# Generalized Geometries Constructed from Differential Forms on Cotangent
Bundle

Radek Suchánek111Department of Mathematics and Statistics, Masaryk University,
Brno, Czech Republic

Abstract. We investigate the landscape of generalized geometries that can be
derived from Monge-Ampère structures. Instead of following the approaches of
Banos, Roubtsov, Kosmann-Schwarzbach, and others, we take a new path, inspired
by the results of Hu, Moraru, and Svoboda. We construct a large family of new
generalized almost geometries derived from non-degenerate 2D symplectic Monge-
Ampère structures and other related geometric objects, such as complex
structures. We demonstrate that, under certain assumptions, non-degenerate
Monge-Ampère structures give rise to quadric surfaces of generalized almost
geometries. Additionally, we discuss the link between Monge-Ampère structures
and Monge-Ampère equations within this framework.

###### Contents

  1. 1 Introduction
  2. 2 Monge-Ampėre structures and equations
    1. 2.1 Monge-Ampėre theory in 2D
    2. 2.2 Field of endomorphisms
  3. 3 Generalized geometry of M-A structures
    1. 3.1 Generalized (almost) structures
      1. 3.1.1 Isotropy, Dirac structures, and integrability
    2. 3.2 Generalized geometry and M-A theory
    3. 3.3 Quadric surfaces of generalized geometries

##  1 Introduction

Construction of generalized structures directly from tensor fields of the
corresponding block representation was considered by many authors [3, 7, 11,
13, 14, 24, 25]. An example of this approach relevant to our case was
Crainic’s paper [7], where the relations for tensor fields defining generic
(integrable) generalized complex structures were described. Using the notion
of a twist of a $2$-form by an endomorphism, Crainic proved the correspondence
between generalized complex structures and Hitchin pairs. By Lychagin’s
results [15, 16, 17], every 2D Monge-Ampère equation can be encoded by a pair
of $2$-forms, which moreover determine a specific endomorphism. Applying
Monge-Ampère theory in the context of generalized complex geometry, Banos
showed in [3] that Monge-Ampère structures of divergence type give rise to
generalized almost complex structures, and integrability of these structures
is connected with the local equivalence problem for 2D symplectic Monge-Ampère
equations with non-vanishing Pfaffian (i.e. elliptic or hyperbolic equations).
This was further used by Kosmann-Schwarzbach and Rubtsov in [14] to deform a
Lie algebroid structure on $\mathbb{T}\left(T^{*}\mathcal{B}\right)$, which
further induces a Courant algebroid structure on
$\mathbb{T}\left(T^{*}\mathcal{B}\right)$ called by the authors Monge-Ampère
Courant algebroid.

In this paper, we explore further the landscape of generalized geometries,
which can be derived from Monge-Ampère structures. There were two apparent
possibilities to do that. Either go into higher dimensions, especially
dimension three, where symplectic classification is still possible or use a
completely different method than Banos did. We took the latter path. Motivated
by the results in [13], where anticommutative pairs and generalized metric
compatible structures are considered, we constructed many new generalized
almost geometries derived from non-degenerate 2D symplectic Monge-Ampère
structures and from other geometric objects, they define (e.g. a complex
structure and a pseudo-Riemannian metric). Some of these generalized almost
geometries can be proven to be integrable by virtue of results in [2, 3, 14,
15, 18]. On the other hand, the standard notion of integrability via Nijenhuis
tensor (or equivalently via Courant involutivity of certain almost Dirac
structures) cannot be applied to many of the geometries constructed by our
method. For example, for non-isotropic cases, one can consider the notion of
weak integrability instead [13, 19].

Original content of the paper. With the help of Monge-Ampère theory [16, 17,
15], we construct a family of generalized almost structures associated to 2D
symplectic M-A structures. We show that, under certain assumptions, non-
degenerate M-A structures give rise to quadric surfaces of generalized almost
geometries. In this framework, we discuss the link between M-A structures and
M-A equations.

##  2 Monge-Ampėre structures and equations

Let $\mathcal{M}$ be a $n$-dimensional symplectic manifold. In this case, $n$
must be an even number. We now define a specific pair of differential forms,
which gives rise to other geometric structures that we will study in the next
sections.

######  Definition 2.1.

A pair
$(\Omega,\alpha)\in\Omega^{2}\left(\mathcal{M}\right)\times\Omega^{n}\left(\mathcal{M}\right)$
is called a Monge-Ampère structure over $\mathcal{M}$ (or just M-A structure),
if $\Omega$ is a symplectic form and

$\displaystyle\Omega\wedge\alpha=0\ .$

If $\mathcal{M}=T^{*}\mathcal{B}$, then $(\Omega,\alpha)$ is called a
symplectic Monge-Ampère structure.

Notation. In our considerations, we will be dealing with symplectic M-A
structures, and the symbol $\Omega$ will always denote the canonical
symplectic form of the cotangent bundle.

###  2.1 Monge-Ampėre theory in 2D

We now consider symplectic 2D M-A structures and show how they look in
canonical symplectic coordinates. We then proceed with the description of the
corresponding M-A operators and equations. We will discuss some unique
features of 2D M-A theory and describe various tensor fields that can be
constructed from (non-degenerate) 2D M-A structures, and which are crucial for
our further considerations.

Let $\dim\mathcal{B}=2$, and recall that $\Omega$ denotes the canonical
symplectic form on the cotangent bundle $T^{*}\mathcal{B}$. In the Darboux
coordinates $x,y,p,q$ ($x,y$ are the base coordinates on $\mathcal{B}$), the
symplectic form writes as

$\displaystyle\Omega=\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}p+\operatorname{\operatorname{d}}y\wedge\operatorname{\operatorname{d}}q\
.$ (1)

Let $\alpha$ be a $2$-form given by

$\displaystyle\begin{split}\alpha&=A\operatorname{\operatorname{d}}p\wedge\operatorname{\operatorname{d}}y+B(\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}p-\operatorname{\operatorname{d}}y\wedge\operatorname{\operatorname{d}}q)\\\
&\qquad+C\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}q+D\operatorname{\operatorname{d}}p\wedge\operatorname{\operatorname{d}}q+E\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}y\
,\end{split}$ (2)

where the coefficients $A,B,C,D,E$ are smooth functions on $T^{*}\mathcal{B}$.
Then $\alpha\wedge\Omega=0$ holds for all possible choice of the coefficients,
and $(\Omega,\alpha)$ is a M-A structure.

Now let $f\colon\mathcal{B}\to\mathbb{R}$ be a smooth function. The
differential of $f$ determines a section,
$\operatorname{\operatorname{d}}f\colon\mathcal{B}\to T^{*}\mathcal{B}$, given
by
$\operatorname{\operatorname{d}}f(x):=\operatorname{\operatorname{d}}_{x}f$,
and we can pullback $\alpha$ onto $\mathcal{B}$ to obtain a top form
$(\operatorname{\operatorname{d}}f)^{*}\alpha\in\Omega^{2}(\mathcal{B})$. Then
the equation

$\displaystyle(\operatorname{\operatorname{d}}f)^{*}\alpha=0$ (3)

defines a nonlinear second-order PDE for $f$ in two variables. If $\alpha$ is
given by (2), then the equation (3) corresponds to 2D symplectic Monge-Ampère
equation

$\displaystyle
Af_{xx}+2Bf_{xy}+Cf_{yy}+D\left(f_{xx}f_{yy}-{f_{xy}}^{2}\right)+E=0\ ,$ (4)

where $f_{xy}:=\frac{\partial^{2}f}{\partial x\partial y}$, and $A,B,C,D,E$
are smooth functions, which depend on $x,y,f_{x},f_{y}$.

In this way, we can represent M-A equations via a pair of differential forms,
the M-A structures. Nonetheless, this representation has a certain ambiguity,
which can be partially removed with the notion of effectivity. The condition
$\Omega\wedge\alpha=O$ can be viewed as a definition of $\alpha$ being
effective.

For a detailed exposition of the theory of Monge-Ampère equations and their
applications, particularly in dimensions two and three, see [15]. For further
details about applications of Monge-Ampère theory, for example in the theory
of complex differential equations, see [4], theoretical meteorology, and
incompressible fluid theory, see [23, 22, 20, 8, 5].

######  Example 2.1.

Let
$\alpha=-\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}q+\operatorname{\operatorname{d}}y\wedge\operatorname{\operatorname{d}}p$.
Then

$\displaystyle(\operatorname{\operatorname{d}}f)^{*}\alpha=-\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}(f_{y})+\operatorname{\operatorname{d}}y\wedge\operatorname{\operatorname{d}}(f_{x})=(-f_{yy}-f_{xx})\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}y\
.$

Hence $(\operatorname{\operatorname{d}}f)^{*}\alpha=0$ amounts to the 2D
Laplace equation $f_{yy}+f_{xx}=0$ and $(\Omega,\alpha)$ is the corresponding
M-A structure. Now let $\alpha=p\operatorname{\operatorname{d}}p\wedge
dy+\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}q$.
Then $(\operatorname{\operatorname{d}}f)^{*}\alpha=0$ describes the von Karman
equation $f_{x}f_{xx}-f_{yy}=0$.

######  Definition 2.2 ([15]).

Pfaffian of a 2D M-A structure, $\operatorname{\operatorname{Pf}}(\alpha)$, is
defined by

$\displaystyle\alpha\wedge\alpha=\operatorname{\operatorname{Pf}}(\alpha)\Omega\wedge\Omega\
.$ (5)

M-A structure is called non-degenerate if the Pfaffian is nowhere-vanishing.
If $\operatorname{\operatorname{Pf}}(\alpha)>0$, then she structure is called
elliptic, and if $\operatorname{\operatorname{Pf}}(\alpha)<0$, then it is
called hyperbolic. We call a non-degenerate structure normalized, if
$|\operatorname{\operatorname{Pf}}(\alpha)|=1$.

######  Remark 2.1.

In the modelling of stably stratified geophysical flows, the Pfaffian is
related with the Rellich’s parameter [26, 9, 21].

For a general 2D symplectic M-A structure, the Pfaffian in canonical
coordinates writes as

$\displaystyle\operatorname{\operatorname{Pf}}(\alpha)=-B^{2}+AC-DE\ .$ (6)

This can be directly checked using coordinate expressions (1) and (2).

######  Example 2.2.

Let $(\Omega,\alpha)$ be the M-A structure described in example 2.1, i.e.
structure for 2D Laplace equation. Then $\operatorname{Pf}(\alpha)=1$. Now
consider the M-A structure for the von Karman equation $f_{x}f_{xx}-f_{yy}=0$.
Then $\operatorname{Pf}(\alpha)=p$. By Lychagin-Rubtsov theorem 2.1 (see
section 2.2), the 2D Laplace equation gives rise to integrable complex
structure, while the von Karman equation does not.

Normalization. A non-degenerate M-A structure can be normalized
$(\Omega,\alpha)\mapsto(\Omega,n(\alpha))$ by

$\displaystyle
n(\alpha):={|\operatorname{\operatorname{Pf}}(\alpha)|}^{-\frac{1}{2}}\alpha$
(7)

Indeed, the Pfaffian of $n(\alpha)$ satisfies
$|\operatorname{\operatorname{Pf}}\left(n(\alpha)\right)|=1$, since

$\displaystyle n(\alpha)\wedge
n(\alpha)={|\operatorname{\operatorname{Pf}}(\alpha)|}^{-1}\alpha\wedge\alpha=\frac{\operatorname{\operatorname{Pf}}(\alpha)}{|\operatorname{\operatorname{Pf}}(\alpha)|}\Omega\wedge\Omega\
,$

which implies

$\displaystyle\operatorname{\operatorname{Pf}}\left(n(\alpha)\right)=\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha)\
.$ (8)

A non-degenerate M-A structure $(\Omega,\alpha)$ and its normalization
$(\Omega,n(\alpha))$ correspond to the same M-A equation, since

$\displaystyle(\operatorname{\operatorname{d}}f)^{*}n(\alpha)=\left.{|\operatorname{\operatorname{Pf}}(\alpha)|}^{-\frac{1}{2}}\right|_{\operatorname{Im}\operatorname{\operatorname{d}}f}(\operatorname{\operatorname{d}}f)^{*}\alpha=0\
.$

######  Remark 2.2.

Notice that we could have rescaled $\alpha$ with an arbitrary non-vanishing
function, which would result in a new M-A structure $(\Omega,\tilde{\alpha})$.
Then, by the same argument as for the normalization, $(\Omega,\tilde{\alpha})$
defines the same M-A equation as $(\Omega,\alpha)$. The main reason we choose
to work with normalized structures is that generalized geometries that we will
construct from $\alpha$ are not invariant with respect to the rescaling. Hence
the normalization condition provides a consistent choice of the representative
in the class $[\alpha]$, where $\tilde{\alpha}\in[\alpha]$, if and only if
$\tilde{\alpha}=e^{h}\alpha$ for some function $h$. Of course, the same
argument also holds for other structures derived from $\alpha$.

###  2.2 Field of endomorphisms

In the following paragraphs, we want to describe how every non-degenerate M-A
structure defines a field of endomorphisms, which square to either
$\operatorname{Id}_{T\left(T^{*}\mathcal{B}\right)}$, or
$-\operatorname{Id}_{T\left(T^{*}\mathcal{B}\right)}$. Before we do so, we
need to fix some notation.

Notation. Tensor $\sigma\in\Gamma\left(T^{*}\mathcal{B}\otimes
T^{*}\mathcal{B}\right)$ can be identified with a
$C^{\infty}(\mathcal{B})$-linear map
$\sigma_{\\#}\colon\Gamma\left(T\mathcal{B}\right)\to\Gamma\left(T^{*}\mathcal{B}\right)$
defined by $\sigma_{\\#}(X):=X\mathbin{\lrcorner}\sigma$. Similarly, if
$\tau\in\Gamma\left(T\mathcal{B}\otimes T\mathcal{B}\right)$, then we denote
by $\tau^{\\#}$ the linear map
$\tau^{\\#}\colon\Gamma\left(T^{*}\mathcal{B}\right)\to\Gamma\left(T\mathcal{B}\right)$,
where $\tau^{\\#}(\xi):=\xi\mathbin{\lrcorner}\tau$. Now consider a non-
degenerate $2$-form $\alpha\in\Omega^{2}(T^{*}\mathcal{B})$, which means that
$\alpha_{\\#}\colon T\mathcal{B}\to T^{*}\mathcal{B}$ is an isomorphism. Using
the inverse $(\alpha_{\\#})^{-1}\colon T^{*}\mathcal{B}\to T\mathcal{B}$, we
define the bivector
$\pi_{\alpha}\in\Gamma\left(\Lambda^{2}TT^{*}\mathcal{B}\right)$ by

$\displaystyle(\pi_{\alpha})^{\\#}:=(\alpha_{\\#})^{-1}\ .$ (9)

For example, if
$\Omega=\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}p+\operatorname{\operatorname{d}}y\wedge\operatorname{\operatorname{d}}q$
is the canonical symplectic form, then
$\pi_{\Omega}=\operatorname{\partial}_{x}\wedge\operatorname{\partial}_{p}+\operatorname{\partial}_{y}\wedge\operatorname{\partial}_{q}$
is the corresponding bivector, where
$\partial_{x}:=\frac{\partial}{\partial_{x}}$ is the $x$-coordinate vector
field (and similarly for the other fields $\partial_{i}$).

When working with matrices, we will use the underline notation to make the
distinction between a morphism and its matrix representation. For example, if
$\rho\in\operatorname{End}\left(T\left(T^{*}\mathcal{B}\right)\right)$, then
corresponding matrix will be denoted $\underline{\rho}$ (which will always be
understood with respect to the canonical coordinates of $\Omega$). Also, we
will be omitting the $\\#$ symbol when dealing with the morphisms derived from
$2$-forms (and $2$-vectors), e.g. the matrix of $\alpha_{\\#}$ will be denoted
simmply $\underline{\alpha}$. Similarly, the matrix of $(\pi_{\alpha})^{\\#}$
will be denoted simply by $\underline{\alpha}^{-1}$ (see defining equation
(9)). When dealing with generalized structures, we write $\mathbb{J}$ for
coordinate-free description, as well as for the coordinate description via
matrices. Since block description of generalized structure either contains
coordinate-free objects or their matrices, and the distinction between the two
is clear in our notation, there is not much space for confusion.

Almost complex and almost product structure. Let
$\rho\in\operatorname{End}\left(TT^{*}\mathcal{B}\right)$ be an endomorphism
defined by

$\displaystyle\rho:={|\operatorname{\operatorname{Pf}}(\alpha)|}^{-\frac{1}{2}}\pi_{\Omega}^{\\#}\circ\alpha_{\\#}\
.$ (10)

If $(\Omega,\alpha)$ is elliptic, then
$\rho^{2}=-\operatorname{Id}_{T\left(T^{*}\mathcal{B}\right)}$, if
$(\Omega,\alpha)$ is hyperbolic, then
$\rho^{2}=\operatorname{Id}_{T\left(T^{*}\mathcal{B}\right)}$. The fact that
$\rho$ is either an almost complex structure on $T^{*}\mathcal{B}$ (if
$\operatorname{\operatorname{Pf}}(\alpha)>0$), or an almost product structure
(if $\operatorname{\operatorname{Pf}}(\alpha)<0$), was proven in [18]. In the
canonical coordinates,

$\displaystyle\underline{\rho}={|\operatorname{\operatorname{Pf}}(\alpha)|}^{-\frac{1}{2}}\begin{pmatrix}B&-A&0&-D\\\
C&-B&D&0\\\ 0&E&B&C\\\ -E&0&-A&-B\end{pmatrix}$ (11)

and, as expected, it follows that

$\displaystyle\underline{\rho}^{2}=\frac{-\operatorname{\operatorname{Pf}}(\alpha)}{|\operatorname{\operatorname{Pf}}(\alpha)|}\operatorname{Id}_{T\left(T^{*}\mathcal{B}\right)}=-\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha)\operatorname{Id}_{T\left(T^{*}\mathcal{B}\right)}\
.$

Notice that $\rho$ is invariant of the normalization, since (8) yields

$\displaystyle\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}\left(n(\alpha)\right)=\operatorname{\operatorname{sgn}}^{2}\operatorname{\operatorname{Pf}}(\alpha)=\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha)\
.$

Integrability. V. Lychagin and V. Rubtsov showed in [18] that there is a
direct link between the local equivalence of M-A equations and integrability
of the $\rho$ structure derived from the corresponding M-A structure
$(\Omega,\alpha)$. Moreover, the integrability condition can be expressed as a
certain closedness condition.

######  Proposition 2.1 (Lychagin-Rubtsov [18]).

A 2D symplectic M-A equation $(\operatorname{\operatorname{d}}f)^{*}\alpha=0$
can be locally transformed via a symplectic transformation to either the
Laplace equation $\Delta f=0$, or the wave equation $\Box f=0$, if and only if
the $\rho$ structure is integrable, which is equivalent to
$\frac{\alpha}{\sqrt{|\operatorname{\operatorname{Pf}}(\alpha)|}}$ being
closed.

##  3 Generalized geometry of M-A structures

We recall some key definitions and objects from generalized geometry. To avoid
any confusion, we formulate these notions for a general smooth manifold
$\mathcal{M}$ and then choose $\mathcal{M}=T^{*}\mathcal{B}$ to investigate
the generalized geometry of symplectic 2D M-A structures and 2D incompressible
fluid flows.

###  3.1 Generalized (almost) structures

A generalized tangent bundle over $\mathcal{M}$ is the vector bundle

${\mathbb{T}\mathcal{M}:=T\mathcal{M}\oplus T^{*}\mathcal{M}}$${\mathcal{M}\
,}$$\scriptstyle{\pi}$ (12)

with the bundle projection $\pi$ defined by the the composition

${\mathbb{T}\mathcal{M}}$${T\mathcal{M}}$${\mathcal{M}\ ,}$

where the left map is the projection on the first factor of the Whitney sum,
and the second map is the tangent bundle projection on $\mathcal{M}$. The
pairing between vector fields and $1$-forms endows $\mathbb{T}\mathcal{M}$
with a non-degenerate, symmetric, bilinear form $\eta$

$\displaystyle\eta\left(\left(X,\xi\right),\left(Y,\zeta\right)\right):=\frac{1}{2}\left(\xi(Y)+\zeta(X)\right)\
,$ (13)

which defines on $\mathbb{T}\mathcal{M}$ a pseudo-Riemannian metric of
signature $(n,n)$. If we choose a coordinate system $(q^{\mu})$ on
$\mathcal{M}$, then the corresponding coordinate vector fields and $1$-forms
define a local basis $\left((\partial_{q^{\mu}},0),(0,dq^{\mu})\right)$ of
$\mathbb{T}\mathcal{M}$. The matrix representation of $\eta$ in this basis is

$\displaystyle\eta=\begin{pmatrix}0&\mathbb{1}\\\ \mathbb{1}&0\end{pmatrix}\
.$

A generalized almost complex structure on $\mathcal{M}$ is a bundle map
$\mathbb{J}\colon\mathbb{T}\mathcal{M}\to\mathbb{T}\mathcal{M}$ such that
$\mathbb{J}^{2}=-\operatorname{Id}_{\mathbb{T}\mathcal{M}}$ and for all
$\left(X,\xi\right),\left(Y,\zeta\right)\in\mathbb{T}\mathcal{M}$

$\displaystyle\eta\left(\mathbb{J}\left(X,\xi\right),\mathbb{J}\left(Y,\zeta\right)\right)=\eta\left(\left(X,\xi\right),\left(Y,\zeta\right)\right)\
,$ (14)

where $\eta$ is the natural inner product (13) [11, 12].

######  Example 3.1.

Let $(\Omega,\alpha)$ be a non-degenerate M-A structure with $\alpha$ closed
(i.e. a pair of symplectic structures). Consider a $(1,1)$-tensor
$A_{\alpha}:=\pi_{\Omega}^{\\#}\circ\alpha_{\\#}$ (c.f. definition (10)). Then
$(\Omega,A_{\alpha})$ defines a Hitchin pair of $2$-forms in the sense of
Crainic [7, 14] since

$\displaystyle\Omega_{\\#}A_{\alpha}=A_{\alpha}^{*}\Omega_{\\#}\ .$

Using the notion of Hitchin pairs, B. Banos showed in [3] that every 2D non-
degenerate M-A structure satisfying for appropriate $\phi\in
C^{\infty}\left(T^{*}\mathcal{B}\right)$ the divergence condition

$\displaystyle\operatorname{\operatorname{d}}(\alpha+\phi\Omega)=0\ ,$

yields an integrable generalized almost structure $\mathbb{J}_{\alpha}$ given
as follows

$\displaystyle\mathbb{J}_{\alpha}=\begin{pmatrix}A_{\alpha}&\pi_{\Omega}^{\\#}\\\
-\left(\Omega_{\\#}+\Omega_{\\#}A_{\alpha}^{2}\right)&-A_{\alpha}^{*}\end{pmatrix}$

The result described in example 3.1 was a key motivation for our further
investigations of other possibilities of constructing generalized (almost)
geometries from Monge-Ampère equations and the corresponding Monge-Ampère
structures. In order to state our result, we need to extend the notion of a
generalized (almost) complex structure by the following definition.

######  Definition 3.1 ([13]).

A generalized almost structure on $\mathcal{M}$ (or a generalized almost
geometry on $\mathcal{M}$) is a bundle map
$\mathbb{J}\colon\mathbb{T}\mathcal{M}\to\mathbb{T}\mathcal{M}$ such that

$\displaystyle\mathbb{J}^{2}=\gamma_{1}\operatorname{id}_{\mathbb{T}M}\ ,$
$\displaystyle\mathbb{J}^{\bullet}\eta=\gamma_{2}\eta\ ,$

where $\gamma_{1},\gamma_{2}\in\\{-1,1\\}$ and
$\mathbb{J}^{\bullet}\eta\left(\left(X,\xi\right),\left(Y,\zeta\right)\right):=\eta\bigl{(}\mathbb{J}\left(X,\xi\right),\mathbb{J}\left(Y,\zeta\right)\bigr{)}$.
Table 1 describes the four possible choices of constants $\gamma_{i}$.

$(\gamma_{1},\gamma_{2})$ |  $(1,1)$ |  $(1,-1)$ |  $(-1,1)$ |  $(-1,-1)$
---|---|---|---|---
type of $\mathbb{J}$ |  GaP |  GaPC |  GaC |  GaAC
Table 1:  Type of a generalized almost structure depending on
$(\gamma_{1},\gamma_{2})$.

The abbreviations stand for generalized almost product (GaP), generalized
almost complex (GaC), generalized almost para-complex (GaPC), and generalized
almost anti-complex (GaAC) structure. A generalized structure is called non-
degenerate, if its eigenbundles are isomorphic to $T\mathcal{M}$ (if
$\mathbb{J}^{2}=\operatorname{Id}_{\mathbb{T}\mathcal{M}}$), or to
$T\mathcal{M}\otimes\mathbb{C}$ (if
$\mathbb{J}^{2}=-\operatorname{Id}_{\mathbb{T}\mathcal{M}}$)

######  Proposition 3.1.

Let $J,P\in\operatorname{End}(T\mathcal{M})$ be an almost complex and almost
product structures, respectively. Let $\alpha\in\Omega^{2}(\mathcal{M})$ be a
non-degenerate $2$-form, and $g\in S^{2}(\mathcal{M})$ a non-degenerate
symmetric bilinear form on $\mathcal{M}$. Suppose that
$\mathbb{J}_{J},\mathbb{J}_{P},\mathbb{J}_{\alpha},\mathbb{J}_{g}\in\operatorname{End}(\mathbbold{T}\mathcal{M})$
are given as follows

$\displaystyle\begin{split}\mathbb{J}_{J}=\begin{pmatrix}J&0\\\ 0&\epsilon
J^{*}\end{pmatrix}\
,\qquad\qquad\mathbb{J}_{\alpha}=\begin{pmatrix}0&\pi_{\alpha}^{\\#}\\\
\epsilon\alpha_{\\#}&0\end{pmatrix}\ ,\\\ \mathbb{J}_{P}=\begin{pmatrix}P&0\\\
0&\epsilon P^{*}\end{pmatrix}\
,\qquad\qquad\mathbb{J}_{g}=\begin{pmatrix}0&\pi_{g}^{\\#}\\\ \epsilon
g_{\\#}&0\end{pmatrix}\ ,\end{split}$ (15)

where $\epsilon\in\\{-1,1\\}$. Then
$\mathbb{J}_{J},\mathbb{J}_{P},\mathbb{J}_{\alpha},\mathbb{J}_{g}$ are
generalized almost structures and they satisfy

$\displaystyle\begin{split}-{\mathbb{J}_{J}}^{2}&={\mathbb{J}_{P}}^{2}=\operatorname{Id}_{\mathbb{T}M}\
,\
\qquad\qquad-\mathbb{J}_{J}^{\bullet}\eta=\mathbb{J}_{P}^{\bullet}\eta=\epsilon\eta\
,\\\
{\mathbb{J}_{\alpha}}^{2}&={\mathbb{J}_{g}}^{2}=\epsilon\operatorname{Id}_{\mathbb{T}M}\
,\qquad\qquad-\mathbb{J}_{\alpha}^{\bullet}\eta=\mathbb{J}_{g}^{\bullet}\eta=\epsilon\eta\
.\end{split}$ (16)

Types of these structures are given in the following table.

$\operatorname{\operatorname{sgn}}{\epsilon}$ |  $\mathbb{J}_{J}$ |  $\mathbb{J}_{P}$ |  $\mathbb{J}_{\alpha}$ |  $\mathbb{J}_{g}$
---|---|---|---|---
$+$ |  GaAC |  GaP |  GaPC |  GaP
$-$ |  GaC |  GaPC |  GaC |  GaAC
Table 2:  Type of a generalized almost structure depending on
$\operatorname{\operatorname{sgn}}\epsilon$.

###### Proof.

Since $-J^{2}=P^{2}=\operatorname{Id}_{T\mathcal{M}}$ and
${J^{*}}^{2}={J^{2}}^{*}$, it is easy to see that
$-{\mathbb{J}_{J}}^{2}={\mathbb{J}_{P}}^{2}=\operatorname{Id}_{\mathbb{T}\mathcal{M}}$.
Now we recall (9), which means

$\displaystyle g_{\\#}\pi_{g}^{\\#}$
$\displaystyle=\operatorname{Id}_{T^{*}\mathcal{M}}=\alpha_{\\#}\pi_{\alpha}^{\\#}\
,$ (17) $\displaystyle\pi_{g}^{\\#}g_{\\#}$
$\displaystyle=\operatorname{Id}_{T\mathcal{M}}=\pi_{\alpha}^{\\#}\alpha_{\\#}\
.$ (18)

This implies
${\mathbb{J}_{\alpha}}^{2}={\mathbb{J}_{g}}^{2}=\epsilon\operatorname{Id}_{\mathbb{T}M}$.
We proceed with the compatibility of $\mathbb{J}_{J}$ with $\eta$. From (13)
we have

$\displaystyle\mathbb{J}_{J}^{\bullet}\eta\left((X,\xi),(Y,\zeta)\right)=\epsilon\left(J^{*}\xi(JY)+J^{*}\zeta(JX)\right)\
.$

Using the definition of the dual map, we arrive at

$\displaystyle\mathbb{J}_{J}^{\bullet}\eta\left((X,\xi),(Y,\zeta)\right)=\epsilon\left(\xi(A^{2}Y)+\zeta(A^{2}X)\right)=-\epsilon\eta\left((X,\xi),(Y,\zeta)\right)\
.$

The computation for $\mathbb{J}_{P}$ is completely analogous with the only
difference coming from $P^{2}=-J^{2}$, resulting in
$\mathbb{J}_{P}^{\bullet}\eta=\epsilon\eta$. The computation for
$\mathbb{J}_{\alpha}$ (and $\mathbb{J}_{g}$) is slightly different. We have

$\displaystyle(\mathbb{J}_{\alpha})^{\bullet}\eta\left((X,\xi),(Y,\zeta)\right)=\frac{\epsilon}{2}\left(\alpha_{\\#}X(\pi_{\alpha}^{\\#}\zeta)+\alpha_{\\#}Y(\pi_{\alpha}^{\\#}\xi)\right)\
.$

Since $\alpha_{\\#}X=\alpha(X,-)$, and due to antisymmetry of $\alpha$,

$\displaystyle\frac{\epsilon}{2}\left(\alpha_{\\#}X(\pi_{\alpha}^{\\#}\zeta)+\alpha_{\\#}Y(\pi_{\alpha}^{\\#}\xi)\right)$
$\displaystyle=\frac{\epsilon}{2}\left(\alpha(X,\pi_{\alpha}^{\\#}\zeta)+\alpha(Y,\pi_{\alpha}^{\\#}\xi)\right)$
$\displaystyle=\frac{-\epsilon}{2}\left((\alpha_{\\#}\pi^{\\#}\zeta)X+(\alpha_{\\#}\pi_{\alpha}^{\\#}\xi)Y\right)\
.$

Using the relations (17) once again, we obtain

$\displaystyle(\mathbb{J}_{\alpha})^{\bullet}\eta\left((X,\xi),(Y,\zeta)\right)=-\epsilon\eta\left((X,\xi),(Y,\zeta)\right)\
.$

The computation for $\mathbb{J}_{g}$ differs from the case of
$\mathbb{J}_{\alpha}$ only in the symmetry of $g$. Thus
$-\mathbb{J}_{J}^{\bullet}\eta=\mathbb{J}_{P}^{\bullet}\eta=\epsilon\eta$, and
$-\mathbb{J}_{\alpha}^{\bullet}\eta=\mathbb{J}_{g}^{\bullet}\eta=\epsilon\eta$.
The table 2 summarizes the resulting type of structures according to
definition 3.1. ∎

######  Example 3.2.

Let $g$ be a non-degenerate symmetric bilinear form. Consider
$\mathbb{A}=\begin{pmatrix}0&\pi_{g}^{\\#}\\\ g_{\\#}&0\\\ \end{pmatrix}.$ w
$\mathbb{A}$ satisfies $\mathbb{A}=-\operatorname{Id}_{\mathbb{T}\mathcal{M}}$
and $\mathbb{A}^{\bullet}\eta=\eta$, so $\mathbb{A}$ is a generalized almost
product structure. The $\pm 1$-eigenbundles of $\mathbb{A}$ are

$\displaystyle E_{\pm}=\\{(X,\pm g_{\\#}X)|X\in\Gamma(T\mathcal{M})\\}\ .$

Indeed, for $x_{+}=(X,g_{\\#}X)\in E_{+}$, and $y_{-}=(Y,-g_{\\#}Y)\in E_{-}$,
it holds $\mathbb{A}x_{+}=(X,g_{\\#}X)=x_{+}$, and
$\mathbb{A}y=(-Y,g_{\\#}Y)=-y_{-}$. Now suppose $x_{+},y_{+}\in E_{+}$. Then

$\displaystyle\eta(x_{+},y_{+})=\frac{1}{2}\left((g_{\\#}X)Y+(g_{\\#}Y)X\right)=g(X,Y)\
.$

Obviously there are $X,Y\in\Gamma(T\mathcal{M})$ such that $g(X,Y)\neq 0$,
thus $E_{+}$ is not totally isotropic, and hence $\mathbb{A}$ is a non-
isotropic structure.

Coordinate description. Since
$\left(\Omega_{\\#}(\partial_{i})\right)(\partial_{j})=\Omega(\partial_{i},\partial_{j})$,
we have

$\displaystyle\underline{\Omega}=\begin{pmatrix}0&\mathbb{1}\\\
-\mathbb{1}&0\end{pmatrix}\ .$

Similarly for the matrix of $\pi_{\Omega}^{\\#}$ we have

$\displaystyle\underline{\pi_{\Omega}}^{\\#}=\underline{\Omega}^{-1}\ .$

The same holds true for arbitrary non-degerate $2$-form $\alpha$. Hecne we
will denote the matrices of $\alpha_{\\#}$ and $\pi_{\alpha}^{\\#}$ simply by
$\underline{\alpha}$ and $\underline{\alpha}^{-1}$, respectively. The
situation does not change when dealing with a symmetric non-degenerate
$2$-tensor $g$. Finally, we recall that for an endomorphism $A$, the matrix of
$A$ and the matrix of $A^{*}$ are related by the transpose
$\underline{A}^{*}=\underline{A}^{T}$. Thus the generalized almost structures
described in lemma 3.1, which are the building blocks for our subsequent
constructions, are written in the canonical coordinates as

$\displaystyle\mathbb{J}_{J}=\begin{pmatrix}J&0\\\ 0&\epsilon
J^{T}\end{pmatrix}\
,\qquad\qquad\mathbb{J}_{\alpha}=\begin{pmatrix}0&\underline{\alpha}^{-1}\\\
\epsilon\underline{\alpha}&0\end{pmatrix}\ ,$
$\displaystyle\mathbb{J}_{P}=\begin{pmatrix}P&0\\\ 0&\epsilon
P^{T}\end{pmatrix}\
,\qquad\qquad\mathbb{J}_{g}=\begin{pmatrix}0&\underline{g}^{-1}\\\
\epsilon\underline{g}&0\end{pmatrix}\ ,$

The Courant bracket. The space of sections
$\Gamma\left(\mathbb{T}\mathcal{M}\right)=\Gamma\left(T\mathcal{M}\right)\oplus\Gamma\left(T^{*}\mathcal{M}\right)$
is equipped with the antisymmetric Courant bracket $[-,-]_{C}$ [6]

$\displaystyle[\left(X,\xi\right),\left(Y,\zeta\right)]_{C}:=\left([X,Y],\mathcal{L}_{X}\zeta-\mathcal{L}_{Y}\xi-\frac{1}{2}\operatorname{\operatorname{d}}(X\mathbin{\lrcorner}\zeta-Y\mathbin{\lrcorner}\xi)\right)\
,$ (19)

where $[X,Y]$ is the Lie bracket and $\mathcal{L}$ is the Lie derivative. Note
that the Courant bracket, which is an extension of the Lie bracket for vector
fields, does not satisfy the Jacobi identity. More importantly for our
considerations, the Courant bracket can be used to define integrability of
isotropic structures on $\mathbb{T}\mathcal{M}$.

####  3.1.1 Isotropy, Dirac structures, and integrability

Every generalized almost structure comes together with the corresponding
subbundles $E_{+},E_{-}$, which are $\pm 1$-eigenbundles if
$\mathbb{J}^{2}=\operatorname{id}_{\mathbb{T}\mathcal{M}}$, and $\pm
i$-eigenbundles if
$\mathbb{J}^{2}=-\operatorname{Id}_{\mathbb{T}\mathcal{M}}$.

######  Definition 3.2.

A subbundle $E\subset\mathbb{T}\mathcal{M}$ is called totally isotropic
(w.r.t. the inner product $\eta$), if for all $x,y\in E:\eta(x,y)=0$. Totally
isotropic $E$ is called an almost Dirac structure on $\mathcal{M}$ if
$\operatorname{rank}E=\operatorname{rank}T\mathcal{M}$. A Dirac structure on
$\mathcal{M}$ is an almost Dirac structure $E$ such that $[E,E]_{C}\subset E$.

######  Remark 3.1.

A Dirac structure on $\mathcal{M}$ can be equivalently defined as a totally
isotropic subbundle $E\subset\mathbb{T}\mathcal{M}$ of maximal rank, which is
involutive with respect to the Courant bracket.

The four possible generalized almost structures determined by definition 3.1
can be divided into two subsets depending on whether the eigenbundles
$E_{\pm}$ are almost Dirac structures or not.

######  Definition 3.3.

Let $\mathbb{J}$ be a generalized almost structure. If the eigenbundles
$E_{\pm}$ are almost Dirac structures, then $\mathbb{J}$ is called isotropic
(with respect to $\eta$). Otherwise $\mathbb{J}$ is called non-isotropic.

The involutivity condition of almost Dirac structures $E_{\pm}$ can serve as a
definition of integrability only for isotropic $\mathbb{J}$ (see remark 3.2
below).

######  Definition 3.4.

An isotropic generalized almost structure
$\mathbb{J}\in\operatorname{End}(\mathbb{T}\mathcal{M})$ is called integrable
if the corresponding eigenbundles $E_{+},E_{-}$ are Dirac structures. An
integrable generalized almost structure is called a generalized structure.

Let $\mathbb{J}\in\operatorname{End}(\mathbb{T}\mathcal{M})$ be an isotropic
generalized almost structure. Then the torsion of $\mathbb{J}$ is defined for
all $x,y\in\Gamma(\mathbb{T}\mathcal{M})$ by

$\displaystyle
N_{\mathbb{J}}(x,y):=[\mathbb{J}x,\mathbb{J}y]_{C}+\mathbb{J}^{2}[x,y]_{C}-\mathbb{J}\left([\mathbb{J}x,y]_{C}+[x,\mathbb{J}y]_{C}\right)\
.$ (20)

This $(1,2)$-tensor
$N_{\mathbb{J}}\colon\Gamma(\mathbb{T}\mathcal{M})\otimes\Gamma(\mathbb{T}\mathcal{M})\to\Gamma(\mathbb{T}\mathcal{M})$
is called (generalized) Nijenhuis tensor. An isotropic $\mathbb{A}$ is
integrable if and only if the coresponding Nijenhuis tensor vanishes:
$N_{\mathbb{A}}(x,y)=0$ for all $x,y\in\Gamma(\mathbb{T}\mathcal{M})$ [7].

######  Remark 3.2.

If a generalized almost structure $\mathbb{J}$ is non-isotropic, then the
eigenbundles $E_{\pm}$ are not totally isotropic (see example 3.2) and the
Courant bracket is not well-defined on them. Thus the torsion (20) for non-
isotropic structures is not well-defined as a tensor. As a consequence, the
notion of integrability, which is usually given either by the condition of
vanishing Nijenhuis tensor or by definition 3.4, cannot be applied to non-
isotropic structures.

######  Remark 3.3.

Authors of [13] considered the notion of weak integrablity, which they defined
for a commuting pair consisting of an indefinite generalized metric
$\mathbb{G}$ and an arbitrary generalized structure $\mathbb{J}$. This notion
of weak integrability can be applied to non-isotropic structures as well but
requires the existence of a generalized Bismut connection222A generalized
Bismut connection associated to a generalized metric is a Courant algebroid
connection that parallelizes the generalized metric [10]. The extension for
indefinite generalized metrics was given in [13]. $\mathfrak{D}$ associated to
a generalized metric $\mathbb{G}$ [10, 13]. By definition, the weak
integrability condition is $\mathfrak{D}\mathbb{J}=0$ [13]. In the case of
generalized Kähler and generalized para-Kähler structures, the integrability
condition introduced in [10] for the case of generalized Kähler structures
(and extended to the case of generalized para-Kähler structures in [13])
implies weak integrability [13].

###  3.2 Generalized geometry and M-A theory

Recall that $\mathcal{B}$ is $2$-dimensional and $\Omega$ denotes the
canonical symplectic form on $T^{*}\mathcal{B}$.

######  Proposition 3.2.

Let
$(\Omega,\alpha)\in\Omega^{2}\left(T^{*}\mathcal{B}\right)\times\Omega^{2}\left(T^{*}\mathcal{B}\right)$
be a non-degenerate Monge-Ampère structure,
$\rho\in\operatorname{End}\left(TT^{*}\mathcal{B}\right)$ the corresponding
endomorphism defined by (10). Then

$\displaystyle\mathbb{J}_{\rho}=\begin{pmatrix}\rho&0\\\
0&\epsilon_{1}\rho^{*}\ ,\end{pmatrix}$
$\displaystyle\mathbb{J}_{\alpha}=\begin{pmatrix}0&\pi_{\alpha}^{\\#}\\\
\epsilon_{2}\alpha_{\\#}&0\end{pmatrix}\ ,$
$\displaystyle\mathbb{J}_{\Omega}=\begin{pmatrix}0&\pi_{\Omega}^{\\#}\\\
\epsilon_{3}\Omega_{\\#}&0\end{pmatrix}\ ,$ (21)

are generalized almost structures.

  1. 1.

If the M-A structure is elliptic, $\operatorname{Pf}(\alpha)>0$, then
$\mathbb{J}_{\rho}$ is a GaPC structure for $\epsilon_{1}=1$, and it is a GaC
structure for $\epsilon_{1}=-1$.

  2. 2.

If the M-A structure is hyperbolic, $\operatorname{Pf}(\alpha)<0$, then
$\mathbb{J}_{\rho}$ is a GaP structure for $\epsilon_{1}=1$, and it is a GaPC
structure for $\epsilon_{1}=-1$.

  3. 3.

The types of $\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}$ are independent of
$\operatorname{\operatorname{sgn}}\left(\operatorname{Pf}(\alpha)\right)$. The
types are determined by the value of $\epsilon_{1}$, and $\epsilon_{2}$,
respectively, and are summarized in the table 2. The two structures never
coincide, $\mathbb{J}_{\alpha}\neq\mathbb{J}_{\Omega}$.

###### Proof.

Firstly notice that the assumption of non-degeneracy of the M-A structure
assures that $\mathbb{J}_{\rho}$ and $\mathbb{J}_{\alpha}$ can be considered.
This can be seen from

$\displaystyle\det\underline{\alpha}={\operatorname{\operatorname{Pf}}(\alpha)}^{2}\
,$ (22)

and recalling that non-degeneracy means that
$\operatorname{\operatorname{Pf}}(\alpha)$ is nowhere vanishing.

For the sake of completeness, we proceed with showing that $\rho$ is either an
almost complex structure if $\operatorname{\operatorname{Pf}}(\alpha)>0$, or
an almost product structure if $\operatorname{\operatorname{Pf}}(\alpha)<0$.
The equation (10) writes
$\underline{\alpha}=\sqrt{|\operatorname{\operatorname{Pf}}(\alpha)|}\underline{\rho}^{T}\underline{\Omega}$,
which implies
$\underline{\rho}=\frac{1}{\sqrt{|\operatorname{\operatorname{Pf}}(\alpha)|}}(\underline{\alpha}\underline{\Omega}^{-1})^{T}$.
Because $\underline{\Omega},\underline{\alpha}$ are antisymmetric matrices,
and $\underline{\Omega}^{-1}=-\underline{\Omega}$, we obtain
${\underline{\rho}}^{2}=\frac{1}{|\operatorname{\operatorname{Pf}}(\alpha)|}(\underline{\Omega}\underline{\alpha})^{2}$.
Since

$\displaystyle\underline{\Omega}=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\
-1&0&0&0\\\ 0&-1&0&0\end{pmatrix}\ ,$
$\displaystyle\underline{\alpha}=\begin{pmatrix}0&E&B&C\\\ -E&0&-A&-B\\\
-B&A&0&D\\\ -C&B&-D&0\end{pmatrix}\ ,$ (23)

we have $(\underline{\Omega}\underline{\alpha})^{2}=(B^{2}-AC+DE)\mathbb{1}$.
Using the coordinate expression for Pfaffian,
$\operatorname{\operatorname{Pf}}(\alpha)=-B^{2}+AC-DE$, we arrive at
${\underline{\rho}}^{2}=-\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha)\mathbb{1}$.
Thus ${\underline{\rho}}^{2}=\mathbb{1}$ for
$\operatorname{\operatorname{Pf}}<0$, which amounts to a hyperbolic M-A
structures, and ${\underline{\rho}}^{2}=-\mathbb{1}$ for
$\operatorname{\operatorname{Pf}}>0$, which defines an elliptic M-A structure.
These two cases are the only possible results for ${\underline{\rho}}^{2}$ due
to the non-degeneracy of the M-A structure $(\Omega,\alpha)$. Now let
$T\in\operatorname{Gl}(4,\mathbb{R})$ be a transformation (acting on a fiber
of $TM$ above some point of $M$), which maps the basis $b$ induced by the
canonical coordinates, to some other basis $\tilde{b}$. Then the matrix of
$\rho$ w.r.t. the $\tilde{b}$ basis satisfies
$\underline{\tilde{\rho}}=T\underline{\rho}T^{-1}$, which implies
${\underline{\tilde{\rho}}}^{\
2}=T{\underline{\rho}}^{2}T^{-1}=\pm\mathbb{1}$. Thus
$\rho^{2}=\pm\operatorname{Id}_{TM}$.

That $\mathbb{J}_{\rho}$ is a GaP structure for $\epsilon_{1}=1$, and a GaPC
structure for $\epsilon_{1}=-1$, now follows from the proposition 3.1.
Similarly, the types of generalized almost structures $\mathbb{J}_{\alpha}$
and $\mathbb{J}_{\Omega}$ are given by the table 2. The fact that this is
independent of $\operatorname{\operatorname{Pf}}(\alpha)$ follows from the
proof of the aforementioned proposition, as well as from the assumption
$\operatorname{\operatorname{Pf}}(\alpha)\neq 0$, which assures that $\alpha$
is non-degenerate, and thus satisfies the assumptions of the proposition.
Indeed, if $\operatorname{\operatorname{Pf}}(\alpha)\neq 0$, then from (22)
follows that $\alpha_{\\#}$ is invertible and hence $\alpha$ is non-
degenerate.

Finally, the definition 2.1 requires $\alpha\wedge\Omega=0$. The non-
degeneracy of the symplectic form $\Omega$ implies that $\alpha$ cannot be
written as a sum of two forms $\alpha=\alpha_{1}+\alpha_{2}$, such that one of
the summands is colinear with $\Omega$. Thus the structures always satisfy
$\mathbb{J}_{\alpha}\neq\mathbb{J}_{\Omega}$. ∎

######  Remark 3.4.

A coordinate-free proof of $\rho$ being either an almost complex structure on
$T^{*}\mathcal{B}$, if $\operatorname{\operatorname{Pf}}(\alpha)>0$, or an
almost product structure if $\operatorname{\operatorname{Pf}}(\alpha)<0$, can
be found in [15].

###  3.3 Quadric surfaces of generalized geometries

Consider a smooth manifold equipped with three almost complex structures
$I,J,K$ satisfying $IJ+JI=0$ and $K=IJ$. The manifold is then called an almost
hypercomplex manifold. On a hypercomplex manifold, there is a $2$-sphere of
almost complex structures
$\\{a_{1}I+a_{2}J+a_{3}K|\sum_{i=1}^{3}{a_{i}}^{2}=1\\}$. This follows from
the fact that the above relations between $I,J,K$ imply anticommutativity of
the triple $\\{I,J\\}=\\{J,K\\}=\\{I,K\\}=0$. We will see that a similar but
much richer situation happens when we consider a pair-wise anticommutative
triple of generalized geometries constructed from tensors associated with 2D
non-degenerate M-A structures.

######  Proposition 3.3.

Let
$(\Omega,\alpha)\in\Omega^{2}\left(T^{*}\mathcal{B}\right)\times\Omega^{2}\left(T^{*}\mathcal{B}\right)$
be a non-degenerate Monge-Ampère structure,
$\rho\in\operatorname{End}(TT^{*}\mathcal{B})$ the corresponding endomorphism
defined by (10). Then the generalized almost structures
$\mathbb{J}_{\rho},\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}$ given by (21)
pair-wise anticommute, if and only if $\epsilon_{1}=-1$ and

$\displaystyle\epsilon_{3}\pi_{\alpha}^{\\#}\Omega_{\\#}+\epsilon_{2}\pi_{\Omega}^{\\#}\alpha_{\\#}=0\
.$ (24)

In canonical coordinates, the condition (24) is equivalent with

$\displaystyle B^{2}-AC+DE=-\epsilon_{2}\epsilon_{3}\ ,$ (25)

where $A,B,C,D,E\in C^{\infty}\left(T^{*}\mathcal{B}\right)$ are coefficients
of $\alpha$ in the canonical basis.

######  Remark 3.5.

In the proof of proposition 3.2, we have shown that
$\operatorname{\operatorname{Pf}}(\alpha)=-B^{2}+AC-DE$. Thus the condition
(25) can be expressed as
$\operatorname{\operatorname{Pf}}(\alpha)=\epsilon_{2}\epsilon_{3}$.

###### Proof.

The anticommutators are

$\displaystyle\\{\mathbb{J}_{\rho},\mathbb{J}_{\alpha}\\}$
$\displaystyle=\begin{pmatrix}0&\rho\pi_{\alpha}^{\\#}+\epsilon_{1}\pi_{\alpha}^{\\#}\rho^{*}\\\
\epsilon_{1}\epsilon_{2}\rho^{*}\alpha_{\\#}+\epsilon_{2}\alpha_{\\#}\rho&0\end{pmatrix}\
,$ $\displaystyle\\{\mathbb{J}_{\rho},\mathbb{J}_{\Omega}\\}$
$\displaystyle=\begin{pmatrix}0&\rho\pi_{\Omega}^{\\#}+\epsilon_{1}\pi_{\Omega}^{\\#}\rho^{*}\\\
\epsilon_{1}\epsilon_{3}\rho^{*}\Omega_{\\#}+\epsilon_{3}\Omega_{\\#}\rho&0\end{pmatrix}\
,$ $\displaystyle\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}$
$\displaystyle=\begin{pmatrix}\epsilon_{3}\pi_{\alpha}^{\\#}\Omega_{\\#}+\epsilon_{2}\pi_{\Omega}^{\\#}\alpha_{\\#}&0\\\
0&\epsilon_{2}\alpha_{\\#}\pi_{\Omega}^{\\#}+\epsilon_{3}\Omega_{\\#}\pi_{\alpha}^{\\#}\end{pmatrix}\
.$

The condition $\\{\mathbb{J}_{\rho},\mathbb{J}_{\alpha}\\}=0$ requires
$\rho\pi_{\alpha}^{\\#}+\epsilon_{1}\pi_{\alpha}^{\\#}\rho^{*}=0$. Multiplying
this equation with $\alpha_{\\#}$ from the right and left (i.e. pre-
composition and composition), we obtain
$\alpha_{\\#}\rho+\epsilon_{1}\rho^{*}\alpha_{\\#}=0$. The dual equation is
$\rho^{*}\alpha_{\\#}^{*}+\epsilon_{1}\alpha_{\\#}^{*}\rho=0$, which, by
antisymmetry of $\alpha$, is equivalent with
$\epsilon_{1}\epsilon_{2}\rho^{*}\alpha_{\\#}+\epsilon_{2}\alpha_{\\#}\rho=0$.
Similarly

$\displaystyle\rho\pi_{\Omega}^{\\#}+\epsilon_{1}\pi_{\Omega}^{\\#}\rho^{*}=0\iff\epsilon_{1}\rho^{*}\Omega_{\\#}+\Omega_{\\#}\rho=0\
.$

Thus the first two anticommutators vanish, if and only if

$\displaystyle\begin{split}\epsilon_{1}\rho^{*}\alpha_{\\#}+\alpha_{\\#}\rho&=0\
,\\\ \epsilon_{1}\rho^{*}\Omega_{\\#}+\Omega_{\\#}\rho&=0\ .\end{split}$ (26)

Working with the canonical coordinates, the system (26) writes

$\displaystyle\frac{-1}{\sqrt{|\operatorname{\operatorname{Pf}}(\alpha)|}}\underline{\alpha}\underline{\Omega}\left(\epsilon_{1}\underline{\alpha}+\underline{\alpha}\right)$
$\displaystyle=0\ ,$
$\displaystyle\frac{-1}{\sqrt{|\operatorname{\operatorname{Pf}}(\alpha)|}}{\underline{\Omega}}^{2}\left(\epsilon_{1}\underline{\alpha}+\underline{\alpha}\right)$
$\displaystyle=0\ .$

Due to non-degeneracy of the M-A structure $(\Omega,\alpha)$, the above
equations hold, if and only if $\epsilon_{1}=-1$. The remaining condition
$\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}=0$ is captured by the equation
(24) since

$\displaystyle\epsilon_{3}\pi_{\alpha}^{\\#}\Omega_{\\#}+\epsilon_{2}\pi_{\Omega}^{\\#}\alpha_{\\#}=0\iff\epsilon_{3}\pi_{\alpha}^{\\#}\Omega_{\\#}+\epsilon_{2}\pi_{\Omega}^{\\#}\alpha_{\\#}=0\
.$

This follows directly from (9). In matrix notation, (24) is equivalent with

$\displaystyle\left(\underline{\alpha}\underline{\Omega}\right)^{2}=-\epsilon_{2}\epsilon_{3}\mathbb{1}\
.$

Recalling (23), we can write $\underline{\alpha}$ and $\underline{\Omega}$ in
the block form

$\displaystyle\underline{\alpha}=\begin{pmatrix}P&Q\\\ -Q^{T}&R\end{pmatrix}$
$\displaystyle\underline{\Omega}=\begin{pmatrix}0&\mathbb{1}\\\
-\mathbb{1}&0\end{pmatrix}\ ,$ (27)

where $P^{T}=-P$ and $R^{T}=-R$. The equation (27) writes

$\displaystyle\begin{pmatrix}Q^{2}-PR&-QP-PQ^{T}\\\
RQ+Q^{T}R&-RP+(Q^{T})^{2}\end{pmatrix}=-\epsilon_{2}\epsilon_{3}\mathbb{1}\ .$

The conditions $-QP-PQ^{T}=RQ+Q^{T}R=0$ are satisfied automatically, since

$\displaystyle P=\begin{pmatrix}0&E\\\ -E&0\end{pmatrix}\ ,$ $\displaystyle
Q=\begin{pmatrix}B&C\\\ -A&-B\end{pmatrix}\ ,$ $\displaystyle
R=\begin{pmatrix}0&D\\\ -D&0\end{pmatrix}\ ,$

where $A,B,C,D,E$ are entries of $\underline{\alpha}$, i.e. coefficients of
$\alpha$ in the canonical basis. We also notice that

$\displaystyle Q^{2}-PR=-\epsilon_{2}\epsilon_{3}\mathbb{1}\iff-
RP+(Q^{T})^{2}=-\epsilon_{2}\epsilon_{3}\mathbb{1}\ ,$

which is due to antisymmetry of $A$ and $C$, and $B^{2}$ being diagonal.
Finally

$\displaystyle Q^{2}-PR=\begin{pmatrix}B^{2}-AC+DE&0\\\
0&B^{2}-AC+DE\end{pmatrix}=-\epsilon_{2}\epsilon_{3}\mathbb{1}\ .$

This concludes the proof. ∎

######  Example 3.3.

Let $(\Omega,\alpha)$ be a non-degenerate M-A structure. We will search for
$2$-forms $\alpha$, which satisfy the condition (24) and commute with the
canonical symplectic form in the sense
$[\underline{\alpha},\underline{\Omega}]=0$, where the matrices are given by
the canonical basis of $\Omega$. The commutativity condition is quite
arbitrary at this point, but becomes more relevant if one is interested in
construction of hyper-(para-)complex structures, or variations of generalized-
Kähler structures on $T\oplus T^{*}$. To obtain even more specific family of
solutions to (24), we further choose $\epsilon_{2}=-\epsilon_{3}$. Then the
condition (24) becomes $\underline{\alpha}^{T}=\underline{\alpha}^{-1}$. Hence
we search for antisymmetric elements of the orthogonal group
$\operatorname{O}(4)$, which commute with $\underline{\Omega}$. If we take
$Q\in\operatorname{O}(2)$, then $\underline{\alpha}:=\begin{pmatrix}0&Q\\\
-Q^{T}&0\end{pmatrix}$ satisfies
$\underline{\alpha}^{T}=-\underline{\alpha}=\underline{\alpha}^{-1}$. The
commutativity with $\underline{\Omega}$ then forces $\det Q=-1$, which amounts
to $Q=\begin{pmatrix}a&b\\\ b&-a\end{pmatrix}$ for $a,b\in\mathbb{R}$ such
that $a^{2}+b^{2}=1$. Comparing this with the general form of
$\underline{\alpha}$ described in (23), we obtain

$\displaystyle\underline{\alpha}=\begin{pmatrix}0&0&B&-A\\\ 0&0&-A&-B\\\
-B&A&0&0\\\ A&B&0&0\end{pmatrix}\ .$

Therefore, the coefficients of $\alpha$ must satisfy $A=-C$, $D=E=0$, and
$\operatorname{\operatorname{Pf}}(\alpha)=-A^{2}-B^{2}=-1$. This is in
concordance with (25). The corresponding Monge-Ampère equation, which is
determined from $(\Omega,\alpha)$ by (3), is

$\displaystyle Af_{xx}+2(1-A^{2})f_{xy}-Af_{yy}=0\ ,$

where $A\in C^{\infty}(B)$. For example, if we choose $|A|=1$, we obtain the
wave equation $f_{xx}=f_{yy}$.

In the above example, we have described a family of non-degenerate M-A
structures (parameterized by smooth functions $A,B$ satisfying
$A^{2}+B^{2}=1$), which, according to proposition 3.3, determine the
corresponding family of anticommutative pairs
$\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}$ of GaPC and GaC structures for
$(\epsilon_{2},\epsilon_{3})=(1,-1)$ (or GaC and GaPC structures for
$(\epsilon_{2},\epsilon_{3})=(-1,1)$). Choosing $\epsilon_{1}=-1$ extends the
pair $\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}$ to a mutually anticommutative
triple of generalized almost structures, with the third structure being the
GaPC structure $\mathbb{J}_{\rho}$ (since
$\operatorname{\operatorname{Pf}}(\alpha)<0$ \- see proposition 3.2). In the
following, we will show how pair-wise anticommutative triples give rise to
certain quadric surfaces of generalized almost structures.

######  Proposition 3.4.

Let
$(\Omega,\alpha)\in\Omega^{2}\left(T^{*}\mathcal{B}\right)\times\Omega^{2}\left(T^{*}\mathcal{B}\right)$
be a non-degenerate M-A structure, and let
$\rho\in\operatorname{End}(TT^{*}\mathcal{B})$ be given by (10). If (24)
holds, then

$\displaystyle\mathbb{A}:=\begin{pmatrix}a_{1}\rho&a_{2}\pi_{\alpha}^{\\#}+a_{3}\pi_{\Omega}^{\\#}\\\
a_{2}\epsilon_{2}\alpha_{\\#}+a_{3}\epsilon_{3}\Omega_{\\#}&-a_{1}\rho^{*}\end{pmatrix}\
,$ (28)

where $a_{i}\in\mathbb{R}\ \forall i$, is a generalized almost structure, if
and only if

$\displaystyle
k:=-\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha){a_{1}}^{2}+{a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3}$
(29)

satisfies $|k|=1$.

  1. 1.

If $(\Omega,\alpha)$ is elliptic, then there are the following quadrics of
generalized almost structures in
$\mathcal{A}:=\operatorname{span}_{\mathbb{R}}\\{\mathbb{J}_{\rho},\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}$:

    1. (a)

for $k=1$, there are two $1$-sheeted hyperboloids, a $2$-sheeted hyperboloid
and, a $2$-sphere of GaPC structures in $\mathcal{A}$,

    2. (b)

for $k=-1$, there are two $2$-sheeted hyperboloids and a $1$-sheeted
hyperboloid of GaC structures in $\mathcal{A}$.

  2. 2.

If $(\Omega,\alpha)$ is hyperbolic, then

    1. (a)

for $k=1$, there are two $2$-sheeted hyperboloids and a $1$-sheeted
hyperboloid of GaPC structures in $\mathcal{A}$,

    2. (b)

for $k=-1$, there are two $1$-sheeted hyperoloids, a $2$-sheeted hyperboloid,
and a $2$-sphere of GaC structures in $\mathcal{A}$.

Before we move to the proof of the proposition 3.4, we also want to emphasize
that statements 1. and 2. are separate cases and should not be mixed together,
for example, when searching for a generalized Kähler structure associated to
$(\Omega,\alpha)$. This is because a given non-degenerate M-A structure has
only one value of
$\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha)$,
distinguishing the elliptic case from the hyperbolic one.

###### Proof.

We have
$\mathbb{A}=a_{1}\mathbb{J}_{\rho}+a_{2}\mathbb{J}_{\alpha}+a_{3}\mathbb{J}_{\Omega}$,
where $\mathbb{J}_{\rho},\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}$ are given by
(21) with $\epsilon_{1}=-1$. Then
$\mathbb{A}^{\bullet}\eta={a_{1}}^{2}\mathbb{J}_{\rho}^{\bullet}\eta+{a_{2}}^{2}\mathbb{J}_{\alpha}^{\bullet}\eta+{a_{3}}^{2}\mathbb{J}_{\Omega}^{\bullet}\eta$
and

$\displaystyle\begin{split}\mathbb{A}^{2}&={a_{1}}^{2}{\mathbb{J}_{\rho}}^{2}+{a_{2}}^{2}{\mathbb{J}_{\alpha}}^{2}+{a_{3}}^{2}{\mathbb{J}_{\Omega}}^{2}\\\
&\quad+a_{1}a_{2}\\{\mathbb{J}_{\rho},\mathbb{J}_{\alpha}\\}+a_{1}a_{3}\\{\mathbb{J}_{\rho},\mathbb{J}_{\Omega}\\}+a_{2}a_{3}\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}\
.\end{split}$ (30)

By proposition 3.3, our assumptions gives
$\\{\mathbb{J}_{\rho},\mathbb{J}_{\alpha}\\}=\\{\mathbb{J}_{\rho},\mathbb{J}_{\Omega}\\}=\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}=0$.
Thus, using relations (16), we obtain

$\displaystyle\mathbb{A}^{2}={a_{1}}^{2}{\mathbb{J}_{\rho}}^{2}+{a_{2}}^{2}{\mathbb{J}_{\alpha}}^{2}+{a_{3}}^{2}{\mathbb{J}_{\Omega}}^{2}\
,$
$\displaystyle\mathbb{A}^{\bullet}\eta={a_{1}}^{2}\mathbb{J}_{\rho}^{\bullet}\eta-\left({a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3}\right)\eta\
.$

By proposition 3.1 and proposition 3.2, the type of generalized almost
structure $\mathbb{J}_{\rho}$ depends on the Pfaffian of the M-A structure
$(\Omega,\alpha)$ as follows

$\displaystyle{\mathbb{J}_{\rho}}^{2}=\begin{cases}\ \ \
\operatorname{Id}_{\mathbb{T}\mathcal{B}}\text{ for
}\operatorname{\operatorname{Pf}}(\alpha)<0\\\
-\operatorname{Id}_{\mathbb{T}\mathcal{B}}\text{ for
}\operatorname{\operatorname{Pf}}(\alpha)>0\end{cases}$
$\displaystyle\mathbb{J}_{\rho}^{\bullet}\eta=\begin{cases}-\eta\text{ for
}\operatorname{\operatorname{Pf}}(\alpha)<0\\\ \ \ \eta\text{ for
}\operatorname{\operatorname{Pf}}(\alpha)>0\end{cases}$

where
$\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha):=\frac{\operatorname{\operatorname{Pf}}(\alpha)}{|\operatorname{\operatorname{Pf}}(\alpha)|}$
(which is well-defined as we assume the M-A structure to be non-degenerate,
$\operatorname{\operatorname{Pf}}(\alpha)\neq 0$). Hence we arrive at

$\displaystyle\begin{split}\mathbb{A}^{2}&=\left(-\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha){a_{1}}^{2}+{a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3}\right)\operatorname{Id}_{\mathbb{T}\mathcal{B}}\
,\\\
\mathbb{A}^{\bullet}\eta&=\left(-\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha){a_{1}}^{2}+{a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3}\right)(-\eta)\
,\end{split}$ (31)

with the immediate consequence that $\mathbb{A}$ is a generalized almost
structure if and only if $|k|=1$. We proceed with the proof of statements 1.
and 2.

1\. $\operatorname{\operatorname{Pf}}(\alpha)>0$: elliptic M-A structures. (a)
For $k=1$, we see from (31) that
$\mathbb{A}^{2}=\operatorname{Id}_{\mathbb{T}\mathcal{B}}$ and
$\mathbb{A}^{\bullet}\eta=-\eta$, i.e. $\mathbb{A}$ is a GaPC structure.
Moreover, positive Pfaffian implies the coefficients of $\mathbb{A}$ must
satisfy $-{a_{1}}^{2}+{a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3}=1$. This
yields two $1$-sheeted hyperboloids, a $2$-sheeted hyperboloid, and a
$2$-sphere, depending on the value of $\epsilon_{2}$ and $\epsilon_{3}$, which
is summarized in table 3.

$\operatorname{\operatorname{sgn}}(\epsilon_{2},\epsilon_{3})$ |  condition on $(a_{1},a_{2},a_{3})$ |  quadric of GaPC structures
---|---|---
$(+,+)$ |  $-{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}=1$ |  $1$-sheeted hyperboloid
$(+,-)$ |  $-{a_{1}}^{2}+{a_{2}}^{2}-{a_{3}}^{2}=1$ |  $2$-sheeted hyperboloid
$(-,+)$ |  $-{a_{1}}^{2}-{a_{2}}^{2}+{a_{3}}^{2}=1$ |  $2$-sheeted hyperboloid
$(-,-)$ |  $-{a_{1}}^{2}-{a_{2}}^{2}-{a_{3}}^{2}=1$ |  $\emptyset$
Table 3:  Quadrics of GaPC structures for
$\operatorname{\operatorname{Pf}}(\alpha)>0$.

(b) If $k=-1$, then
$\mathbb{A}^{2}=-\operatorname{Id}_{\mathbb{T}\mathcal{B}}$ and
$\mathbb{A}^{\bullet}\eta=\eta$, meaning $\mathbb{A}$ is a GaC structure, and
the coefficients $a_{i}$ must satisfy
$-{a_{1}}^{2}+{a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3}=-1$. This
implies the existence of quadrics described in table 4.

$\operatorname{\operatorname{sgn}}(\epsilon_{2},\epsilon_{3})$ |  condition on $(a_{1},a_{2},a_{3})$ |  quadric of GaC structures
---|---|---
$(+,+)$ |  $-{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}=-1$ |  $2$-sheeted hyperboloid
$(+,-)$ |  $-{a_{1}}^{2}+{a_{2}}^{2}-{a_{3}}^{2}=-1$ |  $1$-sheeted hyperboloid
$(-,+)$ |  $-{a_{1}}^{2}-{a_{2}}^{2}+{a_{3}}^{2}=-1$ |  $1$-sheeted hyperboloid
$(-,-)$ |  $-{a_{1}}^{2}-{a_{2}}^{2}-{a_{3}}^{2}=-1$ |  $2$-sphere
Table 4:  Quadrics of GaC structures for
$\operatorname{\operatorname{Pf}}(\alpha)>0$.

2\. $\operatorname{\operatorname{Pf}}(\alpha)<0$: hyperbolic M-A structures.
The situation is very similar to the elliptic case. We again obtain either (a)
for $k=1$ a GaPC structure, or, (b) for $k=-1$ a GaC structure. The main
alteration is that the resulting GaPC/GaC structures differ from those
obtained in the elliptic case. This is simply because for a non-degenerate M-A
structure, by definition,
$\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha)$ is
constant. Thus, the distinction between hyperbolic and elliptic case happens
on the level of the M-A structure, with the expected diference between the
GaPC/GaC structures corresponding to different couples $(\Omega,\alpha)$. The
quadrics of generalized almost structures arising from hyperbolic M-A
structures are given in table 5 for
$k={a_{1}}^{2}+{a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3}=1$, and in
table 6 for
$k={a_{1}}^{2}+{a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3}=-1$.

$\operatorname{\operatorname{sgn}}(\epsilon_{2},\epsilon_{3})$ |  condition on $(a_{1},a_{2},a_{3})$ |  quadric of GaPC structures
---|---|---
$(+,+)$ |  ${a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}=1$ |  $2$-sphere
$(+,-)$ |  ${a_{1}}^{2}+{a_{2}}^{2}-{a_{3}}^{2}=1$ |  $1$-sheeted hyperboloid
$(-,+)$ |  ${a_{1}}^{2}-{a_{2}}^{2}+{a_{3}}^{2}=1$ |  $1$-sheeted hyperboloid
$(-,-)$ |  ${a_{1}}^{2}-{a_{2}}^{2}-{a_{3}}^{2}=1$ |  $2$-sheeted hyperboloid
Table 5:  Quadrics of GaPC structures for $\operatorname{\operatorname{Pf}}(\alpha)<0$. $\operatorname{\operatorname{sgn}}(\epsilon_{2},\epsilon_{3})$ |  condition on $(a_{1},a_{2},a_{3})$ |  quadric of GaC structures
---|---|---
$(+,+)$ |  ${a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}=-1$ |  $\emptyset$
$(+,-)$ |  ${a_{1}}^{2}+{a_{2}}^{2}-{a_{3}}^{2}=-1$ |  $2$-sheeted hyperboloid
$(-,+)$ |  ${a_{1}}^{2}-{a_{2}}^{2}+{a_{3}}^{2}=-1$ |  $2$-sheeted hyperboloid
$(-,-)$ |  ${a_{1}}^{2}-{a_{2}}^{2}-{a_{3}}^{2}=-1$ |  $1$-sheeted hyperboloid
Table 6:  Quadrics of GaC structures for
$\operatorname{\operatorname{Pf}}(\alpha)<0$.

∎

A natural question at this point is whether any of the generalized almost
structures described by proposition 3.4 coincide. In the following
proposition, we will show there are no coincidences between structures
parameterized by different points of either the same quadric or two different
quadrics.

######  Proposition 3.5.

Two different points chosen arbitrarily from the quadrics described in
statement 1. (if $\operatorname{\operatorname{Pf}}(\alpha)>0)$, or statement
2. (if $\operatorname{\operatorname{Pf}}(\alpha)<0)$, of proposition 3.4
represent two different generalized almost structures.

###### Proof.

Let $\mathbb{A},\mathbb{B}$ be given by (28). The difference between the
structures may occur via the choice of coefficients $(a_{i}),(b_{i})$, and the
values of epsilons, $\epsilon_{i}^{\mathbb{A}},\epsilon_{i}^{\mathbb{B}}$. The
coefficients $(a_{i}),(b_{i})$ determine the corresponding
$k_{\mathbb{A}},k_{\mathbb{B}}$, defined by (29). Note that both $\mathbb{A}$
and $\mathbb{B}$ are constructed from a single M-A structure
$(\Omega,\alpha)$, so the Pfaffian is fixed and thus
$\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha)$ is
constant. This means that there will always be only one $\rho$ and only one of
the statements 1. and 2. of proposition 3.4 will apply to a given non-
degenerate M-A structure.

Case 1. Suppose $k_{\mathbb{A}}=k_{\mathbb{B}}$ and
$\epsilon_{i}^{\mathbb{A}}=\epsilon_{i}^{\mathbb{B}}$ for $i=2,3$. Then
$\mathbb{A}$ and $\mathbb{B}$ are $\mathbb{R}$-linear combinations of the same
generalized almost structures, i.e.
$\mathbb{A}=a_{1}\mathbb{J}_{\rho}+a_{2}\mathbb{J}_{\alpha}+a_{3}\mathbb{J}_{\Omega}$
and
$\mathbb{B}=b_{1}\mathbb{J}_{\rho}+b_{2}\mathbb{J}_{\alpha}+b_{3}\mathbb{J}_{\Omega}$.
This means

$\displaystyle\mathbb{A}=\mathbb{B}\quad\iff\quad a_{i}=b_{i}\ \forall i\ .$

Note that this case amounts to choosing the structures $\mathbb{A},\mathbb{B}$
on the same quadric.

Case 2. Suppose $k_{\mathbb{A}}=k_{\mathbb{B}}$ but
$\epsilon_{i}^{\mathbb{A}}\neq\epsilon_{i}^{\mathbb{B}}$ for at least one $i$.
Since the structure $\mathbb{J}_{\rho}$ is fixed for both the $\mathbb{A}$ and
$\mathbb{B}$ (due to the same $\operatorname{\operatorname{Pf}}(\alpha)$), we
have $\mathbb{A}=\mathbb{B}$, if and only if $a_{1}=b_{1}$ and

$\displaystyle\begin{pmatrix}0&a_{2}\pi_{\alpha}^{\\#}+a_{3}\pi_{\Omega}^{\\#}\\\
a_{2}\epsilon_{2}^{\mathbb{A}}\alpha_{\\#}+a_{3}\epsilon_{3}^{\mathbb{A}}\Omega_{\\#}&0\end{pmatrix}=\begin{pmatrix}0&b_{2}\pi_{\alpha}^{\\#}+b_{3}\pi_{\Omega}^{\\#}\\\
b_{2}\epsilon_{2}^{\mathbb{B}}\alpha_{\\#}+b_{3}\epsilon_{3}^{\mathbb{B}}\Omega_{\\#}&0\end{pmatrix}\
.$

Comparing the upper right blocks we get

$\displaystyle(a_{2}-b_{2})\pi_{\alpha}^{\\#}=(b_{3}-a_{3})\pi_{\Omega}^{\\#}\
.$ (32)

By definition of the M-A structure $(\Omega,\alpha)$, we have
$\alpha\wedge\Omega=0$. This implies that the dual bivectors satisfy
$\pi_{\alpha}\wedge\pi_{\Omega}=0$. Consequently, $\pi_{\alpha}^{\\#}$ cannot
be written as a non-zero multiple of $\pi_{\Omega}^{\\#}$, and thus (32)
holds, if and only if $a_{2}=b_{2}$ and $a_{3}=b_{3}$.

Case 3. Suppose $k_{\mathbb{A}}\neq k_{\mathbb{B}}$. Then $\mathbb{A}$ is a
generalized almost structure of a different type than $\mathbb{B}$, which
concludes the proof. ∎

Propositions 3.4 and 3.5 give the following theorem.

######  Theorem 3.1.

Let
$(\Omega,\alpha)\in\Omega^{2}\left(T^{*}\mathcal{B}\right)\times\Omega^{2}\left(T^{*}\mathcal{B}\right)$
be a non-degenerate M-A structure on $T^{*}\mathcal{B}$, and let
$\rho\in\operatorname{End}(TT^{*}\mathcal{B})$ be given by (10). Then
$(\Omega,\alpha)$ defines by (28) a family of generalized almost geometries
$\mathbb{A}\in\operatorname{End}\left(\mathbb{T}T^{*}\mathcal{B}\right)$ on
$T^{*}\mathcal{B}$. They are parameterized by quadric surfaces in
$\mathbb{R}^{3}$, and described in tables 3 \- 6. Two different points of any
of the quadrics parameterize two different geometries.

On the necessity of anticommutativity assumptions. Considering the assumptions
of proposition 3.4, we now focus on the question whether the pair-wise
anticommutativity (which was assumed in the corollary by the choice
$\epsilon_{1}=-1$ in (28) and by the equation
$\epsilon_{3}\pi_{\alpha}^{\\#}\Omega_{\\#}+\epsilon_{2}\pi_{\Omega}^{\\#}\alpha_{\\#}=0$)
is a necessary condition for
$\mathbb{A}=a_{1}\mathbb{J}_{\rho}+a_{2}\mathbb{J}_{\alpha}+a_{3}\mathbb{J}_{\Omega}$
to be a generalized almost structure. We want to consider the situation where
all three generalized almost structures
$\mathbb{J}_{\rho},\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}$ given by (21) are
contributing to $\mathbb{A}$. So we assume $a_{i}\neq 0$ for all $i$. Without
any assumption about the anticommutativity, $\mathbb{A}^{2}$ is described by
(30). If $\mathbb{A}$ is a generalized almost structure, then
$\mathbb{A}^{2}=\pm\operatorname{Id}_{\mathbb{T}\mathcal{B}}$ and by
proposition 3.2

$\displaystyle{a_{1}}^{2}{\mathbb{J}_{\rho}}^{2}+{a_{2}}^{2}{\mathbb{J}_{\alpha}}^{2}+{a_{3}}^{2}{\mathbb{J}_{\Omega}}^{2}=c_{1}\operatorname{Id}_{\mathbb{T}\mathcal{B}}\
,$ (33)

for appropriate $c_{1}\in\mathbb{R}$. This means

$\displaystyle
a_{1}a_{2}\\{\mathbb{J}_{\rho},\mathbb{J}_{\alpha}\\}+a_{1}a_{3}\\{\mathbb{J}_{\rho},\mathbb{J}_{\Omega}\\}+a_{2}a_{3}\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}=c_{2}\operatorname{Id}_{\mathbb{T}\mathcal{B}}\
,$

for some $c_{2}\in\mathbb{R}$. Since both products
$\mathbb{J}_{\rho}\mathbb{J}_{\alpha}$ and
$\mathbb{J}_{\rho}\mathbb{J}_{\Omega}$ have only zeros in the diagonal blocks,
the last equation implies

$\displaystyle\\{\mathbb{J}_{\rho},\mathbb{J}_{\alpha}\\}=0\ ,$
$\displaystyle\\{\mathbb{J}_{\rho},\mathbb{J}_{\Omega}\\}=0\ ,$ $\displaystyle
a_{2}a_{3}\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}=c_{2}\operatorname{Id}_{\mathbb{T}\mathcal{B}}\
.$

By proposition 3.3, the first two equations are satisfied, if and only if
$\epsilon_{1}=-1$ in $\mathbb{J}_{\rho}=\begin{pmatrix}\rho&0\\\
0&\epsilon_{1}\rho^{*}\end{pmatrix}$. Moreover, if $\epsilon_{1}=-1$, then
from proposition 3.1 follows that (33) is equivalent to

$\displaystyle
c_{1}=-\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha){a_{1}}^{2}+{a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3}\
.$

Note that $c_{1}$ coincides with $k$ described in (29). Now let us assume that
the equation
$a_{2}a_{3}\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}=c_{2}\operatorname{Id}_{\mathbb{T}M}$
can be solved. As a consequence of
$\mathbb{A}^{2}=\pm\operatorname{Id}_{\mathbb{T}\mathcal{B}}$, the constants
$c_{1},c_{2}$ must satisfy $|c_{1}+c_{2}|=1$. At the same time, proposition
3.1 implies $\mathbb{A}^{\bullet}\eta=-c_{1}\eta$, and we need $|c_{1}|=1$.

Comming back to the equation
$a_{2}a_{3}\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}=c_{2}\operatorname{Id}_{\mathbb{T}\mathcal{B}}$.
In matrices, this corresponds to

$\displaystyle\begin{split}\epsilon_{2}\epsilon_{3}\mathbb{1}+(\underline{\alpha}\underline{\Omega})^{2}&=\frac{-c_{2}\epsilon_{2}}{a_{2}a_{3}}\underline{\alpha}\underline{\Omega},\\\
\epsilon_{2}\epsilon_{3}\mathbb{1}+(\underline{\Omega}\underline{\alpha})^{2}&=\frac{-c_{2}\epsilon_{2}}{a_{2}a_{3}}\underline{\Omega}\underline{\alpha}\
.\end{split}$ (34)

Since
$(\underline{\alpha}\underline{\Omega})^{2}=(\underline{\Omega}\underline{\alpha})^{2}$,
the system (34) can be solved, if and only if
$[\underline{\alpha},\underline{\Omega}]=0$. In this case, the two equations
coincide. Note that the requirement
$[\underline{\alpha},\underline{\Omega}]=0$ is not present if we want
$\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}=0$ (which corresponds to
$c_{2}=0$). This is because the right-hand side of (34) vanish if we rewrite
$\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}=0$ as a matrix equation.

Thus we arrived at the following conclusion. If we do not assume a priori any
relation between $\mathbb{J}_{\rho},\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}$,
then for $\mathbb{A}$ to be a generalized almost structure,
$\mathbb{J}_{\rho}$ must anticommute with $\mathbb{J}_{\alpha}$ and
$\mathbb{J}_{\Omega}$. The assumption
$\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}=0$ can be replaced with
$\\{\mathbb{J}_{\alpha},\mathbb{J}_{\Omega}\\}=c\operatorname{Id}_{\mathbb{T}\mathcal{B}}$,
with the price given by the necessity of
$[\underline{\alpha},\underline{\Omega}]=0$, as well as the requirement
$|c+k|=1$, which is in addition to $|k|=1$, where $k$ is given by (29).

On the link between M-A structures and M-A equations. We have seen in
propositions 3.2 and 3.4 how a non-degenerate M-A structure gives rise to
various generalized almost structures. We have also described the link between
M-A structures and M-A equations, which is provided by (3). Now we want to use
this link to define generalized geometries associated with a given M-A
equation. To do this, we have to take care of ambiguities (and the
corresponding well-definedness issue) associated with the following
observation.

Consider non-degenerate M-A structures $(\Omega,\alpha)$ and
$(\Omega,\tilde{\alpha})$. By definition,
$\alpha\wedge\Omega=0=\tilde{\alpha}\wedge\Omega$. Moreover, for $h\in
C^{\infty}(\mathcal{B})$

$\displaystyle(\operatorname{\operatorname{d}}f)^{*}(h\alpha)=h|_{\operatorname{Im}\operatorname{\operatorname{d}}f}(\operatorname{\operatorname{d}}f)^{*}\alpha\
,$

where $f\in C^{\infty}(\mathcal{B})$ and
$\operatorname{Im}\operatorname{\operatorname{d}}f$ is understood in the sense
$\operatorname{\operatorname{d}}f\colon B\to
T^{*}\mathcal{B},x\mapsto\operatorname{\operatorname{d}}_{x}f$. Consequently,
if $h$ is everywhere non-zero,

$\displaystyle(\operatorname{\operatorname{d}}f)^{*}\alpha=0\quad\iff\quad(\operatorname{\operatorname{d}}f)^{*}(h\alpha)=0\
.$

Thus two non-degenerate M-A structures give rise to the same M-A equation, if
and only if $\tilde{\alpha}=h\alpha$. Denote by $[\alpha]$ the equivalence
class of $2$-forms satisfying $\alpha\wedge\Omega=0$, where

$\displaystyle\tilde{\alpha}\in[\alpha]\quad\overset{\text{def.}}{\iff}\quad\tilde{\alpha}=h\alpha\
,$

for a non-vanishing $h\in C^{\infty}(T^{*}\mathcal{B}$. Then there is the
following $1-1$ correspondence between equivalence classes and M-A equations

$\displaystyle\\{[\alpha]\ |\
\alpha\in\Gamma\left(\Lambda^{2}T^{*}\mathcal{B}\right)\\}\longleftrightarrow\\{(\operatorname{\operatorname{d}}f)^{*}\alpha=0\
|\ f\in C^{\infty}(\mathcal{B})\\}\ .$

To put this in the context of generalized geometries associated with M-A
structures, we want to see how the change of representative in the equivalence
class $[\alpha]$ changes the family of generalized almost structures given by
proposition (3.2). Recall that $\mathbb{A}$ determined by a non-degenerate M-A
structure $(\Omega,\alpha)$ is defined via three generalized almost structures
given by (21) with $\epsilon_{1}=-1$, and satisfying the condition
$\epsilon_{3}\pi_{\alpha}^{\\#}\Omega_{\\#}+\epsilon_{2}\pi_{\Omega}^{\\#}\alpha_{\\#}=0$.

Consider a non-degenerate M-A structure $(\Omega,\tilde{\alpha})$, where
$\tilde{\alpha}\in[\alpha]$. Firstly, it is clear that $\mathbb{J}_{\Omega}$
is the same for both M-A structures, since $\Omega$ remains the same.
Secondly, we compute the change in $\rho$ and the corresponding
$\mathbb{J}_{\rho}$. Starting with the Pfaffian,

$\displaystyle\tilde{\alpha}\wedge\tilde{\alpha}=h^{2}\operatorname{\operatorname{Pf}}(\alpha)\Omega\wedge\Omega\quad\Rightarrow\quad\operatorname{\operatorname{Pf}}(\tilde{\alpha})=h^{2}\operatorname{\operatorname{Pf}}(\alpha)\
.$ (35)

Now using the definition (10), we have

$\displaystyle\tilde{\rho}=\frac{1}{\sqrt{|\operatorname{\operatorname{Pf}}(\tilde{\alpha})|}}\pi_{\Omega}^{\\#}\tilde{\alpha}_{\\#}=\frac{h}{|h|\sqrt{|\operatorname{\operatorname{Pf}}(\alpha)|}}\pi_{\Omega}^{\\#}\alpha_{\\#}=\operatorname{\operatorname{sgn}}h\rho\
,$ (36)

with the immediate consequence

$\displaystyle\mathbb{J}_{\tilde{\rho}}=\operatorname{\operatorname{sgn}}h\mathbb{J}_{\rho}\
.$ (37)

Thirdly, we focus on $\mathbb{J}_{\alpha}$. We have
$\tilde{\alpha}_{\\#}=h\alpha_{\\#}$ and
$\pi_{\tilde{\alpha}}^{\\#}=\frac{1}{h}\pi_{\alpha}^{\\#}$, which follows from
$C^{\infty}$-linearity of tensor fields and the definition of
$\pi_{\alpha}^{\\#}$. Thus $\alpha\mapsto\tilde{\alpha}=h\alpha$ results in

$\displaystyle\mathbb{J}_{\alpha}=\begin{pmatrix}0&\pi_{\alpha}^{\\#}\\\
\epsilon_{2}\alpha^{\\#}&0\end{pmatrix}\quad\longmapsto\quad\begin{pmatrix}0&\frac{1}{h}\pi_{\alpha}^{\\#}\\\
h\epsilon_{2}\alpha^{\\#}&0\end{pmatrix}=\mathbb{J}_{\tilde{\alpha}}$ (38)

Now we are ready to investigate the anticommutativity of generalized almost
strutures associated with $(\Omega,\tilde{\alpha})$. We have
$\\{\mathbb{J}_{\tilde{\rho}},\mathbb{J}_{\Omega}\\}=\operatorname{\operatorname{sgn}}h\\{\mathbb{J}_{\rho},\mathbb{J}_{\Omega}\\}$,
and thus

$\displaystyle\\{\mathbb{J}_{\tilde{\rho}},\mathbb{J}_{\Omega}\\}=0\quad\iff\quad\epsilon_{1}=-1\
.$

Proceeding with the anticommutator of $\mathbb{J}_{\tilde{\rho}}$ and
$\mathbb{J}_{\tilde{\alpha}}$, we have

$\displaystyle\\{\mathbb{J}_{\tilde{\rho}},\mathbb{J}_{\tilde{\alpha}}\\}=\operatorname{\operatorname{sgn}}h\begin{pmatrix}0&\frac{1}{h}\left(\rho\pi_{\alpha}^{\\#}+\epsilon_{1}\pi_{\alpha}^{\\#}\rho^{*}\right)\\\
h\left(\epsilon_{1}\epsilon_{2}\rho^{*}\alpha_{\\#}+\epsilon_{2}\alpha_{\\#}\rho\right)&0\end{pmatrix}\
,$

which implies

$\displaystyle\\{\mathbb{J}_{\tilde{\rho}},\mathbb{J}_{\tilde{\alpha}}\\}=0\quad\iff\quad\begin{cases}0=\rho\pi_{\alpha}^{\\#}+\epsilon_{1}\pi_{\alpha}^{\\#}\rho^{*}\\\
0=\epsilon_{1}\epsilon_{2}\rho^{*}\alpha_{\\#}+\epsilon_{2}\alpha_{\\#}\rho\end{cases}$

Considering the right side of the equivalence, we have seen in the proof of
proposition 3.3 that the two equations are equivalent, and are satisfied, if
and only if $\epsilon_{1}=-1$. Finally, we consider the anticommutator of
$\mathbb{J}_{\tilde{\alpha}}$ and $\mathbb{J}_{\Omega}$, after which we
discuss the overall dependence of $\mathbb{A}$ on the choice of the
representative in the class $[\alpha]$. The anticommutator is

$\displaystyle\\{\mathbb{J}_{\tilde{\alpha}},\mathbb{J}_{\Omega}\\}=\begin{pmatrix}h\epsilon_{3}\pi_{\alpha}^{\\#}\Omega_{\\#}+\frac{\epsilon_{2}}{h}\pi_{\Omega}^{\\#}\alpha_{\\#}&0\\\
0&h\epsilon_{2}\alpha_{\\#}\pi_{\Omega}^{\\#}+\frac{\epsilon_{3}}{h}\Omega_{\\#}\pi_{\alpha}^{\\#}\end{pmatrix}\
.$

Analogously as with the previous anticommutators, vanishing of
$\\{\mathbb{J}_{\tilde{\alpha}},\mathbb{J}_{\Omega}\\}$ boils down to one
equation

$\displaystyle\\{\mathbb{J}_{\tilde{\alpha}},\mathbb{J}_{\Omega}\\}=0\quad\iff\quad
h^{2}\epsilon_{3}\pi_{\alpha}^{\\#}\Omega_{\\#}+\epsilon_{2}\pi_{\Omega}^{\\#}\alpha_{\\#}=0\
.$

In canonical coordinates, this amounts to

$\displaystyle(\underline{\alpha}\underline{\Omega})^{2}=\frac{-\epsilon_{2}\epsilon_{3}}{h^{2}}\mathbb{1}\
.$ (39)

Now we are ready to compare the situation between $(\Omega,\alpha)$ and
$(\Omega,\tilde{\alpha})$.

From the above discussion we see that there are exactly two obstructions for
$(\Omega,\alpha)$ and $(\Omega,\tilde{\alpha})$ to determine the same family
of generalized almost structures. The first obstruction follows from the
change of $\mathbb{J}_{\alpha}$, described in (38). The second obstruction,
which is a consequence of the first one on the level of anticommutators is
contained in the equation (39). Both obstructions can be avoided by choosing
$h=1$, but this corresponds to leaving the representative of $[\alpha]$
unchanged.

######  Proposition 3.6.

Two non-degenerate M-A structures $(\Omega,\alpha)$ and
$(\Omega,\tilde{\alpha})$, where $\tilde{\alpha}\in[\alpha]$, give rise to the
same family of generalized almost structures determined by proposition 3.4, if
and only if $\tilde{\alpha}=\alpha$.

The anticommutativity between $\mathbb{J}_{\tilde{\rho}}$ and
$\mathbb{J}_{\Omega}$ (as well as $\mathbb{J}_{\tilde{\alpha}}$) is satisfied
by the same condition for all representatives of $[\alpha]$, namely by
choosing $\epsilon_{1}=-1$. Since the change of representative of $[\alpha]$
transforms $\mathbb{J}_{\rho}$ according to (37), we arrive at the following

######  Proposition 3.7.

There is a one-to-one correspondence between 2D symplectic non-degenerate M-A
equations and equivalence classes $[\mathbb{A}]$ of generalized almost
structures given by proposition 3.4 with $a_{2}=0$. Representatives of
$[\mathbb{A}]$ have the same type.

###### Proof.

Based on the discussion preceding the proposition, the only thing to notice is
that $a_{2}=0$ means that $\mathbb{A}$ is constructed only from
$\mathbb{J}_{\rho}$ and $\mathbb{J}_{\Omega}$. This further implies that the
condition (24), which is equivalent to anticommutativity between
$\mathbb{J}_{\alpha}$ and $\mathbb{J}_{\Omega}$, does not have to be satisfied
due to (30). Of course, $a_{2}=0$ also affects the quadrics described in
statements 1. and 2. of the proposition. Some of them will cease to exist
(such as the $2$-sheeted hyperboloid of GaPC structures corresponding to
$\operatorname{Pf}(\alpha)>0$ and $(\epsilon_{2},\epsilon_{3})=(1,-1)$) and
those that survive will become quadric curves instead of surfaces. Finally,
$\tilde{\mathbb{A}}\in[\mathbb{A}]\iff\tilde{a}_{1}=\operatorname{\operatorname{sgn}}ha_{1}$
implies together with (35), that the two representatives of $[\mathbb{A}]$
have the same type. ∎

The notion of normalization (7) yields a unified way of choosing a
representative in $[\alpha]$. This leads to the following definition and the
subsequent theorem.

######  Definition 3.5.

Let $(\operatorname{\operatorname{d}}f)^{*}\alpha=0$ be a 2D symplectic Monge-
Ampère equation satisfying $Pf(\alpha)\neq 0$. A generalized almost geometry
associated with the M-A equation is a generalized almost geometry determined
by the normalized M-A structure $(\Omega,n(\alpha))$ corresponding to the
equation.

Now we are ready to summarize by the following theorem, which is based on
propositions 3.6 and 3.7.

######  Theorem 3.2.

Let $(\operatorname{\operatorname{d}}f)^{*}\alpha=0$ be a 2D symplectic Monge-
Ampère equation on $\mathcal{B}$, such that $Pf(\alpha)\neq 0$ everywhere on
$T^{*}\mathcal{B}$. Then there is a unique family of generalized almost
geometries
$\mathbb{A}\in\operatorname{End}\left(\mathbb{T}T^{*}\mathcal{B}\right)$ on
$T^{*}\mathcal{B}$, determined by theorem 3.1, which are associated with the
equation.

## Conlusion and Outlooks

We have described a construction of generalized almost geometries determined
by non-degenerate 2D symplectic Monge-Ampère structures and the corresponding
PDEs. Inspired by the results in [3, 7, 11, 13, 14, 24, 25], we constructed
many new generalized almost geometries derived from Monge-Ampère structures
and from geometric objects they define. We have shown that non-degenerate
Monge-Ampère structures give rise to quadric surfaces of generalized almost
geometries. We also discussed the link between Monge-Ampère structures and
Monge-Ampère equations in this context. In future work, we will be interested
in constructing generalized geometries in dimensions higher than two,
particularly in dimension three, where the situation is much richer. We will
also focus on studying various notions of integrability, especially the notion
of weak integrability in our context, as well as the closely related notion of
Bismut connections [13, 1, 25]. Particularly interesting would be to find
further links between (weak) integrability of generalized structures and the
local equivalence problem of certain Monge-Apmère PDEs in dimension three.

## References

  * [1] A. Andrada and R. Villacampa.  Bismut connection on Vaisman manifolds.  Mathematische Zeitschrift, 302:1091–1126, 2022.
  * [2] B. Banos.  Integrable geometries and Monge-Ampère equations.  arXiv: Differential Geometry, 2006.
  * [3] B. Banos.  Monge–Ampère equations and generalized complex geometry— The two-dimensional case.  Journal of Geometry and Physics, 57(3):841–853, 2007.
  * [4] B. Banos.  Complex solutions of Monge-Ampère equations.  Journal of Geometry and Physics, 61(11):2187–2198, 2011.
  * [5] B. Banos, V. Rubtsov, and I. Roulstone.  Monge–Ampère Structures and the Geometry of Incompressible Flows.  Journal of Physics A: Mathematical and Theoretical, 49, 10 2016\.
  * [6] T. J. Courant.  Dirac Manifolds.  Transactions of the American Mathematical Society, 319(2):631–661, 1990.
  * [7] M. Crainic.  Generalized complex structures and Lie brackets.  Bulletin of the Brazilian Mathematical Society, New Series, 42:559–578, 2004.
  * [8] S. Delahaies.  Complex and contact geometry in geophysical fluid dynamics.  PhD thesis, University of Surrey, 01 2009.
  * [9] D. G. Dritschel and Á. Viúdez.  A balanced approach to modelling rotating stably stratified geophysical flows.  Journal of Fluid Mechanics, 488:123–150, 2003.
  * [10] M. Gualtieri.  Branes on Poisson varieties.  arXiv: Differential Geometry, 2007.
  * [11] M. Gualtieri.  Generalized complex geometry.  Annals of Mathematics, 174(1):75–123, 2011.
  * [12] N. Hitchin.  Generalized Calabi–Yau Manifolds.  The Quarterly Journal of Mathematics, 54(3):281–308, 09 2003.
  * [13] S. Hu, R. Moraru, and D. Svoboda.  Commuting Pairs, Generalized para-Kähler Geometry and Born Geometry.  arXiv:1909.04646, 2019.
  * [14] Y. Kosmann-Schwarzbach and V. Rubtsov.  Compatible Structures on Lie Algebroids and Monge-Ampère Operators.  Acta Applicandae Mathematicae, 109(1):101–135, Jan 2010.
  * [15] A. Kushner, V. Lychagin, and V. Rubtsov.  Contact Geometry and Nonlinear Differential Equations.  Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2006.
  * [16] V. V. Lychagin.  Contact Geometry and Non-Linear Second-Order Differential Equations.  Russian Mathematical Surveys, 34(1):149–180, February 1979.
  * [17] V. V. Lychagin.  Differential equations on two-dimensional manifolds.  Izv. Vyssh. Uchebn. Zaved. Mat., 5:43–57, 1992.
  * [18] V. V. Lychagin and V. N. Rubtsov.  Local classification of Monge-Ampère differential equations.  Sov. Math., Dokl., 28:328–332, 1983.
  * [19] L. Reidel, F. J. Rudolph, and D. Svoboda.  A Unique Connection for Born Geometry .  Communications in Mathematical Physics, 372:119–150, 2019.
  * [20] I. Roulstone, B. Banos, J. D. Gibbon, and V. Roubtsov.  Kähler geometry and Burgers’ vortices.  Proceedings of Ukrainian National Academy Mathematics, 16(2):303 – 321, 2009.
  * [21] V. Rubtsov.  Geometry of Monge–Ampère Structures, pages 95–156.  Springer International Publishing, Cham, 2019.
  * [22] V. Rubtsov and I. Roulstone.  Holomorphic structures in hydrodynamical models of nearly geostrophic flow.  Proc. R. Soc. Lond. A., 457:1519–1531, 06 2001.
  * [23] V. N. Rubtsov and I. Roulstone.  Examples of quaternionic and Kähler structures in Hamiltonian models of nearly geostrophic flow.  Journal of Physics A: Mathematical and General, 30(4):L63–L68, feb 1997.
  * [24] M. Salvai.  Generalized geometric structures on complex and symplectic manifolds.  Annali di Matematica Pura ed Applicata, 194:1505–1525, 2015.
  * [25] I. Vaisman.  Generalized para-Kähler manifolds.  Differential Geometry and its Applications, 42:84–103, 2015.
  * [26] Á. Viúdez and D. G. Dritschel.  An explicit potential-vorticity-conserving approach to modelling nonlinear internal gravity waves.  Journal of Fluid Mechanics, 458:75–101, 2002.