# Conical Hessian manifolds

Pavel Osipov

###### Abstract

We study some generalizations of Riemannian cones, namely selfsimilar
manifolds. We describe conditions for selfsimilar manifold to be a conical
Riemannian manifold and construct examples of selsfimilar manifolds which are
not conical Riemannian manifolds. Conical Hessian manifold is a radiant
manifold endowed with a selfsimilar Hessian metric. We prove that any conical
Hessian manifold is conical Riemannian. A potential of a Hessian metric can be
described in the case of a complete radiant field. An extensive Hessian
manifold is a radiant manifold endowed with a degenerate Hessian metric such
that a potential of the metric is linear along the radiant vector field. We
construct an extensive Hessian metric on any conical Hessian manifold.

##  1 Introduction

A conical Riemannian manifold $(C,g,\xi)$ is a Riemannian manfold $(C,g)$
endowed with a vector field $\xi$ such that

$D\xi=\text{Id},$

where $D$ is a Levi-Civita connection. If $\xi$ is complete then the manifold
is called a global conical Riemannian manifold. A Riemannian cone is a
Riemannian manifold

${(M\times\mathbb{R}^{>0},t^{2}g_{M}+dt^{2})},$

where $t$ is a coordinate on $\mathbb{R}^{>0}$ and $g_{M}$ is a Riemannian
metric on $M$.

It is well-known that any global conical Riemannian manifold is a Riemannian
cone and any global Riemannian cone is locally isomorphic to a Riemannian cone
(see for example [ACHK]). Riemannian cones have important applications in
supegravity ([ACDM], [ACM], [CDM], [VDMV]). Holonomy and geometry of pseudo-
Riemannian cones are studied in [ACGL] and [ACL].

In section 2, we generalize the definitions of Riemannian cones and conical
Riemannian manifolds.

######  Proposition 1.1.

Let $M\times\mathbb{R}^{>0}$ be a Riemannian manifold. For any
$q\in\mathbb{R}^{>0}$ consider a map $\lambda_{q}:M\times\mathbb{R}^{>0}\to
M\times\mathbb{R}^{>0}$ defined by the rule $\lambda_{q}(x\times t)=x\times
qt$. Then the following conditions are equivalent:

  * (i)

For any $q\in\mathbb{R}^{>0}$ the metric $g$ satisfies
$\lambda_{q}^{*}g=q^{2}g$ .

  * (ii)

The metric $g$ has a form

$g=t^{2}g_{M}+t\text{Sym}(dt\otimes\alpha)+fdt^{2},$

where $t$ is a coordinate on $\mathbb{R}^{>0}$, $g_{M}$ is a Riemannian metric
on $M$, $\alpha$ is a 1-form on $M$, and $f$ is a positive definite function
on $M$.

If a Riemannian metric $g$ on a manifold $M\times\mathbb{R}^{>0}$ satisfies
the conditions of Proposition 1.1 then $g$ is called selfsimilar and the
Riemannian manifold $(M\times\mathbb{R}^{>0},g)$ is called a selfsimilar cone.

A selfsimilar manifold $(M,g,\xi)$ is a Riemannian manifold $(M,g)$ endowed
with a vector field $\xi$ satisfying

$\text{Lie}_{\xi}\ g=2g.$

If $\xi$ is complete then the manifold is called a global selfsimilar
manifold.

Similarly to the case of conical manifolds, we prove that any complete
selfsimilar manifold is a Riemannian cone and any selfsimilar Riemannian
manifold is isometric to a selfsimilar Riemannian cone locally. We describe
conditions for a selfsimilar manifold to be conical Riemannian.

######  Theorem 1.2.

Let $(M,g,\xi)$ be a selfsimilar manifold, $D$ be a Levi-Civita connection on
$M$, and $\theta=\iota_{\xi}g$. Then the following conditions are equivalent

  * (i)

$(M,g,\xi)$ is a conical Riemannian manifold.

  * (ii)

$D\theta=g$.

  * (iii)

$D\theta$ is a symmetric tensor.

  * (iv)

$\theta$ is a closed form.

######  Corollary 1.3.

Let

$g=t^{2}g_{M}+t\text{Sym}(dt\otimes\alpha)+fdt^{2}$

be a selfsimilar metric on $M\times\mathbb{R}^{>0}$. Then the Riemannian
manifold $(M\times\mathbb{R}^{>0},g)$ is a Riemannian cone if and only if

$df=-2\alpha.$

Using Corollary 1.3, we construct a family of examples of selfsimilar
manifolds which are not conical.

######  Proposition 1.4.

Let $(M,\alpha)$ be a contact manifold. Then there exists a Riemannian metric
$g$ on $M\times\mathbb{R}^{>0}$ such that $(M\times\mathbb{R}^{>0},g)$ is a
selfsimilar cone but not a Riemannian cone.

Further, we work with Hessian manifold. A flat affine manifold is a
differentiable manifold equipped with a flat torsion-free connection.
Equivalently, it is a manifold equipped with an atlas such that all
translation maps between charts are affine transformations (see [FGH] or
[Sh]). A Hessian manifold is a flat affine manifold with a Riemannian metric
wich is locally equivalent to a Hessian of a function. Any Kähler metric can
be defined as a complex Hessian of a plurisubharmonic function. Thus, the
Hessian geometry is a real analogue of the Kähler one.

Hessian manifolds have many different application: in supersymmetry ([CMMS],
[CM], [AC]), in convex programming ([N], [NN]), in the Monge-Ampère Equation
([F1], [F2], [Gu]), in the WDVV equations ([T]).

A radiant manifold $(C,\nabla,\xi)$ is a flat affine manifold $(C,\nabla)$
endowed with a vector field $\xi$ satisfying

$\nabla\xi=\text{Id}.$

Equivalently, it is a manifold equipped with an atlas such that all
translation maps between charts are linear transformations (see e.g. [Go]).

A conical Hessian manifold $(C,\nabla,g,\xi)$ is Hessian manifold
$(C,\nabla,g)$ endowed with a vector field $\xi$ such that $(C,g,\xi)$ is a
selfsimilar manifold and $(C,\nabla,\xi)$ is a radiant manifold. If $\xi$ is
complete then $C$ is called a global conical Hessian manifold. If a conical
Hessian manifold has a form
${(M\times\mathbb{R}^{>0},\nabla,g,t\frac{\partial}{\partial t})}$ then we say
that ${(M\times\mathbb{R}^{>0},\nabla,g)}$ is a Hessian cone.

The main result of section 3 is the following.

######  Theorem 1.5.

Let $(C,\nabla,g,\xi)$ be a conical Hessian manifold. Then $(C,g,\xi)$ is a
Riemannian conical manifold.

######  Corollary 1.6.

Any global conical Hessian manifold is a Hessian cone. Any conical Hessian
manifold is locally isomorphic to a Hessian cone.

In section 4, we describe a potential of a Hessian metric on a Hessian cone.

######  Theorem 1.7.

Let $(M\times\mathbb{R}^{>0},g=t^{2}g_{M}+dt^{2})$ be a Hessian cone. Then

$g=\text{Hess}\left(\frac{t^{2}}{2}\right).$

Note that Theorem 1.7 is analogical to a result on Sasakian manifolds. A
Riemannian manifold $(M,g_{M})$ of odd real dimension is called Sasakian if
the metric cone

${(C,g)=(M\times\mathbb{R}^{>0},t^{2}g+dt^{2})}$

is equipped with a dilatation-invariant complex structure, which makes $C$ to
a Kähler manifold (see [BG]). Then the function $t^{2}$ defines a Kähler
potential of $g$ (see e.g. [OV]).

In section 5, we describe a relation of conical Hessian manifolds to a similar
class of manifolds arising in equilibrium thermodynamics. Let $(C,\nabla,\xi)$
be a radiant manifold and $\varphi$ is a function on $C$ such that the
bilinear form $g=\text{Hess}\varphi$ is positive semidefinite. If $\varphi$ is
a linear along $\xi$ function i.e. a function satisfying

$\text{Lie}_{\xi}\varphi=\varphi$

then $(C,\nabla,g,\xi)$ is called extensive Hessian manifold. Equivalently, a
radiant manifold $(C,\nabla,\xi)$ endowed with a Hessian metric $g$ is an
extensive Hessian manifolds if

$\iota_{\xi}g=0$ (1.1)

As we sad above, extensive Hessian manifolds arise in equilibrium
thermodynamics (see [G-A] and [W1]). The equation (1.1) is a coordinate-free
form of the Gibbs-Duhem equation from thermodynamics (see e.g. [W2]).

The definitions of conical Hessian and extensive Hessian manifolds are almost
the same. The difference is the following: a potential of a Hessian metric
along a radiant vector field is linear in the case of extensive Hessian
manifolds and square in the case of conical Hessian manifolds. We construct an
extensive Hessian metric on any conical Hessian manifold.

##  2 Selfsimilar and conical metrics

######  Definition 2.1.

A Riemannian cone is a Riemannian manifold
$(M\times\mathbb{R}^{>0},g=t^{2}g_{M}+dt^{2})$, where $t$ is a coordinate on
$\mathbb{R}^{>0}$ and $g_{M}$ is a Riemannian metric on $M$.

######  Definition 2.2.

A conical Riemannian manifold $(C,g,\xi)$ is a Riemannian manfold $(C,g)$
endowed with a vector field $\xi$ such that

$D\xi=\text{Id},$

where $D$ is a Levi-Civita connection. If $\xi$ is complete then the manifold
is called a global conical Riemannian manifold.

######  Proposition 2.3 ([ACHK]).

  * (i)

The metric cone over any Riemannian manifold is a global conical Riemannian
manifold.

  * (ii)

Conversely, any global conical Riemannian manifold is a metric cone.

  * (iii)

Any conical Riemannian manifold is locally isometric to a metric cone.

######  Proposition 2.4.

Let $M\times\mathbb{R}^{>0}$ be a Riemannian manifold. For any
$q\in\mathbb{R}^{>0}$ consider a map $\lambda_{q}:M\times\mathbb{R}^{>0}\to
M\times\mathbb{R}^{>0}$ defined by the rule $\lambda_{q}(x\times t)=x\times
qt$. Then the following conditions are equivalent:

  * (i)

For any $q\in\mathbb{R}^{>0}$ the metric $g$ satisfies
$\lambda_{q}^{*}g=q^{2}g$ .

  * (ii)

The metric $g$ has a form

$g=t^{2}g_{M}+t\text{Sym}(dt\otimes\alpha)+fdt^{2},$

where $t$ is a coordinate on $\mathbb{R}^{>0}$, $g_{M}$ is a Riemannian metric
on $M$, $\alpha$ is a 1-form on $M$, and $f$ is a positive definite function
on $M$.

###### Proof.

$\text{(i)}\Rightarrow\text{(ii)}$. If $g$ has a form
$g=t^{2}g_{M}+t\text{Sym}(dt\otimes\alpha)+fdt^{2}$ then the condition (i) is
obviously satisfied.

$\text{(ii)}\Rightarrow\text{(i)}$. Let $\pi:M\times\mathbb{R}^{>0}\to M$ be a
projection. Then we have

$T\left(M\times\mathbb{R}^{>0}\right)=\pi^{*}TM\oplus\ker\pi.$

The Riemannian metric $g$ lies in

$\text{Sym}^{2}\left(\pi^{*}TM\oplus\ker\pi\right)^{*}=\text{Sym}^{2}\left(\pi^{*}TM\right)^{*}\oplus\text{Sym}^{2}\left(\pi^{*}TM\otimes\ker\pi\right)^{*}\oplus\text{Sym}^{2}\left(\ker\pi\right)^{*}.$

Let $\pi_{2,0}$, $\pi_{1,1}$, and $\pi_{0,2}$ be projections of
$\text{Sym}^{2}\left(\pi^{*}TM\oplus\ker\pi\right)^{*}$ on the three summands
from the decomposition above. Define $g|_{M\times 1}=g_{M}$. For any
$X_{1},X_{2}\in\pi^{*}TM$ we have

$g(X_{1},X_{2})(m\times t)=\lambda_{t}^{*}g(X_{1},X_{2})(m\times
1)=t^{2}g(X_{1},X_{2})(m\times 1).$

Thus,

$\pi_{2,0}g=t^{2}g_{M}.$ (2.1)

For any $X\in\pi^{*}TM$ and $\frac{\partial}{\partial t}\in\ker\pi$, since

${\lambda_{t}}_{*}\left(t^{-1}\frac{\partial}{\partial
t}\right)=\frac{\partial}{\partial t},$

we have

$g\left(X,\frac{\partial}{\partial t}\right)(m\times
t)=\lambda_{t}^{*}g\left(X,t^{-1}\frac{\partial}{\partial t}\right)(m\times
1)=tg\left(X,\frac{\partial}{\partial t}\right)(m\times 1).$ (2.2)

Define

$\alpha:=\left.\left(\iota_{\frac{\partial}{\partial
t}}g\right)\right|_{M\times 1}.$

Then, by (2.2), we get

$g\left(X,\frac{\partial}{\partial t}\right)(m\times
t)=t\alpha(X)=t\text{Sym}(dt\otimes\alpha(X))\left(X,\frac{\partial}{\partial
t}\right).$

Hence,

$\pi_{1,1}g=t\text{Sym}(dt\otimes\alpha(X)),$ (2.3)

Finally, we have

$g\left(\frac{\partial}{\partial t},\frac{\partial}{\partial
t}\right)(m,t)=\lambda_{t}^{*}g\left(t^{-1}\frac{\partial}{\partial
t},t^{-1}\frac{\partial}{\partial
t}\right)(m,1)=g\left(\frac{\partial}{\partial t},\frac{\partial}{\partial
t}\right)(m,1).$

Therefore, we get

$\pi_{0,2}g=fdt^{2},$ (2.4)

where the function

$f(m)=g\left(\frac{\partial}{\partial t},\frac{\partial}{\partial
t}\right)(m,1)$

is a function on $M$.

Combining (2.1), (2.3), and (2.4), we get

$g=t^{2}g_{M}+t\text{Sym}(dt\otimes\alpha)+fdt^{2}.$

∎

######  Definition 2.5.

Let $(M\times\mathbb{R}^{>0},g)$ be a Riemannian manifold. If the metric $g$
satisfies the conditions of Proposition 2.4 then $g$ is called selfsimilar and
the Riemannian manifold ${(M\times\mathbb{R}^{>0},g)}$ is called a selfsimilar
cone.

######  Definition 2.6.

A selfsimilar manifold $(C,g,\xi)$ is a Riemannian manifold $(M,g)$ endowed
with a field $\xi$ satisfying

$\text{Lie}_{\xi}\ g=2g.$

If $\xi$ is complete then the manifold is called a global selfsimilar
manifold.

######  Lemma 2.7.

Let $M$ be a manifold, $U$ be an open subset of $M\times\mathbb{R}^{>0}$, $t$
be a coordinate on $\mathbb{R}^{>0}$, $\pi:M\times\mathbb{R}^{>0}\to M$ be a
natural projection, and $g$ be a Riemannian metric on $U$. For any
$q\in\mathbb{R}^{>0}$ consider a map $\lambda_{q}:M\times\mathbb{R}^{>0}\to
M\times\mathbb{R}^{>0}$ defined by $\lambda_{q}(x\times t)=x\times qt$. Let
for any $x\in U$ and $q\in\mathbb{R}^{>0}$ satisfying $\lambda_{q}x\in U$ we
have

$(\lambda_{q}^{*}g)(x)=q^{2}g(x).$

Then the metric $g$ on $U$ can be extended to a selfsimilar metric $\tilde{g}$
on $(\pi(U)\cap M)\times\mathbb{R}^{>0}$.

###### Proof.

For any $x\in(\pi(U)\cap M)\times\mathbb{R}^{>0}$ there exists
$q\in\mathbb{R}^{>0}$ such that $\lambda_{q}(x)\in U$. Define $\tilde{g}(x)$
by

$\tilde{g}(x)(X,Y)=q^{-2}\lambda_{q}^{*}g(x)(X,Y)=q^{-2}g(\lambda_{q}x)({\lambda_{q}}_{*}X,{\lambda_{q}}_{*}Y),$

for any $X,Y\in T_{x}(M\times\mathbb{R}^{>0})$. We need to check that the
value $\tilde{g}(x)$ does not depends on the choice of $q$. Let
$q^{\prime}\neq q$ satisfies $\lambda_{q^{\prime}}x\in U$. Then, by the
conditions of the lemma,

$g(\lambda_{q^{\prime}}x)({\lambda_{q^{\prime}}}_{*}X,{\lambda_{q^{\prime}}}_{*}Y)=\lambda_{\frac{q^{\prime}}{q}}^{*}g(\lambda_{q}x)({\lambda_{q}}_{*}X,{\lambda_{q}}_{*}Y)=\left(\frac{q^{\prime}}{q}\right)^{2}g(\lambda_{q}x)({\lambda_{q}}_{*}X,{\lambda_{q}}_{*}Y).$

Therefore,

${q^{\prime}}^{-2}g(\lambda_{q^{\prime}}x)({\lambda_{q^{\prime}}}_{*}X,{\lambda_{q^{\prime}}}_{*}Y)=q^{-2}g(\lambda_{q}x)({\lambda_{q}}_{*}X,{\lambda_{q}}_{*}Y)$

Thus, the value $\tilde{g}(x)$ is correctly defined. Moreover, $\tilde{g}(x)$
is obviously selfsimilar.

∎

######  Proposition 2.8.

  * (i)

Any selfsimilar cone is a global selfsimilar manifold.

  * (ii)

Conversely, any global selfsimilar manifold is a selfsimilar cone.

  * (iii)

Any selfsimilar manifold is locally isometric to a selfsimilar cone.

Note that this proposition is an analogue of Proposition 2.3.

###### Proof.

(i) If $C=M\times\mathbb{R}^{>0}$ be a selfsimilar cone and $t$ be a
coordianate on $\mathbb{R}^{>0}$ then

$\text{Lie}_{t\frac{\partial}{\partial t}}g=\left.\frac{d}{da}\
\lambda_{e^{a}}g\right|_{a=0}=\left.\frac{d}{da}\ e^{2a}g\right|_{a=0}=\ 2g.$

That is, $(C,g,\xi=t\frac{\partial}{\partial t})$ is a selfsimilar manifold.
Moreover, the field $t\frac{\partial}{\partial t}$ is complete. Thus,
$(C,g,\xi=t\frac{\partial}{\partial t})$ is a global selfsimilar manifold

(ii) Let $(C,g,\xi)$ be a selfsimilar manifold. The Lie derivative of a metric
is defined by

$(\text{Lie}_{X}g)(Y,Z)=\text{Lie}_{X}(g(Y,Z))-g(\text{Lie}_{X}Y,\ Z)-g(Y,\
\text{Lie}_{X}Z).$

Therefore,

$\text{Lie}_{\xi}\ (g(\xi,\xi))=(\text{Lie}_{\xi}\ g)(\xi,\xi)=2g(\xi,\xi)>0.$

Thus, the function $t:=|\xi|=\sqrt{g(\xi,\xi)}$ strictly increases along
$\xi$. For any point $p\in C$ denote the integral curve of $\xi$ containing a
point $p$ by $\gamma_{p}$ . Then any integral curve $\gamma_{p}$ is
diffeomorphic to $\mathbb{R}$ because there exists a strictly increasing
function along $\gamma$ ($t$ strictly increases along $\gamma$). Let $s_{p}$
be a coordinate on $\gamma_{p}$ such that

$\frac{\partial}{\partial s_{p}}=\xi.$

Then $t=|\xi|$ satisfies

$\frac{\partial}{\partial
s_{p}}t^{2}=\text{Lie}_{\xi}(g(\xi,\xi))=2g(\xi,\xi)=2t^{2}.$

Therefore,

$t=ae^{s_{p}}.$

where $a\in\mathbb{R}$ is a constant. Moreover $a\in\mathbb{R}^{>0}$ because
$t$ is positive defined. Then $t$ identifies $\gamma$ with $\mathbb{R}^{>0}$
and

$\xi=\frac{\partial}{\partial
s_{p}}=\frac{\partial}{\partial\ln(a^{-1}t)}=\left(\frac{\partial\ln(a^{-1}t)}{\partial
t}\right)^{-1}\frac{\partial}{\partial t}=t\frac{\partial}{\partial t}.$

Denote by

$M:=\\{p\in C|\ t(p)=1\\}$

the level line which is a smooth submanifold of codimension $1$ according to
the inverse function theorem. Since for any $p\in C$, the map
$t|_{\gamma_{p}}:\gamma_{p}\to\mathbb{R}^{>0}$ is an isomorphism, $\gamma_{p}$
has an unique intersection with $M$. Define

$\alpha:C\to M\times\mathbb{R}^{>0}$

by the rule

$\alpha(p)=(\gamma_{p}\cap M)\times t(p).$

For any $p\in C$ the restriction $t|_{\gamma_{p}}$ is an isomorphism.
Moreover,

$C=\bigcup_{p\in C}\gamma_{p}=\bigcup_{p\in M}\gamma_{p}.$

Therefore, $\alpha$ is an isomorphism.

Thus, we have $C=M\times\mathbb{R}^{>0}$ and $\xi=t\frac{\partial}{\partial
t}$, where $t$ is a coordinate on $M$. For any $a\in\mathbb{R}$ we have
$\exp(at\frac{\partial}{\partial t})=\lambda_{e^{a}}$, where
$\lambda_{e^{a}}(m\times t)=m\times e^{a}t$. Then

$\left.\frac{d}{da}\lambda_{e^{a}}g\right|_{a=0}=\text{Lie}_{\xi}\ g=2g.$

Therefore, for any $a_{0}\in\mathbb{R}^{>0}$ we have

$\left.\frac{d}{da}\lambda_{e^{a}}g\right|_{a=a_{0}}=\left.\frac{d}{da}\lambda_{e^{a_{0}+a}}g\right|_{a=0}=\left.\lambda_{e^{a_{0}}}\frac{d}{da}\lambda_{e^{a}}g\right|_{a=0}=2\lambda_{e^{a_{0}}}g.$

That is, we get

$\frac{d}{da}\lambda_{e^{a}}g=2\lambda_{e^{a}}g,$

for any $a\in\mathbb{R}$. Hence, for any $X\in TC$ we have

$\left(\frac{d}{da}\lambda_{e^{a}}g\right)(X,X)=2\lambda_{e^{a}}g(X,X).$

Since $g(X,X)>0$, we can define a smooth function $f_{x}$ on $\mathbb{R}^{>0}$
such that

$\left(\lambda_{e^{a}}g\right)(X,X)=f_{X}(e^{a})g(X,X),$

Thus, we have

$\left(\frac{d}{da}f_{X}\right)(e^{a})=2f_{X}(e^{a}).$

Moreover,

$\left.\left(\frac{d}{da}f_{X}\right)\right|_{a=0}=2.$

Therefore, for any $X$ we have

$f_{X}(e^{a})=e^{2a}.$

That is, for any $X\in TC$, we have

$\lambda_{e^{a}}g(X,X)=e^{2a}g(X,X)$

Thus,

$\lambda_{e^{a}}g=e^{2a}g.$

Hence, for any $q\in\mathbb{R}^{>0}$ we have

$\lambda_{q}g=q^{2}g.$

That is, $M\times\mathbb{R}^{>0}$ is a selfsimilar cone.

(iii) As in the previous part, for any $p\in C$ define

$t(p)=|\xi|(p)=\sqrt{\xi,\xi}\ (p).$

Consider a point $x\in C$. The level set

$M=\\{p\in C|t(p)=t(x)\\}$

is smooth submanifold by the same argument as in (ii). A manifold $C$ contains
a tubular neighborhood $U$ with respect to the field $\xi$. By the definition,
$U$ is diffeomorpic to an open subset $V\subset M\times\mathbb{R}$. The
diffeomorphism $\alpha:V\to U$ is defined by the rule

$\alpha:p\times s\to\exp_{p}(s\xi).$

As in (ii), by solving a differential equation on $t$ along integral curves we
get

$t=ae^{s}$

and

$\xi=t\frac{\partial}{\partial t}$

in $U$. Thus, $U$ is a subset $M\times\mathbb{R}^{>0}$ and
$\xi=t\frac{\partial}{\partial t}$ where $t$ is a coordinate on
$\mathbb{R}^{>0}$. For any $x\in U$ and $q\in\mathbb{R}^{>0}$ satisfying
$\lambda_{q}x\in U$ we have

$(\lambda_{q}^{*}g)(x)=q^{2}g.$

By Lemma 2.7, the metric $g$ on $U$ can be extended to a selfsimilar metric
$\tilde{g}$ on ${(\pi(U)\cap M)\times\mathbb{R}^{>0}}$. Therefore, $U$ is
isomorphic to a part of the selfsimilar cone $\left((\pi(U)\cap
M)\times\mathbb{R}^{>0},\tilde{g},t\frac{\partial}{\partial t}\right)$.

∎

######  Proposition 2.9.

Any conical Riemannian manifold is a selfsimilar manifold and any Riemannian
cone is a selfsimilar cone.

###### Proof.

The proposition is well-known (see e.g. [ACHK]) and immediately follows from
Proposition 3.3 and Proposition 3.8. ∎

######  Theorem 2.10.

Let $(M,g,\xi)$ be a selfsimilar manifold, $D$ be a Levi-Civita connection on
$M$, and $\theta=\iota_{\xi}g$. Then the following conditions are equivalent

  * (i)

$(M,g,\xi)$ is a conical Riemannian manifold.

  * (ii)

$D\theta=g$.

  * (iii)

$D\theta$ is a symmetric tensor.

  * (iv)

$\theta$ is a closed form.

###### Proof.

First of all, the Levi-Civita connection $D$ is compatible with the metric
$g$. That is, for any ${X,Y,Z\in TM}$ we have

$\text{Lie}_{X}(g(Y,Z))=g(D_{X}Y,Z)+g(Y,D_{X}Z).$

Therefore, for any $X,Y\in TM$

$(D\theta)(X,Y)=\text{Lie}_{X}(\theta(Y))-\theta(D_{X}Y)=\text{Lie}_{X}(g(\xi,Y))-g(\xi,D_{X}Y)=g(D_{X}\xi,Y).$
(2.5)

$\text{(i)}\Rightarrow\text{(ii)}$. For any $X\in TM$ we have

$D_{X}\xi=X.$

Combining this with (2.5), we get

$(D\theta)(X,Y)=g(D_{X}\xi,Y)=g(X,Y).$

$\text{(ii)}\Rightarrow\text{(iii)}$. This conclusion is trivial.

$\text{(iii)}\Leftrightarrow\text{(iv)}.$ The connection $D$ is torsion free
hence

$d\theta-\text{Alt}(D\theta)=0,$ (2.6)

where

$\text{Alt}:\Lambda^{1}M\otimes\Lambda^{1}M\to\Lambda^{2}M$

is the inner product. The tensor $D\theta$ is symmetric if and only if

$\text{Alt}(D\theta)=0.$

Combining this with (2.6), we get that the condition

$d\theta=0$

is satisfied if and only if the tensor $D\theta$ is symmetric.

$\text{(iii)}\Rightarrow\text{(i)}.$ Combining the condition (iii) and (2.5),
we get

$g(D_{X}\xi,Y)=g(D_{Y}\xi,X).$ (2.7)

Since $(M,g,\xi)$ is selfsimilar, we have

$2g(X,Y)=\text{Lie}_{\xi}(g)(X,Y)=\text{Lie}_{\xi}(g(X,Y))-g([\xi,X],Y)-g(X,[\xi,Y]).$
(2.8)

The connection $D$ is compatible with $g$ hence

$\text{Lie}_{\xi}(g(X,Y))=g(D_{X}\xi,Y)+g(X,D_{Y}\xi).$ (2.9)

Since connection $D$ is torsion-free,

$g([\xi,X],Y)+g(X,[\xi,Y])=g(D_{\xi}X,Y)-g(D_{X}\xi,Y)+g(X,D_{\xi}Y)-g(X,D_{Y}\xi)$
(2.10)

Combining (2.8), (2.9), and (2.10), we obtain

$2g(X,Y)=g(D_{X}\xi,Y)+g(X,D_{Y}\xi)$

Combining this with (2.7), we get that for any $X,Y\in TM$

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

is satisfied. Thus, for any $X\in TM$ we have

$D_{X}\xi=X.$

∎

######  Corollary 2.11.

Let

$g=t^{2}g_{M}+t\text{Sym}(dt\otimes\alpha)+fdt^{2}$

be a selfsimilar metric on $M\times\mathbb{R}^{>0}$. Then the Riemannian
manifold $(M\times\mathbb{R}^{>0},g)$ is a Riemannian cone if and only if

$df=-2\alpha.$

###### Proof.

We have

$d\iota_{t\frac{\partial}{\partial
t}}g=d(t^{2}\alpha+tfdt)=(2\alpha+df)tdt+t^{2}d\alpha.$

Therefore, if $df=-2\alpha$ then $d\alpha=0$ and
$d\iota_{t\frac{\partial}{\partial t}}g=0$. Hence, the Riemannian manifold
${(M\times\mathbb{R}^{>0},g)}$ is a Riemannian cone by Theorem 2.10.

On the other hand, let $(M\times\mathbb{R}^{>0},g)$ be a Riemannian cone.
Then, by Theorem 2.10,

$(2\alpha+df)tdt+t^{2}d\alpha=d\iota_{\frac{\partial}{\partial t}}g=0.$ (2.11)

Since $f$ and $\alpha$ does not depend on $t$, we have

$\iota_{\frac{\partial}{\partial t}}df=0,\ \ \ \iota_{\frac{\partial}{\partial
t}}d\alpha=0,\ \ \ \text{and}\ \ \iota_{\frac{\partial}{\partial t}}\alpha=0.$

Combining this with (2.11), we get

$0=\iota_{\frac{\partial}{\partial
t}}((2\alpha+df)tdt+t^{2}d\alpha)=(2\iota_{\frac{\partial}{\partial
t}}\alpha+\iota_{\frac{\partial}{\partial
t}}df)tdt+(2\alpha+df)t+t^{2}\iota_{\frac{\partial}{\partial
t}}d\alpha=(2\alpha+df)t.$

Hence,

$(2\alpha+df)t=0,$

and, since $t$ is positive definite,

$2\alpha+df=0.$

∎

######  Lemma 2.12.

Let $(M,g_{M})$ be a Riemannian manifold and $\alpha$ be a $1$-form on $M$.
Then there exists a smooth function $f$ on $M$ such that

$g_{f}:=t^{2}fg_{M}+t\text{Sym}(dt\otimes\alpha)+dt^{2}$

is a positive definite metric on $M\times\mathbb{R}^{>0}$.

###### Proof.

Let $SM$ be a spherical unit tangent bundle on $M$

$SM=\\{X\in TM|\ g_{M}(X,X)=1\\}.$

For any point $p\in M$, we have

$\sup_{X\in T_{p}M}\ \frac{(\alpha(X))^{2}}{g_{M}(X,X)}=\sup_{X\in S_{p}M}\
(\alpha(X))^{2}.$

Since $S_{p}M$ is compact,

$\sup_{X\in S_{p}M}\ (\alpha(X))^{2}<\infty.$

Moreover, $\alpha$ is continuous. Therefore,

$\sup_{X\in S_{p}M}\ (\alpha(X))^{2}$

is continuous function of $p$. Hence, there exists a smooth function $f$ on
$M$ such that

$f(p)>\sup_{X\in S_{p}M}\ (\alpha(X))^{2}=\sup_{X\in T_{p}M}\
\frac{(\alpha(X))^{2}}{g_{M}(X,X)}.$ (2.12)

Any vector field $V\in T(M\times{\mathbb{R}^{>0}})$ has a form

$V=X+a\frac{\partial}{\partial t},$

where $X\in TM$ and $a$ be a function on $M\times\mathbb{R}^{>0}$. We have

$g_{f}(V,V)=g_{f}(X+a\frac{\partial}{\partial t},x+a\frac{\partial}{\partial
t})=t^{2}fg_{M}(X,X)+2at\alpha(X)+a^{2}.$

Combining this with (2.12), we get

$g_{f}(V,V)>t^{2}(\alpha(X))^{2}+2at\alpha(X)+a^{2}=(t\alpha(X)+a)^{2}\geq 0.$

Thus, $g_{f}$ is positive definite.

∎

######  Proposition 2.13.

Let $(M,\alpha)$ be a contact manifold. Then there exists a Riemannian metric
$g$ on $M\times\mathbb{R}^{>0}$ such that $(M\times\mathbb{R}^{>0},g)$ is a
selfsimilar cone but not a Riemannian cone.

###### Proof.

Let $g_{M}$ be a Riemannian metric on $M$. By Lemma 2.12, there exists a
function $f$ on $M$ such that

$g=t^{2}fg_{M}+t\text{Sym}(dt\otimes\alpha)+dt^{2}$

is positive definite. The form $\alpha$ is contact. Hence,

$d\alpha\neq 0.$

Thus, the Riemannian manifold $(M\times\mathbb{R}^{>0},g)$ is a selfsimilar
cone but not a Riemannian manifold by Corollary 2.11.

∎

##  3 Conical Hessian manifolds

######  Definition 3.1.

A flat affine manifold $(M,\nabla)$ is a differentiable manifold $M$ equipped
with a flat torsion-free connection $\nabla$. Equivalently, it is a manifold
equipped with an atlas such that all translation maps between charts are
affine transformations (see e.g. [FGH]). A radiant manifold $(C,\nabla,\xi)$
is a flat affine manifold $(C,\nabla)$ endowed with a vector field $\xi$
satisfying

$\nabla\xi=\text{Id}.$ (3.1)

Equivalently, it is a manifold equipped with an atlas such that all
translation maps between charts are linear transformations (see e.g. [Go]).

######  Proposition 3.2 ([Go]).

Let $(C,\nabla,\xi)$ be a radiant manifold. Then the connection $\nabla$ is
compatible with the flow along $\xi$.

######  Corollary 3.3.

Let $U$ be an open subset of a manifold $M\times\mathbb{R}^{>0}$, $t$ be a
coordinate on $\mathbb{R}^{>0}$, $\pi:M\times\mathbb{R}^{{>0}}$ be a
projection, $\nabla$ be a connection on $U$ such that
$(M,\nabla,t\frac{\partial}{\partial t})$ is a radiant manifold. Then we can
extend $\nabla$ to a connection $\tilde{\nabla}$ on such that
$(\pi(U)\times\mathbb{R}^{>0},\tilde{a}\nabla,t\frac{\partial}{\partial t})$
is a radiant manifold.

###### Proof.

For any $q\in\mathbb{R}^{>0}$ consider a map
$\lambda_{q}:M\times\mathbb{R}^{>0}\to M\times\mathbb{R}^{>0}$ defined by
$\lambda_{q}(x\times t)=x\times qt$. Then the flow along the vector field
$t\frac{\partial}{\partial t}$ during the time $\ln q$ coincides with the map
$\lambda_{q}$. Hence, by proposition 3.2, for any $x\in U$ and
$q\in\mathbb{R}^{>0}$ satisfying $\lambda_{q}x\in U$ we have

$(\lambda_{q}^{*}\nabla)(x)=\nabla(x).$

Then the proof of Corollary 3.3 coincides with the proof of Lemma 2.7. ∎

######  Definition 3.4.

Let $(M,\nabla)$ be a flat affine manifold. If a Rimennian metric $g$ is
locally expressed by a Hessian of a function

$\text{Hess}\varphi=\nabla d\varphi$

then $g$ is called Hessian metric and the triple $(M,\nabla,g)$ is called a
Hessian manifold (see e.g. [Sh]).

######  Definition 3.5.

A conical Hessian manifold $(C,\nabla,g,\xi)$ is Hessian manifold
$(C,\nabla,g)$ endowed with a vector field $\xi$ such that $(C,g,\xi)$ is a
selfsimilar manifold and $(C,\nabla,\xi)$ is a radiant manifold.

If $\xi$ is complete then $C$ is called a global conical Hessian manifold.

######  Definition 3.6.

Let $t$ be a coordinate on $\mathbb{R}^{>0}$ and
${(M\times\mathbb{R}^{>0},\nabla,g,t\frac{\partial}{\partial t})}$ be a
conical Hessian manifold. Then we say that
${(M\times\mathbb{R}^{>0},\nabla,g)}$ is a Hessian cone

######  Theorem 3.7.

Let $(C,\nabla,g,\xi)$ be a conical Hessian manifold. Then $(C,g,\xi)$ is a
Riemannian conical manifold.

###### Proof.

Locally, the Hessian metric has the form $g=\text{Hess}\varphi=\nabla
d\varphi$, where $\varphi$ is a locally defined function. By Theorem 2.10, it
is enough to prove that

$d\iota_{\xi}\nabla d\varphi=0.$

For any $1$-form $\alpha$ we have

$d\alpha(X,Y)=X(\alpha(Y))-Y(\alpha(X))-\alpha[X,Y].$

Hence,

$\begin{split}(d\iota_{\xi}\nabla d\varphi)(X,Y)=X((\iota_{\xi}\nabla
d\varphi)(Y))-Y((\iota_{\xi}\nabla d\varphi)(X))-(\iota_{\xi}\nabla
d\varphi)[X,Y]=\\\ =X((\nabla d\varphi)(Y,\xi))-Y((\nabla
d\varphi)(X,\xi))-(\nabla d\varphi)([X,Y],\xi).\end{split}$ (3.2)

For any $1$-form $\alpha$ and any vector fields $Z,W$, we have

$\nabla\alpha(Z,W)=Z(\alpha(W)-\alpha(\nabla_{Z}W).$

Combining this with (3.2), we get

$\begin{split}(d\iota_{\xi}\nabla
d\varphi)(X,Y)=X(Y(d\varphi(\xi))-X(d\varphi(\nabla_{Y}\xi))-Y(X(d\varphi(\xi))+Y(d\varphi(\nabla_{X}\xi))-\\\
-[X,Y](d\varphi(\xi))+d\varphi(\nabla_{[X,Y]}\xi)=-X(d\varphi(\nabla_{Y}\xi))+Y(d\varphi(\nabla_{X}\xi))+d\varphi(\nabla_{[X,Y]}\xi)\end{split}$

(we have used the identity $XY-YX=[X,Y]$ in the second implication). Combining
this with (3.1), we obtain

$(d\iota_{\xi}\nabla
d\varphi)(X,Y)=-X(d\varphi(Y))+Y(d\varphi(X))+d\varphi({[X,Y]})=-XY(\varphi)+YX(\varphi)+[X,Y](\varphi)=0.$

Thus, we have proved that

$d\iota_{\xi}\nabla d\varphi=0.$

∎

######  Remark 3.1.

According to Theorem 3.7 any Hessian cone admits a form

${\left(M\times\mathbb{R}^{>0},\nabla,g=t^{2}g_{M}+dt^{2}\right)},$

where $t$ is a coordinate on $\mathbb{R}^{>0}$ and $g_{M}$ is a Riemannian
metric on $M$.

######  Lemma 3.8.

Let $(M\times\mathbb{R}^{>0},g)$ be a Rimannian cone,
$\pi:M\times\mathbb{R}^{>0}\to M$ be a projection, $U$ be an open subset of
$M\times\mathbb{R}^{>0}$ such that the exists a connection $\nabla$ on $U$
such that $(U,\nabla,g)$ is a Hessian manifold. Then we can extend $\nabla$ to
a connection $\tilde{\nabla}$ on $\pi(U)\times\mathbb{R}^{>0}$ such that
$(\pi(U)\times\mathbb{R}^{>0},\tilde{\nabla},g)$ is a Hessian cone.

###### Proof.

By corollary 3.3, we can extend $\nabla$ to a connection $\tilde{\nabla}$ on
$\pi(U)\times\mathbb{R}^{>0}$ such that

${\left(\pi(U)\times\mathbb{R}^{>0},\tilde{\nabla},t\frac{\partial}{\partial
t}\right)}$

is a radiant manifold. Since $\tilde{g}$ is selfsimilar
$(\lambda_{q}(U),\nabla\tilde{g})$ is isometric to $(U,q^{2}\tilde{g})$. The
restriction $\tilde{g}|_{U}$ is Hessian therefore
$\tilde{g}|_{\lambda_{q}(U)}$ is Hessian. Thus, $\tilde{g}$ is a Hessian
metric and ${(\pi(U)\times\mathbb{R}^{>0},\tilde{\nabla})}$ is a Hessian cone.
∎

######  Corollary 3.9.

  * (i)

Any global conical Hessian manifold is a Hessian cone.

  * (ii)

Any conical Hessian manifold is locally isomorphic to a Hessian cone.

###### Proof.

(i) Let $(C,\nabla,g,\xi)$ be a global conical Hessian manifold. Using Theorem
3.5. and Proposition 2.3, we get that $(C,g)$ is a Riemannian cone. That is,
we have

$(C,g,\xi)=\left(M\times\mathbb{R}^{>0},t^{2}g_{M}+dt^{2},t\frac{\partial}{\partial
t}\right).$

Therefore, $(C,\nabla,g)=(M\times\mathbb{R}^{>0},\nabla,t^{2}g_{M}+dt^{2})$ is
a Hessian cone.

(ii) Let $(C,\nabla,g,\xi)$ be a conical Hessian manifold. By Theorem 3.2 and
Proposition 2.3. $(C,\nabla,g,\xi)$ is locally isomorphic to a Riemannian
cone. Then, by Lemma 3.8, $C$ is locally isomorphic to a Hessian cone. ∎

##  4 A potential of a Hessian metric on a Hessian cone

######  Proposition 4.1 ([G-A]).

Let $(M,\nabla,\xi)$ be a radiant manifold and $g$ be a Hessian metric on $M$
with respect to $\nabla$. Then

$\text{Lie}_{\xi}g=g+\nabla(\iota_{\xi}g).$

######  Theorem 4.2.

Let $(M\times\mathbb{R}^{>0},g=t^{2}g_{M}+dt^{2})$ be a Hessian cone. Then

$g=\text{Hess}\left(\frac{t^{2}}{2}\right),$

where $t$ be a coordinate on $\mathbb{R}^{>0}$.

###### Proof.

By Theorem 3.7,

$g=t^{2}g_{M}+dt^{2},$

where $g_{M}$ be a metric on $M$ and $t$ be a coordinate on $\mathbb{R}^{>0}$.
Hence

$\iota_{t\frac{\partial}{\partial t}}g=tdt.$

Combining this with proposition 4.1, we get

$\text{Lie}_{t\frac{\partial}{\partial
t}}g=g+\nabla(\iota_{t\frac{\partial}{\partial
t}}g)=g+\nabla\left(tdt\right).$

The metric $g$ is selfsimilar. That is,

$\text{Lie}_{t\frac{\partial}{\partial t}}g=2g.$

Therefore,

$g=\nabla\left(tdt\right)=\nabla
d\left(\frac{t^{2}}{2}\right)=\text{Hess}\left(\frac{t^{2}}{2}\right).$

∎

##  5 Extensive Hessian manifolds

######  Definition 5.1.

Let $(C,\nabla,\xi)$ be a radiant manifold. We say that $(C,\nabla,g,\xi)$ is
extensive Hessian manifold if $g$ is a degenerate Hessian metric and
$\iota_{\xi}g=0$.

######  Proposition 5.2.

Let $(M\times\mathbb{R}^{>0},\nabla,g=t^{2}g_{M}+dt^{2})$ be a Hessian cone
then

$tg_{M}=\text{Hess}(t)$

and $(M\times\mathbb{R}^{>0},\nabla,tg_{M},t\frac{\partial}{\partial t})$ is
an extensive manifold.

###### Proof.

By Theorem 4.2

$g=\text{Hess}\left(\frac{t^{2}}{2}\right)=\nabla
d\left(\frac{t^{2}}{2}\right)(X,Y).$

Then

$g(X,Y)=XY\left(\frac{t^{2}}{2}\right)-\nabla_{X}Y\left(\frac{t^{2}}{2}\right)=X(t)Y(t)+tXY(t)-t\nabla_{X}Y(t)=X(t)Y(t)+t\text{Hess}(t)(X,Y).$

Thus,

$g=dt^{2}+t\text{Hess}(t).$

Therefore,

$t^{2}g_{M}=t\text{Hess}(t).$

and

$tg_{M}=\text{Hess}(t).$

We have

$\iota_{t\frac{\partial}{\partial t}}(tg_{M})=0.$

Moreover, $tg_{M}$ is a positive semi-definite Hessian of a function.
Therefore, $(M\times\mathbb{R}^{>0},\nabla,tg_{M},t\frac{\partial}{\partial
t})$ is an extensive manifold.

∎

Let $(C,\nabla,g,\xi)$ be a conical Hessian manifold. Then there exists a
decomposition

$TC=T_{\xi}C\oplus T_{\xi}^{\perp}C,$

where $T_{\xi}$ is generated by $\xi$. Then above we have

$g=g_{\xi}+g_{\perp}$

with respect to the decomposition above.

######  Theorem 5.3.

Let $(C,\nabla,g,\xi)$ be a conical Hessian manifold and $g_{\perp}$ be as
above. Denote

$\tilde{g}=(g(\xi,\xi))^{-\frac{1}{2}}g_{\perp}.$

Then $(C,\nabla,\tilde{g},\xi)$ is an extensive Hessian manifold.

###### Proof.

$(C,\nabla,g,\xi)$ is locally isomorphic to a Hessian cone
$(M\times\mathbb{R}^{>0},\tilde{\nabla},\tilde{g}=t^{2}g_{M}+dt^{2})$. That
is, for any $p\in C$ there exists neighborhoods $p\in V_{p}\subset C$,
$U_{p}\subset M\times\mathbb{R}^{>0}$ and an isimorphism

$\alpha_{p}:({V_{p}},\ \nabla|_{V_{p}},\ g|_{V_{p}},\
\xi|_{V_{p}})\to\left({U_{p}},\ \tilde{\nabla}|_{U_{p}},\ \tilde{g}|_{U_{p}},\
t\frac{\partial}{\partial t}|_{U_{p}}\right)$

Then we have

$g(\xi,\xi)|_{V_{p}}=\left.\alpha^{*}_{p}\left(\tilde{g}\left(t\frac{\partial}{\partial
t},t\frac{\partial}{\partial
t}\right)\right|_{U_{p}}\right)=\alpha^{*}_{p}(t^{2})|_{U_{p}}.$

and

$g_{\perp}|_{V_{p}}=\alpha^{*}_{p}(t^{2}g_{M}|_{U_{p}})$

Therefore,

$\tilde{g}|_{V_{p}}=g(\xi,\xi)^{-\frac{1}{2}}g_{\perp}|_{U_{p}}=\alpha^{*}_{p}(tg_{M}|_{U_{p}}).$

By Proposition 5.2, $tg_{M}$ is an extensive Hessian metric on
$M\times\mathbb{R}^{>0}$. Hence, $\tilde{g}|_{V_{p}}$ is an extensive Hessian
metric on $V_{p}$. Moreover,

$\bigcup_{p\in C}V_{p}=C.$

Therefore, $\tilde{g}$ is an extensive Hessian metric on $C$ and
$(C,\nabla,\tilde{g},\xi)$ is an extensive Hessian manifold. ∎

## References

  * [AC] D.V. Alekseevsky, V. Cortez, Geometric construction of the r-map: from affine special real to special Kähler manifolds, Comm. Math. Phys. 291 (2009), 579-590.
  * [ACDM] D. V. Alekseevsky, V. Cortés, M. Dyckmanns, T. Mohaupt, Quaternionic Kähler metrics associated with special Kähler manifolds, J.Geom.Phys. 92, (2013), 271-287.
  * [ACGL] D.V. Alekseevsky, V. Cortez, A.S. Galaev, T. Leistner, Cones over pseudo-Riemannian manifolds and their holonomy, Journal für die reine und angewandte Mathematik (Crelle’s Journal) 635 (2009), 23-69.
  * [ACM] D. V. Alekseevsky, V. Cortés, T. Mohaupt, Conification of Kähler and hyper-Kähler manifolds, Comm. Math. Phys. 324 (2013), no. 2, 637-655.
  * [ACHK] D. V. Alekseevsky, V. Cortés, K. Hasegawa, Y. Kamishima, Homogeneous locally conformally Kähler and Sasaki manifolds Int. J. Math. 6 (2015), 1541001 (29 pp).
  * [ACL] , D.V. Alekseevsky, V. Cortez, T. Leistner, Geometry and holonomy of indecomposable cones, arXiv preprint arXiv:1902.02493, (2019).
  * [BG] C. Boyer, K. Galicki, Sasakian geometry, Oxford Univ. Press, (2008).
  * [VDMV] V. Cortez, P. Dempster, T. Mohaupt, O. Vaughan, Special Geometry of Euclidean Supersymmetry IV: the local c-map JHEP 10 (2015) 066.
  * [CDM] V. Cortés, P.-S. Dieterich, T. Mohaupt, ASK/PSK-correspondence and the r-map, Lett. Math. Phys. 108 No. 5 (2018), 1279-1306
  * [CMMS] V. Cortez, C. Mayer, T. Mohaupt, F. Saueressig, Special Geometry of Euclidean Supersymmetry I: Vector Multiplets J. High Energy Phys. 03 (2004) 028.
  * [CM] V. Cortez, T. Mohaupt Special Geometry of Euclidean Supersymmetry III: the local r-map, instantons and black holes J. High Energy Phys. 2009, no. 7, 066, 66 pp.
  * [F1] A. Figalli, On the Monge-Ampère Equation 70e annee, no 1147 (2018)
  * [F2] A. Figalli, The Monge-Ampère Equation and Its Applications EMS Zurich Lectures in Advanced Mathematics 22, (2017), 210 pp.
  * [FGH] D. Fried, W. Goldman, M. Hirsch, Affine manifolds with nilpotent holonomy, Comment. Math. Helvetici 56, (1981), 487-523.
  * [G-A] Garcıa-Ariza M.A., Degenerate Hessian structures on radiant manifolds, Int. J. Geom. Methods Mod. Phys. 15 (2018), 15 pp.
  * [Go] W. M. Goldman, Projective geometry on manifolds, lecture Notes for Mathematics 748B, Spring 1988, University of Maryland.
  * [Gu] C. Gutierrez, The Monge-Ampère Equation Progress in Nonlinear Differential Equations and Their Applications, (2016).
  * [N] A. Nemirovski, Advances in convex optimization: conic programming, Plenary Lecture, International Congress of Mathematicians (ICM), Madrid, Spain (2006).
  * [NN] Nesterov, Y., Nemirovski, A. Interior-point polynomial algorithms in convex programming SIAM Studies in Applied Mathematics, vol.13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994).
  * [OV] L. Ornea M. Verbitsky Sasakian structures on CR-manifolds Geom Dedicata 125 (2007), 159-173.
  * [Sh] H. Shima, The geometry of Hessian structures, World Scientific Publishing Co. Pte. Ltd., Singapore, (2007).
  * [T] B. Totaro, The curvature of a Hessian metric, int. J. of Math. 15(04), (2004).
  * [W1] F. Weinhold, Metric geometry of equilibrium thermodynamics, J. Chem. Phys. 63 (1975), 2479–2483.
  * [W2] F. Weinhold, Thermodynamics and geometry Phys. Today 29, No. 3, 23 (1976)