# Generalized Geometry of 2D Incompressible Fluid Flows

Radek Suchánek
Department of Mathematics and Statistics
Masaryk University, Brno

###### Abstract

We describe a family of generalized almost structures associated with a Monge-
Ampère equation for a stream function of a 2D incompressible fluid flows.
Using an indefinite metric field constructed from a pair of $2$-forms related
to the Monge-Ampère equation, we show the existence of generalized metric
compatible structures in our family of generalized structures. Integrability
of isotropic structures on the level of Dirac structures and differential
forms is discussed.

###### Contents

  1. 1 Introduction
  2. 2 Monge-Ampère theory
    1. 2.1 From M-A structures to M-A equations
    2. 2.2 Endomorphisms and symmetric form
  3. 3 Incompressible fluid flows in 2D
    1. 3.1 Kinematics of incompressible fluids
    2. 3.2 M-A equation for the stream function
    3. 3.3 M-A geometry of incompressible fluid flows
  4. 4 Generalized geometry and M-A theory
    1. 4.1 Generalized (almost) structures
    2. 4.2 Isotropy, Dirac structures, and integrability
  5. 5 Generalized geometry of 2D incompressible fluids
    1. 5.1 Structures generated by anticommutative triples
    2. 5.2 Generalized metric compatible structures
    3. 5.3 Algebras generated by generalized structures
  6. 6 Conclusion

##  1 Introduction

In [7], Crainic described conditions on tensor fields to define a generic
generalized complex structure [12, 11], and showed how Hitchin pairs of
differential forms are connected with these structures. B. Banos showed in [2]
that a certain class of non-linear second order PDEs called Monge-Ampère
equations gives rise to Hitchin pairs and the corresponding generalized
complex geometry and generalized Calabi-Yau geometry [10, 16, 11]. The notion
of integrability then translates to a certain local equivalence problem of
Monge-Ampère equations and Jacobi system of non-linear first order PDEs with
special non-linearity [28, 16]. Motivated by these results, we focus on a
different approach towards construction of generalized geometries and show
that the family of generalized geometries described by Banos can by extended
when dealing with incompressible fluids in 2D. In particular, we prove the
existence of generalized Kähler structure and generalized chiral structures,
as well as generalized hyper-complex and generalized hyper-para-complex
triples. Applications of these structures in the context of twisted
supersymmetric non-linear sigma models were studied in [15].

The idea behind searching for geometric structures connected with Monge-Ampère
equations is two fold. Firstly, if one is interested in concrete examples of
generalized structures with specific properties, Monge-Ampère theory provides
numerous examples of geometric objects from which the generalized structures
can be constructed [17, 16, 28]. Secondly, investigating geometry of equations
may lead to better understanding and possible improvement of certain models.
Example of usefulness of this approach is provided in [28, 25, 26, 1, 8, 22,
24].

##  2 Monge-Ampère theory

All our manifolds will be real and smooth. Let $\mathcal{B}$ be (a real,
smooth) two dimensional manifold. Denote by $x,y$ (local) coordinates on
$\mathcal{B}$. In the following definition, we will describe a subclass of
nonlinear second-order PDEs, with the nonlinearities given by minors and
determinant of the hessian matrix.

######  Definition 2.1.

Monge-Ampère equation on $\mathcal{B}$ is the following nonlinear second-order
PDE

$\displaystyle
Af_{xx}+2Bf_{xy}+Cf_{yy}+D\left(f_{xx}f_{yy}-{f_{xy}}^{2}\right)+E=0\ ,$ (1)

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

######  Example 2.1.

We provide a few standard examples of M-A equations. From (1) we see that the
2D Laplace equation $f_{xx}+f_{yy}=0$, as well as the 2D wave equation
$f_{xx}-f_{yy}=0$, are constant coefficients M-A equations. These are
particularly interesting in view of proposition 2.1. Next examples are
$\det\operatorname{Hess}(f)=1$ and $\det\operatorname{Hess}(f)=-1$, which will
play a key role in this paper, as they arise from the Navier-Stokes equations
for the incompressible 2D fluid flows. Moreover, by virtue of proposition 2.1,
these equations are equivalent with the 2D Laplace and wave equations via a
symplectic transformation. Let us end this short overview of M-A equations
with the von Karman equation $f_{x}f_{xx}-f_{yy}=0$. We will see in example
2.3 that the von Karman equation has, in a certain sense, very different
behaviour then the previous examples.

Let $T^{*}\mathcal{B}\xrightarrow{\pi}\mathcal{B}$ be the cotangent bundle
over $\mathcal{B}$. Throughout this paper, $\Omega$ will always denote the
canonical symplectic form on the cotangent bundle. In the following, we will
show how M-A equations can be encoded by a certain pair of differential
$2$-forms on $T^{*}\mathcal{B}$, called M-A structure. We start with the
following observation. In the Darboux coordinates $x,y,p,q$ ($x,y$ are the
base coordinates on $\mathcal{B}$), the canonical symplectic form writes as

$\displaystyle\Omega=\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}p+\operatorname{\operatorname{d}}y\wedge\operatorname{\operatorname{d}}q\
.$ (2)

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

$\displaystyle\begin{split}\alpha&=A\operatorname{\operatorname{d}}p\wedge\operatorname{\operatorname{d}}y+B(\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}p-\operatorname{\operatorname{d}}y\wedge\operatorname{\operatorname{d}}q)\\\
&\qquad+C\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}q+D\operatorname{\operatorname{d}}p\wedge\operatorname{\operatorname{d}}q+E\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}y\
,\end{split}$ (3)

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.

######  Definition 2.2 ([17, 18]).

A pair
$(\Omega,\alpha)\in\Omega^{2}\left(T^{*}\mathcal{B}\right)\times\Omega^{2}\left(T^{*}\mathcal{B}\right)$
is called a 2D Monge-Ampère structure over $T^{*}\mathcal{B}$ (or just M-A
structure), if $\Omega$ is a symplectic form and the effectivity condition
$\alpha\wedge\Omega=0$ is satisfied.

######  Remark 2.1.

Note that the coefficients $A,B,\ldots,E$ in (3) are functions on
$T^{*}\mathcal{B}$, while the coefficients $A,B,\ldots,E$ given in (1) denoted
functions on $\mathcal{B}$. We often denote by the same symbol a function
$H\in C^{\infty}\left(T^{*}\mathcal{B}\right)$, which depends on $x,y,p,q$,
and its restriction to the graph of $\operatorname{\operatorname{d}}f$, which
depends on $x,y,f_{x},f_{y}$. That is, we write $H=H(x,y,p,q)$, as well as
$H=H(x,y,f_{x},f_{y})\equiv(\operatorname{\operatorname{d}}f)^{*}H$.
Similarly, we write $F=F(x,y)\in C^{\infty}\mathcal{B}$ as well as
$F=F(x,y,0,0)\in C^{\infty}\left(T^{*}\mathcal{B}\right)$. The domain of a
function under consideration will always be clear from the context or stated
explicitly.

###  2.1 From M-A structures to M-A equations

We will now describe the link between M-A structures and M-A equations [20,
21]. This will justify the nomenclature of definition 2.2 and also explain the
seemingly unnatural factor of $2$ in the second summand of (1). 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

$\displaystyle(\operatorname{\operatorname{d}}f)^{*}\alpha=0$ (4)

defines a nonlinear second order PDE for $f$. If $\alpha$ is given by (3),
then the equation (4) corresponds exactly to (1). For a detailed exposition on
the theory of Monge-Ampère equations and their applications, particularly in
dimensions two and three, see [17]. For further details about applications of
Monge-Ampère theory, for example in the theory of complex differential
equations see [3], theoretical meteorology, and incompressible fluid theory,
see [27, 26, 25, 8, 1].

######  Example 2.2.

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 2.3 ([17]).

Pfaffian of a 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
$|\operatorname{\operatorname{Pf}}(\alpha)|=1$.

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

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

######  Example 2.3.

Let $(\Omega,\alpha)$ be the M-A structure described in example 2.2, 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 $\operatorname{Pf}(\alpha)=p$. In section 2.2, we will see that the 2D
Laplace equation give rise to integrable complex structure, while the von
Karman equation does not.

######  Remark 2.2.

In the modelling of stably stratified geophysical flows, the Pfaffian is
related with the the Rellich’s parameter [29, 9, 28].

A non-degenerate M-A structure can be normalized
$(\Omega,\alpha)\mapsto(\Omega,n(\alpha))$ by
$n(\alpha):={|\operatorname{\operatorname{Pf}}(\alpha)|}^{-\frac{1}{2}}\alpha$.
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

$\displaystyle\operatorname{\operatorname{Pf}}\left(n(\alpha)\right)=\operatorname{\operatorname{sgn}}\operatorname{\operatorname{Pf}}(\alpha)\
.$ (7)

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 2.3.

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 some of the generalized geometries
constructed 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 $h$.

###  2.2 Endomorphisms and symmetric form

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
$\pi_{\alpha}\in\Gamma\left(\Lambda^{2}T\left(T^{*}\mathcal{B}\right)\right)$
by

$\displaystyle(\pi_{\alpha})^{\\#}:=(\alpha_{\\#})^{-1}\ .$ (8)

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
(8)). When dealing with generalized structures, we write $\mathbb{J}$ for
coordinate-free description, as well as for the coordinate description via
matrices. Nevertheless there is no space for confusion, 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.

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

$\displaystyle\rho:={|\operatorname{\operatorname{Pf}}(\alpha)|}^{-\frac{1}{2}}\pi_{\Omega}^{\\#}\circ\alpha_{\\#}\
.$ (9)

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 [20]. In the
canonical coordinates,

$\displaystyle\underline{\rho}={|\operatorname{\operatorname{Pf}}(\alpha)|}^{-\frac{1}{2}}\begin{pmatrix}B&-A&0&-D\\\
C&-B&D&0\\\ 0&E&B&C\\\ -E&0&-A&-B\end{pmatrix}$ (10)

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 (7) 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 [20] that there is a
direct link between 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 [20]).

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.

Symmetric bilinear form. Every M-A structure defines the following symmetric
bilinear form on $T^{*}\mathcal{B}$ [17, 26]

$\displaystyle g(X,Y):=\frac{2(X\mathbin{\lrcorner}\alpha\wedge
Y\mathbin{\lrcorner}\Omega+Y\mathbin{\lrcorner}\alpha\wedge
X\mathbin{\lrcorner}\Omega)\wedge\pi^{*}\operatorname{vol}}{\Omega\wedge\Omega}\
,$ (11)

where $X,Y\in\Gamma\left(T(T^{*}\mathcal{B})\right)$, $\mathbin{\lrcorner}$ is
the interior product, and
$\pi^{*}\operatorname{vol}\in\Omega^{2}\left(T^{*}\mathcal{B}\right)$ is the
pullback of a locally chosen top form
$\operatorname{vol}\in\Omega^{2}\left(\mathcal{B}\right)$ along the cotangent
bundle projection $\pi\colon T^{*}\mathcal{B}\to\mathcal{B}$. Since in our
considerations the symplectic form is fixed, we see that the above symmetric
field is parameterized by a $2$-form, and the definition (11) amounts to a
mapping $\Omega^{2}\left(T^{*}M\right)\to S^{2}\left(T^{*}\mathcal{B}\right)$,
given by $\alpha\mapsto g$. The matrix of $g$ in canonical coordinates is

$\displaystyle\underline{g}=\begin{pmatrix}2C&-2B&D&0\\\ -2B&2A&0&D\\\
D&0&0&0\\\ 0&D&0&0\end{pmatrix}\ ,$ (12)

where $A,B,C,D$ are the coefficients of $\alpha$ as in (3). Notice the
independence of $\underline{g}$ on $E$. Moreover, it is clear that

$\displaystyle\det\underline{g}\neq 0\quad\iff\quad D\neq 0\ .$

Observe that the coordinate vector fields $\frac{\partial}{\partial
p},\frac{\partial}{\partial q}$ span a $2$-dimensional totally isotropic
subspace of $T(T^{*}\mathcal{B})$. Since
$\operatorname{rank}T(T^{*}\mathcal{B})=4$, it follows that $g$ has signature
$\left(2,2\right)$. Applications of the symmetric form (11) in theoretical
meteorology and the corresponding theory of fluid flows were discussed in [1,
20, 26]. Further information about the properties of $g$, as well as of the
corresponding pullback metric $(df)^{*}g$, in the context of Ricci-flatness
condition and Hessian structures was discussed in our previous work [14].

##  3 Incompressible fluid flows in 2D

The motivation for us to investigate the stream equation arising from the
kinematics of incompressible 2D fluid flows are the works of Banos, Delahaies,
Roubtsov, and Roulstone [1, 8, 25, 27]. In these papers was demonstrated that
modern differential geometric approach offers a new framework for
understanding and studying semi-geostrophic theory and related models. In
particular, the authors have shown the possibility to model coherent
structures in large-scale atmosphero-ocean flows via a hierarchy of
approximations to the Navier-Stokes equations and the related incompressible
fluid flows models.

###  3.1 Kinematics of incompressible fluids

Starting point is the system

$\displaystyle\operatorname{\partial}_{t}\vec{v}+(\vec{v}\cdot\vec{\nabla})\vec{v}$
$\displaystyle=\nu\Delta\vec{v}-\vec{\nabla}P\ ,$ (13)
$\displaystyle\vec{\nabla}\cdot\vec{v}$ $\displaystyle=0$ (14)

where $\vec{v}$ is the velocity field,
$\vec{v}=\left(a_{t}(x,y),b_{t}(x,y)\right)$ ($t$ is understood as a
parameter), $\nu\in\operatorname{\mathbb{R}}$ is the viscosity (which we
assume constant), $P=P_{t}\left(x,y)\right)$ is the pressure field, $\cdot$ is
the dot product,
$\vec{\nabla}=\left(\operatorname{\partial}_{x},\operatorname{\partial}_{y}\right)$,
and the expression $\Delta\vec{v}$ refers to the vector Laplace operator
$\Delta\vec{v}:=\vec{\nabla}\left(\vec{\nabla}\cdot\vec{v}\right)-\vec{\nabla}\times\left(\vec{\nabla}\times\vec{v}\right)$.
Since we are working with $2D$ vectors, the vector product is understood as
the composition

${\operatorname{\mathbb{R}}^{2}}$${\operatorname{\mathbb{R}}^{3}}$${\operatorname{\mathbb{R}}^{3}}$${\operatorname{\mathbb{R}}^{3}}$${\operatorname{\mathbb{R}}^{2}\
,}$$\scriptstyle{\iota}$$\scriptstyle{\vec{\nabla}\times(-)}$$\scriptstyle{\vec{\nabla}\times(-)}$$\scriptstyle{\operatorname{\operatorname{pr}}}$

where $\iota(x,y)=(x,y,0)$ and
$\operatorname{\operatorname{pr}}(x,y,z)=(x,y)$. The Navier-Stokes equation
(13) and the incompressibility constraint (14) together (locally) describe the
kinematics of 2D incompressible fluids flows.

###  3.2 M-A equation for the stream function

Applying $\vec{\nabla}\cdot(-)$ on (13) we obtain

$\displaystyle\operatorname{\partial}_{t}\vec{\nabla}\cdot\vec{v}+\vec{\nabla}\cdot\left((\vec{v}\cdot\vec{\nabla})\vec{v}\right)=\nu\vec{\nabla}\cdot\left(\Delta\vec{v}\right)-\Delta
P\ .$ (15)

Using the incompressibility constraint we have
$\vec{\nabla}\cdot\operatorname{\partial}_{t}\vec{v}=\operatorname{\partial}_{t}\vec{\nabla}\cdot\vec{v}=0$
and

$\displaystyle\vec{\nabla}\cdot\left((\vec{v}\cdot\vec{\nabla})\vec{v}\right)$
$\displaystyle=\left(\operatorname{\partial}_{x}a\right)^{2}+\left(\operatorname{\partial}_{y}b\right)^{2}+2\operatorname{\partial}_{y}a\operatorname{\partial}_{x}b\
,$
$\displaystyle=-2\operatorname{\partial}_{x}a\operatorname{\partial}_{y}b+2\operatorname{\partial}_{y}a\operatorname{\partial}_{x}b\
.$

Hence the left-hand side of the equation (15) can be written as

$\displaystyle\vec{\nabla}\cdot\operatorname{\partial}_{t}\vec{v}+\vec{\nabla}\cdot\left((\vec{v}\cdot\vec{\nabla})\vec{v}\right)=-2\det\begin{pmatrix}\operatorname{\partial}_{x}a&\operatorname{\partial}_{y}a\\\
\operatorname{\partial}_{x}b&\operatorname{\partial}_{y}b\end{pmatrix}$

The term $\nu\vec{\nabla}\cdot\left(\Delta\vec{v}\right)$, appearing on the
right-hand side of (15), vanish since

$\displaystyle\vec{\nabla}\cdot\left(\Delta\vec{v}\right)=\vec{\nabla}\cdot\left[\vec{\nabla}\times\left(\vec{\nabla}\times\vec{v}\right)\right]=\operatorname{\partial}_{xxy}b-\operatorname{\partial}_{xyy}a+\operatorname{\partial}_{xyy}a-\operatorname{\partial}_{xxy}b=0$

The above computation leads to the following Poisson equation as a consequence
of the system of equations (13), (14)

$\displaystyle\Delta
P=2\det\begin{pmatrix}\operatorname{\partial}_{x}a&\operatorname{\partial}_{y}a\\\
\operatorname{\partial}_{x}b&\operatorname{\partial}_{y}b\end{pmatrix}\ .$
(16)

The fluid incompressibility equation (14) can be solved by
$\vec{v}=\left(f_{y},-f_{x}\right)$. This implies that (16) can be written as

$\displaystyle\det\operatorname{Hess}(f)=\frac{1}{2}\Delta P\ .$ (17)

Instead of viewing the above equation as a Poisson equation for $P$, one can
instead interprete it as a Monge-Ampère equation for a function $f$ of two
variables, $f=f(x,y)$, called the stream function. This equation was studied
by Roulston, Banos, Gibbon, and Roubtsov, in the context of semigeostropic
models [25, 23], where the relationship between
$\operatorname{\operatorname{sgn}}\Delta P$ and the balance between the rate
of strain and the enstrophy of the flow was discussed. In order to proceed
further with our geometric approach, we assume $\Delta P\neq 0$.

###  3.3 M-A geometry of incompressible fluid flows

We now move toward the description of the M-A equation (17) via a pair of
$2$-forms $(\Omega,\alpha)$ forming a normalized M-A structure. The
corresponding pair of differential $2$-forms is given by the canonical
symplectic form (2) and

$\displaystyle\alpha=\sqrt{\frac{2}{|\Delta
P|}}\operatorname{\operatorname{d}}p\wedge\operatorname{\operatorname{d}}q-\frac{\Delta
P}{\sqrt{|2\Delta
P|}}\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}y\ ,$
(18)

where $\Delta P\neq 0$ is viewed as a function on $T^{*}\mathcal{B}$, which
coincides with $\Delta P$ from equation (13) (see remark 2.1). The pair
clearly satisfy $\alpha\wedge\Omega=0$, making it an M-A structure. This is in
concordance with general situation described in section 2. Moreover,

$\displaystyle(\operatorname{\operatorname{d}}f)^{*}\alpha$
$\displaystyle=\left(\sqrt{\frac{2}{|\Delta
P|}}(f_{xx}f_{yy}-{f_{xy}}^{2})-\frac{\Delta P}{\sqrt{|2\Delta
P|}}\right)\operatorname{\operatorname{d}}x\wedge\operatorname{\operatorname{d}}y\
.$

This implies that $(\operatorname{\operatorname{d}}f)^{*}\alpha=0$ amounts to
$\det\operatorname{Hess}(f)=\frac{\Delta P}{2}$, and thus the above M-A
structure corresponds to the M-A equation (17). From (18) we read the
coefficients of $\alpha$

$\displaystyle A=0\ ,B=0\ ,C=0\ ,D=\sqrt{\frac{2}{|\Delta P|}}\
,E=\frac{-\Delta P}{\sqrt{|2\Delta P|}}\ .$

By direct computation, or with the help of (6), we see that the Pfaffian is

$\displaystyle\operatorname{\operatorname{Pf}}(\alpha)=\operatorname{\operatorname{sgn}}\Delta
P\ .$

Hence the M-A structure is normalized in the sense of definition 2.3.
Consequently, the matrix of endomorphism $\rho$ is

$\displaystyle\underline{\rho}={|2\Delta
P|}^{-\frac{1}{2}}\begin{pmatrix}0&0&0&-2\\\ 0&0&2&0\\\ 0&-\Delta P&0&0\\\
\Delta P&0&0&0\end{pmatrix}\ .$ (19)

Depending on the $\operatorname{\operatorname{sgn}}\Delta P$, we obtain the
following cases

$\displaystyle\rho^{2}=\begin{cases}-\operatorname{Id}_{T\left(T^{*}\mathcal{B}\right)}&\
\ \ \text{for $\Delta P>0$}\ ,\\\ \ \ \
\operatorname{Id}_{T\left(T^{*}\mathcal{B}\right)}&\ \ \ \text{for $\Delta
P<0$}\ ,\end{cases}$ (20)

Let us proceed with the question of integrability. By virtue of proposition
(2.1), the integrability of $\rho$ is equivalent to
$\operatorname{\operatorname{d}}\left({|\operatorname{\operatorname{Pf}}(\alpha)|}^{-\frac{1}{2}}\alpha\right)=0$,
where $\alpha$ is given by (18). Recall that the pressure is assumed to be a
function of $x,y$ only, $P=P(x,y)$, hence we have

$\displaystyle\operatorname{\operatorname{d}}\left({|\operatorname{\operatorname{Pf}}(\alpha)|}^{-\frac{1}{2}}\alpha\right)=\left(-\frac{\sqrt{2}}{2}|\Delta
P|^{-\frac{3}{2}}\operatorname{\operatorname{sgn}}\Delta
P\right)\operatorname{\operatorname{d}}\left(\Delta
P\right)\wedge\operatorname{\operatorname{d}}p\wedge\operatorname{\operatorname{d}}q\
.$

Since $\Delta P$ is a $0$-form (hence cannot be closed), the last equation
implies

$\displaystyle\rho\text{ is integrable}\quad\iff\quad\Delta P\text{ is
constant.}$

Indefinite metric field. To finish this section, we give the matrix
description of the symmetric bilinear form (11) corresponding to (17). The
matrix of $g$ for a generic $\alpha$ is displayed in (12), which in our case
becomes the following pseudo-metric field with split signature $(2,2)$

$\displaystyle\underline{g}=\sqrt{\frac{2}{|\Delta
P|}}\begin{pmatrix}0&\mathbb{1}\\\ \mathbb{1}&0\end{pmatrix}\ .$ (21)

##  4 Generalized geometry and M-A theory

In this section we want to describe generalized structures associated with the
equation for the stream function (17). For the sake of completeness, 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 $\det\operatorname{Hess}(f)=1$.

###  4.1 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}$ (22)

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 $\eta$

$\displaystyle\eta\bigl{(}\left(X,\xi\right),\left(Y,\zeta\right)\bigr{)}:=\frac{1}{2}\left(\xi(Y)+\zeta(X)\right)\
,$ (23)

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 $\mathbb{J}$ over $\mathcal{M}$ 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 $\left(X,\eta\right),\left(Y,\theta\right)\in\mathbb{T}\mathcal{M}$

$\displaystyle\langle\mathbb{J}\left(X,\eta\right),\mathbb{J}\left(Y,\theta\right)\rangle=\langle\left(X,\eta\right),\left(Y,\theta\right)\rangle$
(24)

where $\eta$ is the natural inner product (23) [11, 12].

######  Example 4.1.

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 (9)). Then
$(\Omega,A_{\alpha})$ defines a Hitchin pair of $2$-forms in the sense of
Crainic [7, 16] since

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

Using the notion of Hitchin pairs, B. Banos [2] showed in 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 4.1 was a key motivation for our further
investigations of other possiblities of constructing generalized (almost)
geometries from Monge-Ampère equations and the corresponding Monge-Ampère
structures. Starting with the M-A equation for the stream function (17) with
specifically chosen $\Delta P$, we will show that the family of generalized
structures constructed with the help of M-A theory can be significantly
enlarged. In order to state our result, we need to extend the notion of
generalized (almost) complex structure by the following definition.

######  Definition 4.1 ([15]).

A generalized almost structure $\mathbb{J}$ over $\mathcal{M}$ 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 $\mathbb{J}^{\bullet}\eta$ is
understood in the sense of (24). Table 1 describes the four possible choice 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
$\mathbb{J}^{2}=-\operatorname{Id}_{\mathbb{T}\mathcal{M}}$)

######  Example 4.2.

A family of examples of generalized almost structures comes from non-
degenerate tensor fields of rank $2$. Let
$J\in\operatorname{Aut}\left(T\mathcal{M}\right)$ be an almost complex
structure, $\alpha\in\Omega^{2}\left(\mathcal{M}\right)$ a non-degenerate
$2$-form, and $g\in S^{2}\left(\mathcal{M}\right)$ a non-degenerate symmetric
bilinear form. We have seen in the previous sections that M-A theory is a
natural source of the above three object. Let $\epsilon\in\\{-1,1\\}$ and
consider

$\displaystyle\mathbb{J}$ $\displaystyle=\begin{pmatrix}J&0\\\ 0&\epsilon
J^{*}\end{pmatrix}\ ,$ $\displaystyle\mathbb{K}$
$\displaystyle=\begin{pmatrix}0&\pi_{\alpha}^{\\#}\\\
\epsilon\alpha_{\\#}&0\end{pmatrix}\ ,$ $\displaystyle\mathbb{G}$
$\displaystyle=\begin{pmatrix}0&\pi_{g}^{\\#}\\\ \epsilon
g_{\\#}&0\end{pmatrix}\ .\ $ (25)

Then $\mathbb{J}$ is a GaAC stucture for $\epsilon=1$, and a GaC structure for
$\epsilon=-1$, while $\mathbb{K}$ is a GaPC structure for $\epsilon=1$, and a
GaC structure for $\epsilon=-1$. For $\mathbb{G}$ we have a GaP in the case
$\epsilon=1$ and a GaAC if $\epsilon=-1$.

We see that starting with non-degenerate rank $2$ tensor fields, one can
construct isotropic, as well as non-isotropic structures. We will end this
example with the demonstration of non-isotropy of generalized almost product
structure. Take $\mathbb{G}$ with $\epsilon=1$. Then the $\pm 1$-eigenbundles
of $\mathbb{G}$ are

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

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

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

Obviously there are $X,Y\in\Gamma\left(T\mathcal{M}\right)$ such that
$g(X,Y)\neq 0$, thus $E_{+}$ is not totally isotropic, and hence $\mathbb{G}$
is a non-isotropic structure. Nontheless, we can still speak about
(non-)degeneracy of a non-isotropic structures.

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]

$\displaystyle[\left(X,\xi\right),\left(Y,\zeta\right)]_{C}:=\left([X,Y],\mathcal{L}_{X}\zeta-\mathcal{L}_{Y}\xi-\frac{1}{2}\operatorname{\operatorname{d}}(X\mathbin{\lrcorner}\zeta-Y\mathbin{\lrcorner}\xi)\right)\
,$ (26)

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

###  4.2 Isotropy, Dirac structures, and integrability

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

######  Definition 4.2.

A subbundle $E\subset\mathbb{T}M$ 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 $M$ if
$\operatorname{rank}E=\operatorname{rank}TM$. A Dirac structure on $M$ is an
almost Dirac structure $E$ such that $[E,E]_{C}\subset E$.

######  Remark 4.1.

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

The four possible generalized almost structures determined by definition 4.1
can be divided into two subsets depending on whether the eigenbundles
$E_{\pm}$ are almost Dirac structures or not.

######  Definition 4.3.

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 4.2
below).

######  Definition 4.4.

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

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

$\displaystyle
N_{\mathbb{J}}(x,y):=[\mathbb{J}x,\mathbb{J}y]_{C}+\mathbb{J}^{2}[x,y]_{C}-\mathbb{J}\left([\mathbb{J}x,y]_{C}+[x,\mathbb{J}y]_{C}\right)\
.$ (27)

This $(1,2)$-tensor
$N_{\mathbb{J}}\colon\Gamma(\mathbb{T}M)\otimes\Gamma(\mathbb{T}M)\to\Gamma(\mathbb{T}M)$
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 $x,y\in\Gamma(\mathbb{T}M)$ [7].

######  Remark 4.2.

If a generalized almost structure $\mathbb{J}$ is non-isotropic, then the
eigenbundles $E_{\pm}$ are not totally isotropic (see example 4.2) and the
Courant bracket is not well-defined on them. Moreover, for non-isotropic
structures, the torsion (27) is not a tensor. As a consequence, the notion of
integrability, which is usually given either by the condition of vanishing
Nijenhuis tensor, or by definition 4.4, cannot be applied for non-isotropic
structures. Nontheless, there are other means of defining integrability of
non-isotropic structures. For example, the authors of [15] defined weak
integrablity using the notion of generalized Bismut connection [10].

##  5 Generalized geometry of 2D incompressible fluids

We have seen in section 3.3 that, by virtue of the proposition 2.1, the
endomorphism $\rho$ is an integrable structure, if and only if $\Delta P$ is
constant, nowhere vanishing function. This also implies
$\operatorname{\operatorname{d}}\alpha=0$. In the following proposition, we
will focus on the case $\Delta P>0$ and, for the convenience choose $\Delta
P=2$. In this situation, the equation for the stream function (17) becomes

$\displaystyle\det\operatorname{Hess}(f)=1\ .$ (28)

The block matrices of $\rho$ and $\alpha_{\\#}$ are
$\underline{\rho}=\begin{pmatrix}0&A\\\ A&0\end{pmatrix}$ and
$\underline{\alpha}=\begin{pmatrix}A&0\\\ 0&-A\end{pmatrix}$, where
$A=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$. We have
$\underline{\rho}^{2}=\underline{\alpha}^{2}=-\mathbb{1}$, and
$\underline{\rho}^{T}=-\underline{\rho}$. Notice that this relations change
when $\Delta P=-2$. Of course, in all cases we have
$\underline{\alpha}^{T}=-\underline{\alpha}$, since $\alpha$ is a $2$-form.

######  Proposition 5.1.

Let $(\Omega,\alpha)$ be the normalized M-A structure determined by the
equation (17) with $\Delta P=2$. Let $\rho$ be the corresponding endomorphism
(9). Then the following
$\mathbb{J}_{i}\colon\mathbb{T}\mathcal{B}\to\mathbb{T}\mathcal{B}$

$\displaystyle\mathbb{J}_{1}=\begin{pmatrix}\underline{\rho}&0\\\
0&\epsilon_{1}\underline{\rho}\end{pmatrix}\ ,$
$\displaystyle\mathbb{J}_{2}=\begin{pmatrix}0&\underline{\alpha}\\\
\epsilon_{2}\underline{\alpha}&0\end{pmatrix}\ ,$
$\displaystyle\mathbb{J}_{3}=\begin{pmatrix}0&\underline{\Omega}\\\
\epsilon_{3}\underline{\Omega}&0\end{pmatrix}\ ,$ (29)

where $\epsilon_{i}\in\\{-1,1\\}$, are generalized almost structures
structures.

  1. 1.

If $\epsilon_{1}=1$, then $\mathbb{J}_{1}$ is isotropic and integrable. If
$\epsilon_{1}=-1$, then $\mathbb{J}_{1}$ is non-isotropic.

  2. 2.

The structures $\mathbb{J}_{2},\mathbb{J}_{3}$ are isotropic and integrable.

The types of these structures depending on $\epsilon_{i}$ are described in
table 2.

$\operatorname{\operatorname{sgn}}{\epsilon_{i}}$ |  $\mathbb{J}_{1}$ |  $\mathbb{J}_{2}$ |  $\mathbb{J}_{3}$
---|---|---|---
$+$ |  GC |  GC |  GC
$-$ |  GaAC |  GPC |  GPC
Table 2:  Generalized structure associated with
$\det\operatorname{Hess}(f)=1$.

###### Proof.

The choice of the triple $(\mathbb{J}_{i})$ corresponds to the example (4.2)
with respect to the M-A equation $\det\operatorname{Hess}(f)=1$. This implies

$\displaystyle(\mathbb{J}_{1})^{2}=-\mathbb{1}\ ,$
$\displaystyle(\mathbb{J}_{2})^{2}=\epsilon_{2}(-\mathbb{1})\ ,$
$\displaystyle(\mathbb{J}_{3})^{2}=\epsilon_{3}(-\mathbb{1})\ ,$ (30)

and

$\displaystyle(\mathbb{J}_{1})^{*}\eta=\epsilon_{1}\eta\ ,$
$\displaystyle(\mathbb{J}_{2})^{*}\eta=\epsilon_{2}\eta\ ,$
$\displaystyle(\mathbb{J}_{3})^{*}\eta=\epsilon_{3}\eta\ ,$ (31)

where
$(\mathbb{J}_{i})^{*}\underline{\eta}={\mathbb{J}_{i}}^{T}\underline{\eta}\mathbb{J}_{i}$.
Due to our choice of $\Delta P$, $\rho$ is an integrable almost complex
structure (a complex structure). If $\epsilon_{1}=1$, this implies that
$\mathbb{J}_{1}$ is a generalized complex structure. If $\epsilon_{1}=-1$,
then $\mathbb{J}_{1}$ is a generalized almost anti-complex structure, which is
non-isotropic. Furthermore, both $\alpha$ and $\Omega$ are closed. Thus for
all choices of $\epsilon_{2},\epsilon_{3}$, the structures $\mathbb{J}_{2}$
and $\mathbb{J}_{3}$ are integrable. Therefore $\mathbb{J}_{2}$ and
$\mathbb{J}_{3}$ are (independently) either GC or GPC structures. ∎

###  5.1 Structures generated by anticommutative triples

Let $\\{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}\\}$ be a triple of pair-
wise anticommuting generalized almost structures, given by (29). The
anticommutators are

$\displaystyle\\{\mathbb{J}_{1},\mathbb{J}_{2}\\}$
$\displaystyle=\begin{pmatrix}0&-(\epsilon_{1}-1)\underline{\Omega}\\\
\epsilon_{2}(\epsilon_{1}-1)\underline{\Omega}&0\end{pmatrix}\ ,$
$\displaystyle\\{\mathbb{J}_{2},\mathbb{J}_{3}\\}$
$\displaystyle=\begin{pmatrix}-(\epsilon_{2}-\epsilon_{3})\underline{\rho}&0\\\
0&(\epsilon_{2}-\epsilon_{3})\underline{\rho}\end{pmatrix}\ ,$
$\displaystyle\\{\mathbb{J}_{1},\mathbb{J}_{3}\\}$
$\displaystyle=\begin{pmatrix}0&-(1-\epsilon_{1})\underline{\alpha}\\\
\epsilon_{3}(1-\epsilon_{1})\underline{\alpha}&0\end{pmatrix}\ ,$

and we observe

$\displaystyle\\{\mathbb{J}_{i},\mathbb{J}_{j}\\}$ $\displaystyle=0\ \forall
i,j\quad\iff\quad\epsilon_{1}=1,\epsilon_{2}=\epsilon_{3}\ .$

This implies that there are exactly two possibilities in which the pair-wise
anticommutivity occurs. In both cases $\mathbb{J}_{1}$ is a GC structure, and
either $\mathbb{J}_{2},\mathbb{J}_{3}$ are a pair of GC structures, or a pair
of GPC structures. The first case consisting of a triple of GC structures has
some further properties which are described in the proposition 5.2 and the
subsequent corollary. In both cases, any of the two structures generate the
third one and $\mathbb{J}_{1}$ is a fixed GC structure.

######  Proposition 5.2.

Let $\\{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}\\}$ be given by (29).
Then the structures pair-wise anticommute,

$\displaystyle\\{\mathbb{J}_{i},\mathbb{J}_{i+1}\\}=0\ ,\ i=1,2,3\ ,$

if, and only if, $\epsilon_{1}=\epsilon_{2}=\epsilon_{3}=1$, or,
$\epsilon_{1}=-\epsilon_{2}=-\epsilon_{3}=1$.

  1. 1.

Let $\epsilon_{1}=\epsilon_{2}=\epsilon_{3}=1$. Then each $\mathbb{J}_{i}$ is
a GC structure. Moreover, there is a $2$-sphere of GaC structures in
$\operatorname{span}_{\mathbb{R}}\\{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}\\}$.

  2. 2.

Let $\epsilon_{1}=-\epsilon_{2}=-\epsilon_{3}=1$. Then $\mathbb{J}_{1}$ is a
GC structure and $\mathbb{J}_{2},\mathbb{J}_{3}$ are GPC structures. Moreover,
there is a one-sheeted hyperboloid of GaPC structures and a two-sheeted
hyperboloid of GaC structures in
$\operatorname{span}_{\mathbb{R}}\\{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}\\}$.

In both cases we have

$\displaystyle\mathbb{J}_{i}\mathbb{J}_{i+1}=\mathbb{J}_{i+2}\mod{3}\ ,\
i=1,2,3\ .$ (32)

###### Proof.

Let $\mathbb{A}$ be an $\operatorname{\mathbb{R}}$-linear combination of the
almost generalized structures $\mathbb{J}_{i}$

$\displaystyle\mathbb{A}=a_{1}\mathbb{J}_{1}+a_{2}\mathbb{J}_{2}+a_{3}\mathbb{J}_{3}\
.$ (33)

From the mutual anticommutativity of the triple, we get

$\displaystyle\mathbb{A}^{2}$
$\displaystyle=\left({a_{1}}^{2}+{a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3}\right)(-\mathbb{1})\
,$ $\displaystyle\mathbb{A}^{*}\eta$
$\displaystyle=({a_{1}}^{2}+{a_{2}}^{2}\epsilon_{2}+{a_{3}}^{2}\epsilon_{3})\eta\
.$

Suppose $\epsilon_{1}=\epsilon_{2}=\epsilon_{3}=1$. Then for any
$(a_{1},a_{2},a_{3})\in\operatorname{\mathbb{R}}^{3}$ such that
${a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}=1$ we have $\mathbb{A}^{2}=-\mathbb{1}$
and $\mathbb{A}^{*}\eta=\eta$. Thus $\mathbb{A}$ is a GaC structure. Now
suppose $\epsilon_{1}=-\epsilon_{2}=-\epsilon_{3}=1$. Then

$\displaystyle\mathbb{A}^{2}$
$\displaystyle=\left(-{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}\right)\mathbb{1}\ ,$
(34) $\displaystyle\mathbb{A}^{*}\eta$
$\displaystyle=\left({a_{1}}^{2}-{a_{2}}^{2}-{a_{3}}^{2}\right)\eta\ .$ (35)

Thus for $-{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}=1$ we have
$\mathbb{A}^{2}=\mathbb{1}$ and $\mathbb{A}^{*}\eta=-\eta$, so $\mathbb{A}$ is
a GaPC structure. Similarly, for $-{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}=-1$ we
have $\mathbb{A}^{2}=-\mathbb{1}$ and $\mathbb{A}^{*}\eta=\eta$, hence
$\mathbb{A}$ is a GaC structure. The remaining statements of the proposition
were discussed before the proposition. ∎

######  Definition 5.1 ([5, 13]).

A generalized almost hyper-complex structure is a triple
$\\{\mathbb{A},\mathbb{B},\mathbb{C}\\}$ of GaC structures such that
$\mathbb{A}\mathbb{B}\mathbb{C}=\mathbb{-1}$. A generalized almost hyper-para-
complex structure is a triple $\\{\mathbb{A},\mathbb{B},\mathbb{C}\\}$,
consisting of a GaC structure and a pair of GaPC structures such that
$\mathbb{A}\mathbb{B}\mathbb{C}=\mathbb{1}$. The structures are integrable, if
$\\{\mathbb{A},\mathbb{B},\mathbb{C}\\}$ are integrable. In this case we call
the triple generalized hyper-complex and generalized hyper-para-complex
structure, respectively.

######  Remark 5.1.

In [13], the notion of a generalized hyper-complex structure is named
hypercomplex structure on a Courant algebroid (in our case, the Courant
algebroid is the generalized tangent bundle).

######  Proposition 5.3.

The triplet of pair-wise anticommuting generalized structures given by (29)
satisfies

$\displaystyle\mathbb{J}_{1}\mathbb{J}_{2}\mathbb{J}_{3}=\epsilon_{3}(\mathbb{-1})\
.$ (36)

  1. 1.

If $\epsilon_{3}=1$, then $\\{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}\\}$
is a generalized hyper-complex structure on $\mathcal{B}$.

  2. 2.

If $\epsilon_{3}=-1$, then
$\\{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}\\}$ is a generalized hyper-
para-complex structure on $\mathcal{B}$.

###### Proof.

From proposition (5.2) we know, independently of the value of $\epsilon_{3}$,
that $\mathbb{J}_{1}\mathbb{J}_{2}=\mathbb{J}_{3}$ and $\epsilon_{1}=1$, which
means that all the three structures are integrable. Due to (30), we have
$\mathbb{J}_{1}\mathbb{J}_{2}\mathbb{J}_{3}=\epsilon_{3}(-\mathbb{1})$. If
$\epsilon_{3}=1$, then $\epsilon_{2}=1$, and thus
$\\{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}\\}$ is a triple of GC
structures (cf. table (2)). Moreover,
$\mathbb{J}_{1}\mathbb{J}_{2}\mathbb{J}_{3}=\mathbb{-1}$, so the triple is a
generalized hyper-complex structure. On the other hand, if $\epsilon_{3}=-1$,
then $\epsilon_{2}=-1$, and thus
$\\{\mathbb{J}_{1},\mathbb{J}_{2},\mathbb{J}_{3}\\}$ contains a GC structure
and a pair of GPC structures (cf. table (2)). Moreover,
$\mathbb{J}_{1}\mathbb{J}_{2}\mathbb{J}_{3}=\mathbb{1}$, so the triple is a
generalized hyper-para-complex structure. ∎

###  5.2 Generalized metric compatible structures

We say that a generalized almost product structure
$\mathbb{G}\in\operatorname{End}\left(T\mathcal{M}\right)$ is a generalized
(pseudo-)metric, if the eigenbundles of $\mathbb{G}$ are isomorphic to
$T\mathcal{M}$, that is, if $\mathbb{G}$ is non-degenerate.

######  Example 5.1.

The main example of an indefinite generalized metric can be constructed from
an indefinite metric (11) as $\mathbb{G}=\begin{pmatrix}0&\pi_{g}^{\\#}\\\
g_{\\#}&0\end{pmatrix}$.

######  Definition 5.2 ([15]).

A generalized Kähler structure is a commuting pair of a generalized metric and
a GaC structure. A generalized chiral structure is a commuting pair of a
generalized metric and a GaP structure.

######  Proposition 5.4.

The M-A equation $\det\operatorname{Hess}(f)=\frac{1}{2}\Delta P$ defines a
generalized Kähler structure for $\Delta P=2$, and a generalized chiral
structure for $\Delta P=-2$.

###### Proof.

Let $\mathbb{G}$ and $\mathbb{J}$ be given by the following block matrices

$\displaystyle\mathbb{G}$ $\displaystyle=\begin{pmatrix}0&\underline{g}\\\
\underline{g}&0\end{pmatrix}\ ,$ $\displaystyle\mathbb{J}$
$\displaystyle=\begin{pmatrix}\underline{\rho}&0\\\
0&\operatorname{\operatorname{sgn}}\left(\Delta
P\right)\underline{\rho}\end{pmatrix}\ ,$

where $\underline{g}=\begin{pmatrix}0&\mathbb{1}\\\ \mathbb{1}&0\end{pmatrix}$
corresponds to (21), with $|\Delta P|=2$, and $\rho$ is given by (21)

$\displaystyle\underline{\rho}$ $\displaystyle=\begin{pmatrix}0&A\\\
\operatorname{\operatorname{sgn}}\left(\Delta P\right)A&0\end{pmatrix}\ ,$

where $A=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$ Notice that $\mathbb{G}$ and
$\mathbb{J}$ correspond to structures from (25) with $\epsilon=1$, and where
$\rho$ is also allowed to be an almost product structure. Then

$\displaystyle[\mathbb{G},\mathbb{P}]=0\quad\iff\quad\begin{cases}\
[\underline{g},\underline{\rho}]=0&\text{ for }\Delta P=2\\\
\\{\underline{g},\underline{\rho}\\}=0&\text{ for }\Delta P=-2\\\ \end{cases}$
(37)

A direct computation shows that $[\underline{g},\underline{\rho}]=0$ is true
for $|\Delta P|=2$ (i.e. independently of the sign of Pfaffian). Thus
$\mathbb{G}$ commutes with $\mathbb{J}$ for both values of $\Delta P$.
Moreover, $\underline{\rho}$ is a complex structure for $\Delta P=2$ and a
product structure for $\Delta P=-2$, which further implies that $\mathbb{J}$
is a GC structure for $\Delta P=2$ and a GP structure for $\Delta P=-2$. Hence
for $\Delta P=2$, the pair $(\mathbb{G},\mathbb{J})$ is a generalized Kähler
structure, while for $\Delta P=-2$, the pair is a generalized chiral
structure. ∎

###  5.3 Algebras generated by generalized structures

There are multiple algebraic structures we can associate to a triple of
generalized structures (29). The main point of the following paragraph is to
discuss the possibility of studying M-A equations and M-A structures on the
level of algebraic structures determined by their generalized geometries. The
hypothesis which we want to investigate in our future work is whether the
algebras defined below are invariant with respect to certain transformations.
For example, one might study invariance with respect to symplectomorphisms,
which provide notion of equivalence between M-A equations (see proposition
2.1).

Let us denote by $\mathcal{A}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}$ the
free, associative, unital $\mathbb{R}$-algebra, given by the real vector space
generated by the generalized almost structures
$\\{\mathbb{J}_{i}\\}_{i=1}^{3}$ with fixed values $(\epsilon_{i})_{i=1}^{3}$,
and equipped with the product of matrices (or composition of operators, if we
work coordinate-free)

$\displaystyle\mathcal{A}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}:=\left(\mathbb{R}[\mathbb{J}_{i}]_{i=1}^{3},(\epsilon_{i})_{i=1}^{3},\cdot\right)\
.$ (38)

We will omit the $\cdot$ symbol when writing product of elements from
$\mathcal{A}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}$. We further consider two
other bilinear operations on this algebra, the commutator and the
anticommutator.

Jordan algebra. By a Jordan algebra we mean a commutative (possibly non-
associative) algebra which satisfies $(AB)(AA)=A(B(AA))$ [4]. From every
associative $k$-algebra111Where $k$ is a field of characteristic not equal to
$2$ $(\mathcal{A},\cdot)$ we can construct a Jordan $k$-algebra by
substituting the algebra product with half of the anticommutator

$\displaystyle A\bullet B:=\frac{1}{2}\\{A,B\\}\ .$ (39)

The resulting Jordan algebra is called a special Jordan algebra. Let us denote
the special Jordan algebra constructed this way from
$\mathcal{A}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}$ by
$\operatorname{Jor}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}$, i.e.

$\displaystyle\operatorname{Jor}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}:=\left(\mathbb{R}[\mathbb{J}_{i}]_{i=1}^{3},(\epsilon_{i})_{i=1}^{3},\bullet\right)\
.$ (40)

Lie algebra. We will denote the Lie algebra generated by the generalized
structures $\\{\mathbb{J}_{i}\\}_{i=1}^{3}$ with fixed epsilons
$(\epsilon_{i})_{i=1}^{3}$ by
$\operatorname{Lie}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}$, i.e.

$\displaystyle\operatorname{Lie}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}:=\left(\mathbb{R}[\mathbb{J}_{i}]_{i=1}^{3},(\epsilon_{i})_{i=1}^{3},[-,-]\right)\
.$ (41)

Poisson algebra. We have two Poisson algebra structures at hand. One of them
is coming from a Lie algebra, the other one from a Jordan algebra.

  1. 1.

Since the underlying algebra
$\mathcal{A}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}$ is associative and the
Lie bracket is given by the commutator, we automatically have a Poisson
algebra at hand

$\displaystyle\operatorname{Poiss}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}:=\left(\mathcal{A}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3},[-,-]\right)\
.$ (42)

Indeed, for every
$\mathbb{A},\mathbb{B},\mathbb{C}\in\mathcal{A}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}$
we have

$\displaystyle[\mathbb{A},\mathbb{B}\mathbb{C}]=[\mathbb{A},\mathbb{B}]\mathbb{C}+\mathbb{B}[\mathbb{A},\mathbb{C}]\
.$ (43)

  2. 2.

The second Poisson algebra is given by considering the (non-associative)
Jordan algebra $\operatorname{Jor}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3}$,
equipped with the commutator

$\displaystyle\operatorname{Poiss}(\mathbb{J}_{i},\epsilon_{i},\bullet)_{i=1}^{3}:=\left(\operatorname{Jor}(\mathbb{J}_{i},\epsilon_{i})_{i=1}^{3},[-,-]\right)\
.$ (44)

Indeed, for every
$\mathbb{A},\mathbb{B},\mathbb{C}\in\mathbb{R}[\mathbb{J}_{i}]_{i=1}^{3}$ we
have

$\displaystyle[\mathbb{A},\mathbb{B}\bullet\mathbb{C}]=[\mathbb{A},\mathbb{B}]\bullet\mathbb{C}+\mathbb{B}\bullet[\mathbb{A},\mathbb{C}]\
.$ (45)

######  Remark 5.2.

In the second case of Poisson algebra comming from a Jordan algebra, one might
wonder whether we have a Lie-Jordan algebra at hand, i.e. an algebraic
structure which is both, a Lie algebra, a Jordan algebra, and the two bilinear
operations are compatible in the following sense

$\displaystyle[\mathbb{A},\mathbb{B}\bullet\mathbb{C}]$
$\displaystyle=[\mathbb{A},\mathbb{B}]\bullet\mathbb{C}+\mathbb{B}\bullet[\mathbb{A},\mathbb{C}]\
,$ (46) $\displaystyle q^{2}[[\mathbb{A},\mathbb{C}],\mathbb{B}]$
$\displaystyle=(\mathbb{A}\bullet\mathbb{B})\bullet\mathbb{C}-\mathbb{A}\bullet(\mathbb{B}\bullet\mathbb{C})\
,$ (47)

where $q\in\mathbb{R}$ (or $q\in k$, if we consider a different underlying
field). As we have stated above, the first axiom is satisfied. The second
axiom is satisfied if, and only if $q=\frac{\sqrt{-1}}{2}$.

######  Remark 5.3.

V. Lychagin developed in [19] all the necessary definitions of algebraic
structures, such as Lie algebras and Jordan algebras, defined for a system of
two dimensional non-linear first order partial differential equations with a
specific type of non-linearity. A subclass of these systems are called Jacobi
systems. In our future work, we will be interested in extension of our
algebraic setup to such systems, based on the results in [19].

##  6 Conclusion

With the help of Monge-Ampère theory [18, 19, 17], we have constructed a
family of generalized structures associated to a Monge-Ampère equation (17),
which was derived from the Navier-Stokes system for 2D incompressible fluid
flows. Distinguishing between the hyperbolic and elliptic case, we have shown
that many of these generalized structures are integrable and give rise to
further composite structures such as hyper-complex and hyper-para-complex.
Introducing an indefinite generalized metric defined with the help of pseudo-
metric (21), we have also found generalized metric compatible structures,
namely generalized Kähler and generalized chiral structures.

Our future work will be focused on the study of other M-A equations and their
relation to generalized geometry, with the aim to go into higher dimensions.
Particularly interesting seems to be dimension three, as the situation there
is much richer, yet the symplectic classification of differential forms is
still possible. In this context, we want to further investigate the
correspondences between M-A equations, families of generalized structures, and
algebras constructed from these structures. We are especially curious which
kind of equivalence notions naturally present in the M-A theory are able to
survive when passing to either generalized structures, or various algebras we
have linked with them. Yet another direction to investigate is the possiblity
of extending this framework to the Jacobi systems and other more general
situations of first or second order PDEs with special types of non-
linearities, since a framework for doing so has already been developed [19].

## Acknowledgments

First of all I want to express my wholehearted gratitude to my beloved wife
and son for their continuous support and patience. This paper was written as a
part of my PhD research under the cotutelle agreement between the Masaryk
University, Brno, Czech Republic, and the University of Angers, France. I am
grateful for hospitality and support of these institutions during the research
period. I am also grateful for the funding provided by the Czech Ministry of
Education, and by the Czech Science Foundation under the project GAČR EXPRO
GX19-28628X. I want to express my gratefulness to my supervisors Volodya
Rubtsov and Jan Slovák for their supervision over my research project. I
especially want to thank Volodya Rubtsov for his valuable remarks and
suggestions of improvement of this work. The results of this paper were
reported at the Winter School and Workshop Wisla 22, a European Mathematical
Society event organized by the Baltic Institute of Mathematics.

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