) $\gamma(u,y)=s\left(u,\frac{\partial T(x,u,y)}{\partial u}\right)\mbox{
and }\gamma(x,u)=t\left(u,\frac{\partial T(x,u,y)}{\partial u}\right).$

One can check that the Calabi functions $\mathcal{C}_{n}$ and $T$ satisfy the
identity

$\frac{\partial}{\partial
x_{k}^{i}}T(x_{k-1},x_{k},x_{k+1})=\frac{\partial\mathcal{C}_{n}}{\partial
x_{k}^{i}}$

for $k=1,\ldots,n$, where we identify $x_{0}$ with $x_{n}$ and $x_{n+1}$ with
$x_{1}$. Now (26) implies that

$s\left(x_{k-1},\frac{\partial\mathcal{C}_{n}}{\partial
x_{k-1}}\right)=\gamma(x_{k-1},x_{k})=t\left(x_{k},\frac{\partial\mathcal{C}_{n}}{\partial
x_{k}}\right),$

which means that

$\left\\{\left(x_{k},\frac{\partial\mathcal{C}_{n}}{\partial
x_{k}}\right),k=1,\ldots,n\right\\}$

is an $n$-cycle in $T^{\ast}U\cap V$. We have thus proved the following
theorem.

######  Theorem 6.1.

The Lagrangian manifold of $n$-cycles of the local symplectic groupoid
constructed in Theorem 3.1 has a generating function whose germ around the
diagonal $M_{n}$ of $M^{n}$ is given by the cyclic $n$-point Calabi function
$\mathcal{C}_{n}$.

##  7\. A critical point of the function $u\mapsto T(x,u,y)$

In Section 6 we constructed the mapping $\gamma:V_{\Gamma}\to M$, where
$V_{\Gamma}$ is a neighborhood of the diagonal $M_{2}$ in $M^{2}$. It
satisfies (11), (12), and (13). We also defined the Calabi function
$T=T(x,u,y)$ in terms of a local potential of the form $\Omega$. The function
$T$ does not depend on the choice of local potentials and is defined on some
neighborhood $W_{T}$ of the diagonal $M_{3}$ in $M^{3}$.

In this section we will show that if $x$ and $y$ are close, then $\gamma(x,y)$
is a nondegenerate critical point of the function $u\mapsto T(x,u,y)$ with
zero critical value, $T(x,\gamma(x,y),y)=0$. Heuristically it means that for
some constant $C$ the oscillatory integral operator

$f(x)\mapsto C\hbar^{-m}\int e^{\frac{i}{\hbar}(\Phi(x,y)-\Phi(y,y))}f(y)\,dy$

is an approximate projector.

Let $(x,y)\mapsto(z(x,y),\tilde{z}(x,y))$ be the mapping from $V_{\Gamma}$ to
$W_{\Gamma}$ inverse to $\pi\times\tilde{\pi}|_{W_{\Gamma}}:W_{\Gamma}\to
V_{\Gamma}$. Thus, $\pi(z(x,y))=x$ and $\tilde{\pi}(\tilde{z}(x,y))=y$. We
have $z(x,x)=z_{x}$ and $\tilde{z}(x,x)=\tilde{z}_{x}$.

######  Lemma 7.1.

  1. (i)

There is a neighborhood $V^{\prime}$ of $M_{2}$ in $V_{\Gamma}$ such that

(27) $(\gamma(x,y),y)\in V_{\Gamma}\mbox{ and
}\gamma(\gamma(x,y),y)=\gamma(x,y)$

for any $(x,y)\in V^{\prime}$.

  2. (ii)

There is a neighborhood $V^{\prime\prime}$ of $M_{2}$ in $V_{\Gamma}$ such
that

(28) $(x,\gamma(x,y))\in V_{\Gamma}\mbox{ and
}\gamma(x,\gamma(x,y))=\gamma(x,y)$

for any $(x,y)\in V^{\prime\prime}$.

###### Proof.

Suppose that $(x,y)\in V_{\Gamma}$. Then
$\gamma(x,y)=\lambda(z(x,y))=\tilde{\lambda}(\tilde{z}(x,y))$. It follows from
the equalities
$\lambda(z_{\gamma(x,y)})=\gamma(x,y)=\tilde{\lambda}(\tilde{z}(x,y))$ that
the range of the mapping

(29) $V_{\Gamma}\ni(x,y)\mapsto(z_{\gamma(x,y)},\tilde{z}(x,y))$

lies in $\Gamma$. The mapping (29) maps $(x,x)$ to $(z_{x},\tilde{z}_{x})$ for
all $x\in M$ and thus it maps $M_{2}$ to $Z_{\hat{E}}$. Denote by $V^{\prime}$
the inverse image of $W_{\Gamma}$ with respect to the mapping (29). It is easy
to check that $V^{\prime}$ satisfies the conditions of part (i) of the lemma.
Part (ii) can be proved similarly. ∎

We define mappings $\alpha,\beta:V_{\Gamma}\to M^{2}$ as follows:

$\alpha:(x,y)\mapsto(\gamma(x,y),y)\mbox{ and
}\beta:(x,y)\mapsto(x,\gamma(x,y)).$

Lemma 7.1 implies that

(30) $\displaystyle\alpha(V^{\prime})\subset V_{\Gamma}\mbox{ and
}\alpha\circ(\alpha|_{V^{\prime}})=\alpha|_{V^{\prime}};$
$\displaystyle\beta(V^{\prime\prime})\subset V_{\Gamma}\mbox{ and
}\beta\circ(\beta|_{V^{\prime\prime}})=\beta|_{V^{\prime\prime}}.$

If $(x,y)\in V^{\prime}$, then $\beta\alpha(x,y)=(\gamma(x,y),\gamma(x,y))$
and if $(x,y)\in V^{\prime\prime}$, then
$\alpha\beta(x,y)=(\gamma(x,y),\gamma(x,y))$. Observe that
$\epsilon\gamma(x,y)=(\gamma(x,y),\gamma(x,y))$ for $(x,y)\in V_{\Gamma}$.
Hence,

(31)
$\alpha\circ(\beta|_{V^{\prime\prime}})=\epsilon\gamma|_{V^{\prime\prime}}\mbox{
and }\beta\circ(\alpha|_{V^{\prime}})=\epsilon\gamma|_{V^{\prime}}.$

Let ${x_{0}}$ be an arbitrary point in $M$ and $U$ be a contractible
neighborhood of ${x_{0}}$ in $M$. One can shrink $U$ around ${x_{0}}$ so that
if $\alpha^{\ast}$ is applied to (14), it will follow from (30) and (31) that

$\alpha^{\ast}\Omega=\alpha^{\ast}\beta^{\ast}\Omega=\Omega$

on $U^{2}$. Similarly, we can assume that $\beta^{\ast}\Omega=\Omega$ on
$U^{2}$. Let $\Phi$ be a potential of $\Omega$ on $U$, so that

$\Omega=d_{x}d_{y}\Phi=dd_{y}\Phi,$

where $d=d_{x}+d_{y}$ is the decomposition of the exterior derivative with
respect to the product structure on $M^{2}$. Shrinking $U$ further, we can
assume that $\Phi(\gamma(x,y),y),\Phi(x,\gamma(x,y))$, and
$\Phi(\gamma(x,y),\gamma(x,y))$ are defined for all $x,y\in U$. We obtain from
$\alpha^{\ast}\Omega=\Omega$ that

$d(\alpha^{\ast}d_{y}\Phi-d_{y}\Phi)=0.$

It is easy to check that the closed form $\alpha^{\ast}d_{y}\Phi-d_{y}\Phi$ is
of type (0,1) with respect to the product structure on $U^{2}$ and vanishes on
the diagonal of $U^{2}$. Therefore, it vanishes identically on $U^{2}$.
Similarly, the form $\beta^{\ast}d_{x}\Phi-d_{x}\Phi$ also vanishes on
$U^{2}$. Thus,

(32) $\alpha^{\ast}d_{y}\Phi=d_{y}\Phi\mbox{ and
}\beta^{\ast}d_{x}\Phi=d_{x}\Phi$

on $U^{2}$. It follows from (31) that

$T(x,\gamma(x,y),y)=\Phi-\alpha^{\ast}\Phi-\beta^{\ast}\Phi+\alpha^{\ast}\beta^{\ast}\Phi$

for all $x,y\in U$. Differentiating this equality, we get from (32) that

$\displaystyle
d(T(x,\gamma(x,y),y))=d_{x}\Phi-\alpha^{\ast}d_{x}\Phi-\beta^{\ast}d_{x}\Phi+\alpha^{\ast}\beta^{\ast}d_{x}\Phi+$
$\displaystyle
d_{y}\Phi-\alpha^{\ast}d_{y}\Phi-\beta^{\ast}d_{y}\Phi+\alpha^{\ast}\beta^{\ast}d_{y}\Phi=$
$\displaystyle-\alpha^{\ast}d_{x}\Phi+\alpha^{\ast}\beta^{\ast}d_{x}\Phi-\beta^{\ast}d_{y}\Phi+\beta^{\ast}\alpha^{\ast}d_{y}\Phi=$
$\displaystyle-\alpha^{\ast}(d_{x}\Phi-\beta^{\ast}\Phi)-\beta^{\ast}(d_{y}\Phi-\alpha^{\ast}\Phi)=0.$

Since the function $T(x,\gamma(x,y),y)$ vanishes on the diagonal of $U^{2}$
and is constant, it vanishes identically on $U^{2}$. Since $U^{2}$ is a
neighborhood of the diagonal point $({x_{0}},{x_{0}})\in M_{2}$ in $M^{2}$ and
${x_{0}}\in M$ is arbitrary, we arrive at the following statement.

######  Lemma 7.2.

There exists a neighborhood $V_{0}$ of $M_{2}$ in $V_{\Gamma}$ such that
$(x,\gamma(x,y),y)\in W_{T}$ and the identity $T(x,\gamma(x,y),y)=0$ holds for
any $(x,y)\in V_{0}$.

######  Lemma 7.3.

There exists a neighborhood $V_{1}$ of $M_{2}$ in $V_{\Gamma}$ such that
$(x,\gamma(x,y),y)\in W_{T}$ and the point $\gamma(x,y)$ is a nondegenerate
critical point of the function $u\mapsto T(x,u,y)$ for any $(x,y)\in V_{1}$.

###### Proof.

Assume that $U$ is a neighborhood of an arbitrary point ${x_{0}}$ in $M$ such
that $U^{2}\subset V^{\prime}\cap V^{\prime\prime}$, where $V^{\prime}$ and
$V^{\prime\prime}$ are as in Lemma 7.1. We additionally assume that $U$ lies
in a contractible coordinate chart $(\hat{U},\\{x^{i}\\})$, is such that
$\gamma(U^{2})\subset\hat{U}$, and $\Phi$ is a potential of the form $\Omega$
on $\hat{U}^{2}$. It will be convenient to denote $\gamma(x,y)$ by
$\bar{\gamma}$. It follows from (32) that in coordinates

$(\partial^{1}_{i}\Phi)(x,\bar{\gamma})=(\partial^{1}_{i}\Phi)(x,y)\mbox{ and
}(\partial^{2}_{i}\Phi)(\bar{\gamma},y)=(\partial^{2}_{i}\Phi)(x,y).$

Applying $\alpha^{\ast}$ to the first identity and $\beta^{\ast}$ to the
second one, we get from Lemma 7.1 that

$(\partial^{1}_{i}\Phi)(\bar{\gamma},\bar{\gamma})=(\partial^{1}_{i}\Phi)(\bar{\gamma},y)\mbox{
and
}(\partial^{2}_{i}\Phi)(\bar{\gamma},\bar{\gamma})=(\partial^{2}_{i}\Phi)(x,\bar{\gamma}).$

Using these identities, we obtain that

$\displaystyle\frac{\partial}{\partial
u^{i}}T(x,u,y)|_{u=\bar{\gamma}}=(\partial^{2}_{i}\Phi)(x,\bar{\gamma})-(\partial^{1}_{i}\Phi)(\bar{\gamma},\bar{\gamma})-$
$\displaystyle(\partial^{2}_{i}\Phi)(\bar{\gamma},\bar{\gamma})+(\partial^{1}_{i}\Phi)(\bar{\gamma},y)=0.$

Thus, $\bar{\gamma}=\gamma(x,y)$ is a critical point of the function $u\mapsto
T(x,u,y)$ for all $x,y\in U$. Now we will show that there exists a
neighborhood $U_{1}$ of ${x_{0}}$ in $U$ such that this critical point is
nondegenerate for all $x,y\in U_{1}$. Writing the form $\Omega$ on $U$ in
local coordinates as

$\Omega=\frac{1}{2}\omega_{kl}(\gamma)d\gamma^{k}\wedge d\gamma^{l},$

we obtain from Lemma 4.2 that

(33) $\omega_{kl}(\gamma)\frac{\partial\gamma^{k}}{\partial
x^{i}}\frac{\partial\gamma^{l}}{\partial x^{j}}=0\mbox{ and
}\omega_{kl}(\gamma)\frac{\partial\gamma^{k}}{\partial
y^{i}}\frac{\partial\gamma^{l}}{\partial y^{j}}=0.$

Consider the matrices

$A:=((\partial^{1}_{i}\gamma^{k})(x,x)),B:=((\partial^{2}_{i}\gamma^{k})(x,x)),\mbox{
and }C:=(\omega_{kl}(x)),$

where $x\in U$. Since $\gamma(x,x)=x$, we get that $A+B=1$ (the identity
matrix). We derive from (33) that

$A^{t}CA=0\mbox{ and }B^{t}CB=0,$

whence it follows that

$C=A^{t}C+CA.$

Therefore, $B=C^{-1}A^{t}C$ and $BA=C^{-1}(A^{t}CA)=0$, which implies that $A$
and $B$ are complementary projectors.

Remark It can be seen from (33) that for a fixed $x$ both projectors $A$ and
$B$ are of rank $m$.

We have that

$\Omega=\omega_{kl}(\gamma)\frac{\partial\gamma^{k}}{\partial
x^{i}}\frac{\partial\gamma^{l}}{\partial y^{j}}dx^{i}\wedge
dy^{j}=(\partial^{1}_{i}\partial^{2}_{j}\Phi)(x,y)dx^{i}\wedge dy^{j}$

on $U$. Therefore,

$\omega_{kl}(\gamma)\frac{\partial\gamma^{k}}{\partial
x^{i}}\frac{\partial\gamma^{l}}{\partial
y^{j}}=(\partial^{1}_{i}\partial^{2}_{j}\Phi)(x,y).$

Setting $y=x$, we get

(34) $A^{t}CB=((\partial^{1}_{i}\partial^{2}_{j}\Phi)(x,x)),$

whence

(35) $B^{t}CA=-((\partial^{1}_{j}\partial^{2}_{i}\Phi)(x,x)).$

We want to show that if $x$ and $y$ are close, then the critical point
$u=\gamma(x,y)$ of the function $u\mapsto T(x,u,y)$ is nondegenerate, that is,
the Hessian matrix

$((\partial^{2}_{i}\partial^{2}_{j}T)(x,\gamma(x,y),y))$

is nondegenerate. Here $\partial^{2}_{i}$ is the partial derivative with
respect to $u^{i}$. It suffices to show that it holds true if $x=y$, that is,
the matrix

$((\partial^{2}_{i}\partial^{2}_{j}T)(x,x,x))$

is nondegenerate. We have

$\displaystyle\frac{\partial^{2}T(x,u,y)}{\partial u^{i}\partial
u^{j}}=(\partial^{2}_{i}\partial^{2}_{j}\Phi)(x,u)-(\partial^{1}_{i}\partial^{1}_{j}\Phi)(u,u)-(\partial^{1}_{i}\partial^{2}_{j}\Phi)(u,u)-$
$\displaystyle(\partial^{2}_{i}\partial^{1}_{j}\Phi)(u,u)-(\partial^{2}_{i}\partial^{2}_{j}\Phi)(u,u)+(\partial^{1}_{i}\partial^{1}_{j}\Phi)(u,y),$

whence

$\displaystyle(\partial^{2}_{i}\partial^{2}_{j}T)(x,x,x)=(\partial^{2}_{i}\partial^{2}_{j}\Phi)(x,x)-(\partial^{1}_{i}\partial^{1}_{j}\Phi)(x,x)-(\partial^{1}_{i}\partial^{2}_{j}\Phi)(x,x)-$
$\displaystyle(\partial^{2}_{i}\partial^{1}_{j}\Phi)(x,x)-(\partial^{2}_{i}\partial^{2}_{j}\Phi)(x,x)+(\partial^{1}_{i}\partial^{1}_{j}\Phi)(x,x)=$
$\displaystyle-(\partial^{1}_{i}\partial^{2}_{j}\Phi)(x,x)-(\partial^{1}_{j}\partial^{2}_{i}\Phi)(x,x).$

Using that $A+B=1$, we get from (34) and (35) that

$\displaystyle((\partial^{2}_{i}\partial^{2}_{j}T)(x,x,x))=-A^{t}CB+B^{t}CA=$
$\displaystyle-A^{t}C+A^{t}CA+CA-A^{t}CA=-A^{t}C+CA=$ $\displaystyle
C(A-C^{-1}A^{t}C)=C(A-B).$

Since $A$ and $B$ are complementary projectors, $A-B$ is invertible.
Therefore, the Hessian matrix $((\partial^{2}_{i}\partial^{2}_{j}T)(x,x,x))$
is nondegenerate.

∎

Lemmas 7.2 and 7.3 imply the following theorem.

######  Theorem 7.1.

There exists a neighborhood $V_{T}$ of $M_{2}$ in $V_{\Gamma}$ such that
$(x,\gamma(x,y),y)\in W_{T}$ and $\gamma(x,y)$ is a nondegenerate critical
point of the function $u\mapsto T(x,u,y)$ with zero critical value for all
$(x,y)\in V_{T}$.

Discussion

The geometric objects considered in this paper are germs of real smooth
submanifolds of a vector bundle containing its zero section or of $M^{n}$
containing the diagonal $M_{n}$. One can generalize the results of this paper
in two possible ways. Instead of fields of real Lagrangian submanifolds, one
can consider fields of Hörmander’s complex Lagrangian ideals. We expect that
then one can use germs of complex almost analytic manifolds introduced in [10]
in order to construct a local symplectic groupoid over an almost-Kähler
manifold from two transversal fields of Lagrangian ideals. For this groupoid,
the form $\Omega$, its local potentials, and the corresponding Calabi
functions will be complex-valued. We expect that this groupoid should
correspond to Berezin quantization via covariant symbols on general symplectic
manifolds (see [4] and references therein). One can also consider formal
symplectic groupoids introduced in [3] and [8], where germs of submanifolds
are replaced with their $\infty$-jets. Such formal symplectic groupoids over
almost-Kähler manifolds were constructed in [7].

## References

  * [1] Cabrera, A.: Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization, arXiv:2011.02321 [math.SG].
  * [2] Calabi, E.: Isometric imbeddings of complex manifolds, Annals of Math, Second Series, 58, no. 1, 1–23 (1953).
  * [3] Cattaneo, A., Dherin, B., Felder, G.: Formal Symplectic Groupoid, Commun. Math. Phys. 253, 645–674 (2005).
  * [4] Charles, L.: Quantization of Compact Symplectic Manifolds, J. Geom. Anal. 26, 2664–2710 (2016).
  * [5] Coste, A., Dazord, P., Weinstein, A.: Groupoides Symplectiques, Pub. Dépt. Math. Lyon, 2, 1–62 (1987).
  * [6] Hawkins, E.: A groupoid approach to quantization, J. Symplect. Geom. 6, 61–125 (2008).
  * [7] Karabegov, A.V.: On the Dequantization of Fedosov’s Deformation Quantization, Lett. Math. Phys. 65, 133–146 (2003).
  * [8] Karabegov, A.: Formal Symplectic Groupoid of a Deformation Quantization, Commun. Math. Phys. 258, 223–256 (2005).
  * [9] Karasev, M.: Analogues of the objects of Lie group theory for nonlinear Poisson brackets, Math. USSR Izvest.  28, 497–527 (1987).
  * [10] Melin A., Sjöstrand, J.: Fourier integral operators with complex-valued phase functions, in Lecture Notes in Math., 459 Springer, Berlin, 1975, 120–223.
  * [11] Weinstein, A.: Symplectic groupoids and Poisson manifolds, Bull. Am. Math. Soc. (N.S.) 16, 101–103 (1987).
  * [12] Zakrzewski, S.: Quantum and classical pseudogroups, I and II, Commun. Math. Phys. 134, 347–395 (1990).


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# Long geodesics on convex surfaces

Arseniy Akopyan  Anton Petrunin

###### Abstract

We review the theory of intrinsic geometry of convex surfaces in the Euclidean
space and prove the following theorem: if the boundary surface of a convex
body $K$ contains arbitrarily long closed simple geodesics, then $K$ is an
isosceles tetrahedron.

##  1 Introduction

The goal of this note is to introduce the reader to the theory of intrinsic
geometry of convex surfaces. We illustrate the power of the tools by proving a
theorem on convex surfaces containing an arbitrarily long closed simple
geodesic.

Let us remind that a curve in a surface is called _geodesic_ if every
sufficiently short arc of the curve is length minimizing; if in addition it
has no self intersections, we call it _simple geodesic_. A tetrahedron with
equal opposite edges will be called _isosceles_.

1.1. Theorem. Assume that the boundary surface $\Sigma$ of a convex body $K$
in the Euclidean space $\mathbb{E}^{3}$ admits an arbitrarily long simple
closed geodesic. Then $K$ is an isosceles tetrahedron.

This gives an affirmative answer to the question asked by Vladimir Protasov;
he proved the statement in assumption that $\Sigma$ is the boundary surface of
a convex polyhedron, see [11, 12].

The axiomatic method of Alexandrov geometry allows to work with metrics of
convex surfaces directly, without approximating it first by smooth or
polyhedral metric. Such approximations destroy the closed geodesics on the
surface; therefore it is hard (if at all possible) to apply approximations in
the proof of our theorem. On the other hand, a proof in the smooth or
polyhedral case usually admits a _translation_ into _Alexandrov’s language_ ;
such translation makes the result more general. In fact, our proof resembles a
translation of the proof given by Protasov.

Note that the main theorem implies in particular that a smooth convex surface
does not have arbitrarily long simple closed geodesics. However we do not know
a proof of this corollary which is essentially simpler than the one presented
below.

##  2 An overview of the theory

$a$$b$$c$$\gamma$

The geometry of intrinsic metric on convex surfaces can be considered as
generalization of Euclidean plane geometry, where instead of equality in the
cosine rule we have the following inequality

$c^{2}\leqslant a^{2}+b^{2}-2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}a{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}b{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}\cos\gamma.$

This approach provides a uniform way to work with smooth and singular
surfaces.

As the main reference, we will use the classical book of Alexandrov [2]; it is
available in Russian and German. The books [5] or [10] of Busemann and
Pogorelov should work as well and they are available in English.

As a motivation, we recommend an other classical book of Alexandrov [3] — it
is fun to read, it worth reading and once red, it should be easy to read it
should also be easy to read most papers on the subject including [2]. Some
related topics discussed in popular books; see for example [6, Lectures 20, 24
and 25].

### Angles

By a _surface_ we mean a compact $2$-dimensional manifold (possibly with non-
empty boundary), which is equipped with _geodesic_ metric. A metric on the
surface $\Sigma$ is called geodesic if any two points $p,q\in\Sigma$ can be
joined by a curve with length $|p-q|_{\Sigma}$; we denote by $|p-q|_{\Sigma}$
the distance from $p$ to $q$ in $\Sigma$. This curve will be denoted by $[pq]$
and called a _minimizing geodesic_ from $p$ to $q$.

A triple of points $x,y,z\in\Sigma$, together with a choice of three
minimizing geodesics $[xy]$, $[yz]$, $[zx]$ will be called a _triangle_ and
denoted by $[xyz]$. Many different triangles with vertices $x$, $y$ and $z$
may exist, any of which can be denoted by $[xyz]$.

$z$$\bar{z}$$y$$\bar{y}$$x$

A triangle $[\tilde{x}\tilde{y}\tilde{z}]$ in the Euclidean plane
$\mathbb{E}^{2}$ with the same side lengths as $[xyz]$ is called a _model
triangle_ of $[xyz]$; this relation will be written as
$[\tilde{x}\tilde{y}\tilde{z}]=\tilde{\triangle}(xyz)$. The angle
$\measuredangle[\tilde{x}\,{}^{\tilde{y}}_{\tilde{z}}]$ of the model triangle
$[\tilde{x}\tilde{y}\tilde{z}]$ is called _model angle_ of the triangle
$[xyz]$ at $x$ and denoted by $\text{\hbox
to0.0pt{$\tilde{\text{\phantom{n}}}$\hss}{$\measuredangle$}}(x\,{}^{y}_{z})$.

A pair of geodesics $[xy]$ and $[xz]$ with common endpoint $x$ is called a
_hinge_ and is denoted by $[x\,{}^{y}_{z}]$. The angle measure
$\measuredangle[x\,{}^{y}_{z}]$ of the hinge is defined as the limit of model
angles for the triangles sliding along the sides of hinge to its vertex. That
is,

$\measuredangle[x\,{}^{y}_{z}]=\lim_{\bar{y},\bar{z}\to
x}\left\\{\,\left.{\text{\hbox
to0.0pt{$\tilde{\text{\phantom{n}}}$\hss}{$\measuredangle$}}(x\,{}^{\bar{y}}_{\bar{z}})}\vphantom{\bar{y}\in\left]xy\right],\bar{z}\in\left]xz\right]}\,\right|\,{\bar{y}\in\left]xy\right],\bar{z}\in\left]xz\right]}\,\right\\},$

where $\left]xy\right]=[xy]\backslash\\{x\\}$. In general, the angle of a
hinge maybe undefined, but as you will see it will be defined every time we
need it. Note that if $p\in\left]xy\right[=[xy]\backslash\\{x,y\\}$ then
$\measuredangle[p\,{}^{x}_{y}]=\pi$.

### Comparison

2.1. Comparison property. We say that a hinge $[x\,{}^{y}_{z}]$ satisfies the
_comparison property_ if the angle $\measuredangle[x\,{}^{y}_{z}]$ is defined
and

$\measuredangle[x\,{}^{y}_{z}]\geqslant\text{\hbox
to0.0pt{$\tilde{\text{\phantom{n}}}$\hss}{$\measuredangle$}}(x\,{}^{y}_{z}){}{}.$

Two hinges $[p\,{}^{x}_{z}]$ and $[p\,{}^{y}_{z}]$ will be called
_supplementary_ , if they share a side $[pz]$ and $p\in\left]xy\right[$.

$x$$y$$z$$p$

2.2. Supplementary property. We say that the _suplementary property_ holds for
two supplementary hinges $[p\,{}^{x}_{z}]$ and $[p\,{}^{y}_{z}]$ if the angles
$\measuredangle[p\,{}^{x}_{z}]$ and $\measuredangle[p\,{}^{y}_{z}]$ are
defined and

$\measuredangle[p\,{}^{x}_{z}]+\measuredangle[p\,{}^{y}_{z}]=\pi.$

We will say that a surface $\Sigma$ has _non-negative curvature in the sense
of Alexandrov_ if the angles of all hinges in $\Sigma$ are defined and satisfy
the comparison and supplementary properties.111It is not known, if the
comparison property for all hinges in $\Sigma$ imply the supplementary
property.

The geometry of such surfaces is very specific. For example, by the comparison
property, if a hinge has vanishing angle then one of its sides lies in the
other. In particular, geodesics in $\Sigma$ can not bifurcate. The following
theorem plays the central role in the theory.

2.3. Globalization theorem. Assume that any point of the surface $\Sigma$ has
a neighborhood $U$ such that the angles of all hinges in $U$ are defined and
satisfy the comparison and supplementary properties. Then these properties
hold for all hinges; that is, $\Sigma$ has non-negative curvature in the sense
of Alexandrov.

This theorem was proved by Alexandrov [4] and generalized since then many
times; an amusing proof was given recently by Urs Lang and Viktor Schroeder in
[8].

Further, assume that the surface $\Sigma$ has non-negative curvature in the
sense of Alexandrov, $\gamma$ is a geodesic in $\Sigma$ parametrized by
length, and let $p$ be an arbitrary point on $\Sigma$. Note that the function
$f(t)=|p-\gamma(t)|_{\Sigma}$ is $1$-Lipschitz. In particular, the function
$f$ is differentiable almost everywhere.

$p$$\gamma(t)$$\varphi_{+}$$\varphi_{-}$

Let us denote by $\varphi_{\pm}(t)$ the angles between the positive and
negative directions of $\gamma$ and a geodesic $[\gamma(t)\,p]$; see the
diagram. From the definition of angle, via the triangle inequality, one gets
the following

$\pm f^{\prime}(t)\leqslant-\cos[\varphi_{\pm}(t)]$

at any $t$ for which the derivative $f^{\prime}(t)$ is defined (see [2, XI §2
(7)]).

By the supplementary property $\varphi_{-}+\nobreak\varphi_{+}=\pi$; hence
$\cos\varphi_{+}+\cos\varphi_{-}=0$. Therefore the two inequalities above
imply so called _first variation formula_

$f^{\prime}(t)=-\cos[\varphi_{+}(t)]$ (${*}$)

for any $t$, where the derivative $f^{\prime}(t)$ is defined.

Further, for any surface $\Sigma$ with non-negative curvature in the sense of
Alexandrov, the Kirszbraun extension theorem holds. That is, any distance non-
expanding map from a subset of $\Sigma$ to the Euclidean plane can be extended
to a distance non-expanding map defined on whole $\Sigma$; see [7, 1]. (In
fact, this statement could be used to define the spaces with non-negative
curvature in the sense of Alexandrov.)

Applying the Kiszbraun theorem for three-point sets, one gets the following
area comparison property which will be important to us.

2.4. Area comparison. Assume $\Sigma$ is a surface with non-negative curvature
in the sense of Alexandrov and a triangle $[xyz]$ in $\Sigma$ bounds a open
set $\Delta$ homeomorphic to a disc. Then

$\mathop{\rm area}\nolimits\Delta\geqslant\mathop{\rm
area}\nolimits\tilde{\triangle}(xyz).$

An earlier proof of this theorem by slicing $\Delta$ into small triangles is
given in [2, X §1].

### Convex surfaces

Recall that the intrinsic distance between points $x$ and $y$ on a surface
$\Sigma$ in $\mathbb{E}^{3}$, is defined as the greatest lower bound for the
lengths of curves connecting $x$ to $y$ in $\Sigma$.

2.5. Comparison theorem. The boundary surface of any convex body in
$\mathbb{E}^{3}$, if equipped with the induced intrinsic metric, is a sphere
with a non-negatively curved metric in the sense of Alexandrov.

The converse of this theorem also holds if one considers convex plane figure
as a degenerate convex body [2, III §3]. The _boundary surface_ of a flat
convex figure has to be defined as its doubling; that is, two copies of the
figure glued along the boundary — it will look like a surface if you can walk
on both sides of the figure, but can not pass from one side directly to an
other side.

2.6. Theorem. Any surface $\Sigma$ with non-negative curvature in the sense of
Alexandrov which is homeomorphic to the sphere is isometric to the boundary
surface of a convex body in $\mathbb{E}^{3}$, which possibly degenerates to a
flat figure.

It turns out that $\Sigma$ defines the convex body up to congruence. This is a
very hard theorem; it was first proved by Pogorelov in [9]. We will use the
following weaker statement proved by Alexandrov earlier [2, VI §5], [3, 3.3];
its proof goes essentially the same way as Cauchy’s rigidity theorem.

2.7. Rigidity theorem. Convex polyhedrons in $\mathbb{E}^{3}$ with isometric
boundary surfaces are congruent.

### Curvature measure

Let $\Sigma$ be a surface with non-negative curvature in the sense of
Alexandrov. Assume a triangle $[xyz]$ bounds an open set $\Delta$ in $\Sigma$
which is convex and homeomorphic to a disc. By convex we mean that any
minimizing geodesic with endpoints in $\Delta$ lie completely in $\Delta$. In
this case we define the _curvature_ of $\Delta$ as the _angle excess_ of the
triangle $[xyz]$; that is,

$\kappa(\Delta)=\measuredangle[x\,{}^{y}_{z}]+\measuredangle[y\,{}^{z}_{x}]+\measuredangle[z\,{}^{x}_{y}]-\pi.$

The functional $\kappa$ uniquely extends to non-negative measure, the so
called _curvature measure_ , defined on all Borel subsets in the interior of
$\Sigma$.

It turns out that any geodesic without its end-points has vanishing curvature.
This can be proved by covering the interior of geodesic by two thin triangles
with small excess as shown in the picture. (This requires some work, but
simple.)

Further, the curvature of an interior point in $\Sigma$ is $2{{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}}\pi$ minus total angle around it. The later
can be seen from the picture — to find the curvature of the central vertex one
has to subtract excesses of three small triangles from the excess of the big
one. (The existence of such configuration also requires some work.) A point of
non-zero curvature is called _singular_.

For curvature measure, an analog of the Gauss–Bonnet formula holds; in
particular, if $\Sigma$ is a surface with non-negative curvature in the sence
of Alexandrov then

  * $\diamond$

If $\Sigma$ is homeomorphic to the sphere then

$\kappa(\Sigma)=4{{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}}\pi$

  * $\diamond$

If a closed geodesic cuts from $\Sigma$ a disc $\Delta$ then

$\kappa(\Delta)=2{{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}}\pi.$

  * $\diamond$

If a closed broken geodesic cuts from $\Sigma$ a disc $\Delta$ then

$\kappa(\Delta)=2{{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}}\pi+(\vartheta_{1}-\pi)+\dots+(\vartheta_{n}-\pi),$

where $\vartheta_{1},\dots,\vartheta_{n}$ are the inner angles at the corners
of $\Delta$.

##  3 Four singular points

The following lemma is the key to the proof of Theorem 1. A point in a surface
is called _flat_ if it admits a flat neighborhood; that is, a neighborhood
isometric to an open subset of the plane. Equivalently, the curvature measure
is vanishing in a neighborhood of this point. Note that for a polyhedral
surface all points except the vertexes are flat.

3.1. Lemma. Assume that the boundary surface $\Sigma$ of a convex body $K$ in
$\mathbb{E}^{3}$ admits an arbitrarily long simple closed geodesic. Then
$\Sigma$ contains $4$ singular points with curvature $\pi$ and the rest of it
is flat.

In the proof we will show that it is possible to find four non-intersecting
open sets of arbitrarily small diameter, such that each of them has curvature
arbitrary close to $\pi$. Passing to the limit we get the statement of the
lemma.

By cutting the surface $\Sigma$ along a sufficiently long closed simple
geodesic, we get two discs. The key step is to show that each of these discs
is “long” and “thin”. Then we show that their “ends” form four required sets
with big curvature. To be more precise, the four ends will have small
perimeter, but according to the following claim, it implies small diameter as
well. In fact, this claim holds for any metric on the sphere, not necessary
the closed convex surface.

3.2. Claim. For any $\varepsilon>0$ there is a $\delta>0$, such that any
simple curve in $\Sigma$ of length smaller than $\delta$ bounds a region of
diameter at most $\varepsilon$.

Proof. We will come to contradiction by showing that there is a point which
cuts $\Sigma$ into connected non-empty parts. Suppose there is a sequence of
curves $\gamma_{n}$ which cuts $\Sigma$ into open regions $A_{n}$ and $B_{n}$
of diameter at least $\varepsilon$ each and such that $\mathop{\rm
length}\nolimits\gamma_{n}\to 0$ as $n\to\infty$. By compactness of $\Sigma$,
we can pass to a subsequence of $\gamma_{n}$ which converges to a point,
denote it by $p$.

Note that for large $n$, each of the regions $A_{n}$ and $B_{n}$ contains a
disk of radius $\tfrac{\varepsilon}{3}$; label their centers by $a_{n}$ and
$b_{n}$ respectively. Indeed, if $n$ is large, then $\gamma_{n}$ lies in
$\tfrac{\varepsilon}{6}$ neighborhood of $p$. Take $a_{n}$ and $b_{n}$ to be
points in the regions maximizing the distance to $p$, note that these
distances should be greater than $\tfrac{\varepsilon}{2}$. Now, note that if
disk of radius $\tfrac{\varepsilon}{3}$ centered at $a_{n}$ intersects
$\gamma_{n}$, then distance from $a_{n}$ to $p$ will be less then
$\tfrac{\varepsilon}{3}+\tfrac{\varepsilon}{6}=\tfrac{\varepsilon}{2}$.

Pass to a subsequence of $\gamma_{n}$ so that $a_{n}$ and $b_{n}$ converge,
denote by $a$ and $b$ their limits. Note that for large $n$ the domains
$A_{n}$ and $B_{n}$ contain the disks of radius $\tfrac{\varepsilon}{4}$
centered at $a$ and $b$ correspondingly.

Since for all large $n$, any path from $a$ to $b$ has to cross $\gamma_{n}$
and the sequence $\gamma_{n}$ converges to $p$, we obtain that any path from
$a$ to $b$ contains $p$. That is, $p$ is a cut point of $\Sigma$, a
contradiction. ∎

Proof of Lemma 3. According to the comparison theorem (2) the surface $\Sigma$
has non-negative curvature in the sense of Alexandrov. Cut $\Sigma$ along a
sufficiently long closed simple geodesic, we get two discs. Choose one of the
discs, say $D$. Note that $D$ is locally convex in $\Sigma$; that is,
sufficiently short minimizing geodesic with the ends in $D$ has to lie in $D$.

Equip $D$ with the intrinsic metric further denoted by $|{*}-{*}|_{D}$. By the
globalization theorem (2), $D$ has non-negative curvature in the sense of
Alexandrov.

$p$$q$$\gamma_{1}(t)$

$\geqslant\tfrac{\pi}{3}$

Choose a pair of points $p,q\in\partial D$ which maximize the distance
$|p-q|_{D}$. Clearly,

$|x-p|_{D}\leqslant|p-q|_{D},\quad|x-q|_{D}\leqslant|p-q|_{D}$

for any other point $x\in\partial D$. By the comparison property (2)

$\measuredangle[x\,^{p}_{q}]\geqslant\tfrac{\pi}{3}.$ (${**}$)

The points $p$ and $q$ divide $\partial D$ into two arcs, say $\gamma_{1}$ and
$\gamma_{2}$; let us parametrize them by arc length from $p$ to $q$. By the
first variation formula (${*}$ ‣ 2), for almost all $t$ we have

$\tfrac{d}{dt}|p-\gamma_{i}(t)|_{D}=-\cos\varphi,$

where $\varphi$ is the angle between the direction of $\gamma_{i}$ at
$x=\gamma_{i}(t)$ and the geodesic $[xp]$. The same way

$\tfrac{d}{dt}|q-\gamma_{i}(t)|_{D}=-\cos\psi,$

where $\psi$ is the angle between the direction of $\gamma_{i}$ at
$x=\nobreak\gamma_{i}(t)$ and the geodesic $[xq]$. By (${**}$ ‣ 3) and the
supplementary property (2) we have $\varphi-\psi\geqslant\tfrac{\pi}{3}$.
Therefore

$\tfrac{d}{dt}\left(|p-\gamma_{i}(t)|_{D}-|q-\gamma_{i}(t)|_{D}\right)=\cos\psi-\cos\varphi\geqslant\tfrac{1}{2}.$

The function $t\mapsto|p-\gamma_{i}(t)|_{D}-|q-\gamma_{i}(t)|_{D}$ ranges from
$-|p-q|_{D}$ to $|p-q|_{D}$. Therefore

$|p-q|_{D}\geqslant\tfrac{1}{4}{{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}}\mathop{\rm length}\nolimits\gamma_{i}$

and therefore

$|p-q|_{D}\geqslant\tfrac{1}{8}{{\hskip 0.5pt\cdot\nobreak\hskip
0.5pt}}\mathop{\rm length}\nolimits[\partial D].$

That is, if the geodesic was long then $D$ has large diameter.

Fix small $\varepsilon>0$ and a positive $\delta\ll\varepsilon$. (The value
$\delta=\varepsilon{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\sin\varepsilon$
will do. It means that in any Euclidean triangle with one side $\varepsilon$
and another $\delta$ the angle opposite to the side of length $\delta$ is less
than $\varepsilon$.)

Let $x$ and $y$ be the first points along $\gamma_{1}$ and $\gamma_{2}$
respectively at distance $\varepsilon$ from $p$. If $|q-x|_{D}>|q-y|_{D}$,
move the point $y$ along $\gamma_{2}$ toward $p$ until $|q-x|_{D}=|q-y|_{D}$;
since $|p-q|_{D}$ is maximal, this distance will be achieved. In case
$|q-x|_{D}<|q-y|_{D}$ do the same for the point $x$. Now we have

$|q-x|_{D}=|q-y|_{D}\geqslant|p-q|_{D}-\varepsilon.$

By the area comparison (2)

$\mathop{\rm area}\nolimits[\tilde{\triangle}qxy]\leqslant\mathop{\rm
area}\nolimits[\triangle qxy]\leqslant\mathop{\rm area}\nolimits\Sigma.$

Since $\tilde{\triangle}qxy$ is isosceles, its area is larger than
$\tfrac{1}{10}{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}|x-y|_{D}{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}|x-q|_{D}$; this holds since $\varepsilon$ is
small. Applying
$(\smash{\raisebox{-2.15277pt}{\begin{tabular}[]{@{}cc@{}}\lx<EMAIL_ADDRESS>*&*\end{tabular}}})$, we get

$\displaystyle|x-y|_{D}$ $\displaystyle\leqslant 100{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}\frac{\mathop{\rm
area}\nolimits\Sigma}{\mathop{\rm length}\nolimits[\partial D]}.$

Therefore, if $D$ has large perimeter then we can assume that
$|x-y|_{D}<\delta$.

Cut $D$ by $[xy]$ and consider the part (a lune) $L_{p}$ with the point $p$ in
it. Note that the curvature of $L_{p}$ is $\alpha+\beta$, where $\alpha$ and
$\beta$ are the angles as on the diagram. By the comparison property
$\alpha\geqslant\tilde{\measuredangle}(x\,^{p}_{y})$ and
$\beta\geqslant\tilde{\measuredangle}(y\,^{p}_{x})$. By Gauss–Bonnet formula,
the curvature of $L_{p}$ is at least
$\pi-\tilde{\measuredangle}(p\,^{x}_{y})>\pi-\varepsilon$.

$D$$L_{p}$$p$$q$$x$$y$$\gamma_{1}$$\gamma_{2}$$\alpha$$\beta$

Assuming $\partial D$ is long, we can find a lune $L_{p}$ with perimeter at
most $2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\varepsilon+\delta$, such that
curvature $L_{p}$ is at least $\pi-\varepsilon$. If $\varepsilon$ is small, by
Lemma 3, $L_{p}$ has small diameter, say
$\varepsilon^{\prime}=\varepsilon^{\prime}(\varepsilon)$.

Using the same construction for $p$ and $q$ in the disc $D$, and for the other
disc, we get four lunes in $\Sigma$, each of diameter at most
$\varepsilon^{\prime}$, and each with curvature at least $\pi-\varepsilon$. By
Gauss–Bonnet formula, the remaining curvature is at most $4{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}\varepsilon$.

Since $\varepsilon>0$ is arbitrary, it follows that support of curvature
measure can be covered by four sets with arbitrarily small diameter; that is,
support of $\kappa$ is a $4$-point set. Clearly each of these points has
curvature $\pi$, the remaining part of $\Sigma$ has vanishing curvature and
therefore flat. ∎

##  4 Isosceles tetrahedron

4.1. Lemma. Assume that a closed convex surface in $\mathbb{E}^{3}$ has $4$
singular points with curvature $\pi$. Then it bounds an isosceles tetrahedron.

$x$$z$$y$

Proof. Denote the surface by $\Sigma$ and by $p$, $x$, $y$ and $z$ the
singular points.

Connect $p$ to the $x$, $y$ and $z$ by three minimizing geodesics $[px]$,
$[py]$ and $[pz]$. Note that these geodesics intersect only at the common
point $p$.

Let us cut $\Sigma$ along three geodesics $[px]$, $[py]$ and $[pz]$. Develop
the obtained flat surface on the plane. Since each point $x$, $y$ and $z$ have
curvature $\pi$, the pairs of cuts at these points open straight. Therefore,
the development forms a triangle; denote it by $\triangle$. The points $x$,
$y$ and $z$ correspond to the midpoints of the sides of $\triangle$ and the
point $p$ correspond to the vertices of $\triangle$.

It follows that $\Sigma$ is isometric to the boundary surface of isosceles
tetrahedron. The statement now follows from the Pogorelov’s uniqueness
theorem, mentioned in the theory overview.

Alternatively one could apply the following exercise and the rigidity theorem
(2) which has a much simpler proof. ∎

4.2. Exercise. Assume that a convex body $K$ is bounded by a closed flat
surface with finite number of singular points. Show that $K$ is a polyhedron.

Acknowledgment. We want to thank Kristof Huszar and Sergei Tabachnikov for
valuable comments on the preliminary version of this note.

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Arseniy Akopyan,

Institute of Science and Technology Austria,

Am Campus 1, 3400 Klosterneuburg, Austria.

Email<EMAIL_ADDRESS>
Anton Petrunin,

Mathematics Department,

Pennsylvania State University,

University Park, PA 16802, USA

Email<EMAIL_ADDRESS>


where the change of $i\to-i$ is a consequence of the time reversal operation.
Following (A.6), the operators (say $\hat{\mathcal{O}}$) that act on the state
vectors in the total Hilbert space become the direct-sum of the operators
corresponding to $\mathcal{H}_{I}$ and $\mathcal{H}_{II}$ as

$\hat{\mathcal{O}}=\hat{\mathcal{O}}_{I}\oplus\hat{\mathcal{O}}_{II}=\begin{pmatrix}\hat{\mathcal{O}}_{I}&0\\\
0&\hat{\mathcal{O}}_{II}\end{pmatrix}$ (A.9)

As we see in (A.9) the direct-sum of operators mean extending the matrix
operators into block diagonal form which act on the higher dimensional state
vectors $|\psi\rangle$ as written in (A.4). From (A.9), we deduce that states
belonging to the direct-sum Hilbert space, evolve according to the following
equation which we call as direct-sum Schrödinger equation

$i\frac{\partial}{\partial t}\begin{pmatrix}|\psi_{I}\rangle&\\\
|\psi_{II}\rangle\end{pmatrix}=\begin{pmatrix}\hat{H}&0\\\
0&-\hat{H}\end{pmatrix}\begin{pmatrix}|\psi_{I}\rangle&\\\
|\psi_{II}\rangle\end{pmatrix}\,,$ (A.10)

which is simply the direct-sum of the Schödinger equations I (A.1) and II
(A.3), respectively. It is worth to note that (A.4) is explicitly
$\mathcal{P}\mathcal{T}$ symmetric, because the direct-sum state vector
$|\psi\rangle$ is $\mathcal{P}\mathcal{T}$ invariant. For physical systems,
whose Hamilitonians are $\mathcal{P}\mathcal{T}$ symmetric, the formulation of
standard Schrödinger equation (3.1) would not yield $\mathcal{P}\mathcal{T}$
symmetric states but direct-sum Schrödinger equation does yield
$\mathcal{P}\mathcal{T}$ symmetric states. Nevertheless, if we restrict to
$\mathcal{P}\mathcal{T}$ symmetric physical systems we cannot see any
difference between results of standard Schrödinger equation (3.1) and the
direct-sum Schrödinger equation. However, in the context of physical systems
whose Hamiltonian breaks the $\mathcal{P}\mathcal{T}$ symmetry (but remain
Hermitian) one might see differences in results, which is for the moment
beyond the scope of the current investigation.

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