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%\leftheadtext{K. S. CHANG}
%\rightheadtext{Resume of Papers of H. Shima}
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${}$
\bigrm{
\centerline{Resume of Papers of H. Shima II}
}
\vskip 20pt
\bigtype
\centerline{
{November, 18, 1999}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 40pt
\topmatter
\abstract
The purpose of this talk
is to introduce the works of H. Shima,
mainly concentrated on affinely flat structures.
\endabstract
\endtopmatter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip 20pt
\definition{Definition} A {\it locally flat} linear connection $D$
on a connected smooth manifold $M$
if its torsion and curvature tensors vanish identically.
Then for each point $p \in M$, there exists a local coordinate system
$\{ x^1, \cdots, x^n \}$ in a neighborhood of $p$ such that
$D_{\pa \over \pa x^i} {\pa \over \pa x^j} = 0$,
which we call an {\it affine local coordinate} system.
A Riemannian metric $\text{g}$ on $M$ is said to be {\it locally Hessian}
with respect to $D$,
if for each point $p \in M$, there exists a real valued $C^\infty$ function
$\phi$ on a neighborhood of $p$ such that
$$\text{g} = D^2 \phi,$$
that is,
$$\text{g} = \sum_{ij} {{\pa^2\phi} \over {\pa x^i \pa x^j}}d x^i d x^j.$$
If $\phi$ is defined over $M$,
the metric $\text{g}$ is called {\it Hessian} metric on $M$.
A locally flat manifold with a (locally) Hessian metric is called a
(locally) {\it Hessian manifold}.
\enddefinition
\vskip 10pt
Let $M$ be an affine homogeneous manifold of a connected Lie group $G$.
Assume that $M$ admits a locally flat linear connection $D$ and
a volume element $\nu$ which are invariant under $G$.
If $\nu = K dx^1 \wedge \cdots \wedge dx^n$,
then let
$$\alpha = \sum_i {\pa \log K \over \pa x^i} d x^i,$$
$$D\alpha = \sum_{ij} {\pa^2 \log K \over \pa x^i \pa x^j} d x^i d x^j.$$
\vskip 20pt
Let $\Om$ be an affine homogeneous proper convex domain in $(\R^n, D)$.
Then there exists an invariant volume element
$\nu = K dx^1 \wedge \cdots \wedge dx^n$,
and letting $\alpha = \sum_i {\pa \log K \over \pa x^i} d x^i,$
we obtain the canonical bilinear form
$$D\alpha = \sum_{ij} {\pa^2 \log K \over \pa x^i \pa x^j} d x^i d x^j,$$
which defines an invariant Riemannian metric on $\Om$.
\vskip 10pt
$M$ flat affine manifold,
$\{x^1, \cdots,x^n \}$ local affine coordinate.
g Riemannian metric on $M$ is {\it Hessian}
if and only if
for all $p \in M$, $\exists \phi \in C^\infty(U_p)$
such that
$\text{g}_{ij} = {\pa^2 \phi \over \pa x^i \pa x^j},$
where $\phi$ is called a {\it primitive} of g on $M$.
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
\ref\key S1
\by \ H. Shima
\paper On locally symmetric homogeneous domains of completely reducible linear Lie groups
\jour Math. Ann.
\vol 217
\yr 1975
\pages 93 -- 95
\endref
}
\vskip 30pt
\bigtype
\proclaim{Theorem}
Let $\Om = G/H$ be a locally symmetric homogeneous domain in $\R^n$,
acted by a completely reducible linear Lie group $G$. Then
\roster
\item"(i)" $\Om$ is a $\omega$-domain and the canonical bilinear form
$D\alpha$ of $\Om$ is non-degenerate.
\item"(ii)" Let $(p,q)$ be the signature of $D\alpha$.
Then there exists a $p$-dimensional subspace $W$ of $\R^n$ such that
$\Om^+ = \Om \cap W$ is a homogeneous self-dual cone,
and $q$ is equal to the difference between the dimension of a maximal compact
subgroup of $G$ and the dimension of a maximal compact subgroup
of the identity component $H_0$ of $H$.
\item"(iii)" If $H_0$ is compact, then there exists a maximal compact subgroup $N$
of $G$ containing $H_0$ such that $\Om = N \Om^+$.
In particular, $H$ is a maximal compact subgroup of $G$
if and only if
$\Om$ is a homogeneous self-dual cone.
\endroster
\endproclaim
%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%
\bigsm{
\ref\key S2
\by \ H. Shima
\paper Symmetric spaces with invariant locally
Hessian structures
\jour J. Math. Soc. Japan
\vol 29, No. 3
\yr 1977
\pages 581 -- 589
\endref
}
\vskip 30pt
\bigtype
Let $G$ be a connected Lie group, $H$ be a closed Lie subgroup of $G$.
$(G,H)$ is called a {\it symmetric} pair
if and only if
there exists an involutive automorphism $\sigma$ of $G$ such that
$(H_\sigma)_0 \subset H \subset H_\sigma$,
where $H_\sigma = \{ g \in G \mid \sigma(g) = g \}$.
If, in addition, $H$ contains no non-trivial normal subgroup of $G$,
then $(G,H)$ is said to be an {\it effective symmetric} pair.
\vskip 30pt
\proclaim{Theorem}
Let $(G,H)$ be an effective symmetric pair.
If $M = G/H$ admits a locally Hessian structure $(D,\text{g})$
such that $D$ and g are invariant under $G$,
then
$M$ is affinely diffeomorphic and isometric with respect to
$D$ and g respetively
to a direct product
$$M_0 \times M_1 \times \cdots \times M_r,$$
where $M_0$ is a flat Riemannian manifold and
the universal covering manifold of $M_i \ (1 \geq i \leq r)$
is an irreducible homogeneous self-dual convex cone
with a canonical locally Hessian structure.
\endproclaim
%%%%%%%%%%%%%%
\vfil\newpage
%%%%%%%%%%%%%%
Note a canonical locally Hessian structure
if $D\alpha = D^2 \log K$.
\vskip 30pt
\proclaim{Proposition}
Let $G$ be a connected Lie group, $H$ be a closed subgroup of $G$
(not necessarily symmetric pair).
If $G/H$ admits an invariant locally Hessian structure,
then
$G$ is not semisimple.
\endproclaim
\vskip 30pt
\proclaim{Proposition}
Under the same assumption as in Theorem,
assume further that $f$ is a faithful representation of $\fg$ in $T_o(G/H) = \R^n$.
Then
the universal covering manifold of $G/H$
is an irreducible homogeneous self-dual convex cone.
\endproclaim
%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%
\bigsm{
\ref\key H1
\by \ \ H. Shima
\paper On certain locally flat homogeneous manifolds of solvable Lie groups
\jour Osaka J. Math.
\vol 13
\yr 1976
\pages 213 -- 229
\endref
}
\vskip 30pt
\bigtype
\proclaim{Proposition}
Let $M$ be a connected $C^{\infty}-$ manifold with locally flat linear connection $D$,
and a Riemannian metric $\text{\rm g}$.
Let $\gamma$ be a cotangent bundle valued 1-form on $M$ defined by
$$\gamma(X)(Y) = \text{\rm g}(X,Y), \ \ \text{ for } X, Y \in \frak X (M).$$
The cotangent bundle being locally flat, we may consider
the exterior differential $\ul d$ for cotangent bundle valued forms on $M$.
Then the following conditions are equivalent:
\roster
\item"(i)" $\text{\rm g}$ is locally Hessian with respect to $D$.
\item"(ii)" For each affine coordinate system $\{ x^1, \cdots, x^n \}$,
the components $\text{\rm g}_{ij}$ of $\text{\rm g}$ satisfy the relations
$$ {{\pa \text{\rm g}_{ij}} \over {\pa x^k}} = {{\pa \text{\rm g}_{ik}} \over {\pa x^j}}
\ \ (1 \geq i, j, k \leq n). $$
\item"(iii)" $(D_Z \text{\rm g})(X,Y) = (D_Y\text{\rm g})(X,Z)$
for all vector fields $X, Y, Z$ on $M$.
\item"(iv)" $\ul d \gamma = 0$.
\endroster
In addition to these equivalent conditions, assume further
$H^1(M,\R) = \{ 0 \}$ and that $D$ is flat.
Then $\text{\rm g}$ is a Hessian metric.
\endproclaim
%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%
Koszul :
Let $M$ be a homogeneous manifold with an invariant flat linear connection $D$ and
an invariant volume element $\nu$.
Then the bilinear form $D\alpha$ is positive definite
if and only if
$M$ is an affine homogeneous proper convex domain.
\vskip 30pt
\proclaim{Theorem}
Let $G$ be a connected solvable Lie group and $M$ an orientable smooth manifold
on which $G$ acts simply transitively.
Suppose that $M$ admits a locally flat linear connection $D$ and
a locally Hessian metric $\text{\rm g}$ with respect ot $D$,
which are invariant under $G$.
Let $\nu$ be the volume element defined by $\text{\rm g}$.
If the bilinear form $D\alpha$ determined by $\nu$ is non-degenerate,
then $D\alpha$ is positive definite.
\endproclaim
\vskip 30pt
Combined with the Koszul's result,
\proclaim{Corollary}
Under the same assumptions as in Theorem, assume further that $D$ is flat.
Then $M$ is an affine homogeneous proper convex domain.
\endproclaim
%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%
\definition{Definition}
A left-symmetric algebra $A$ is called {\it elementary}
if $A$ satisfies the following conditions:
\roster
\item"(E.1)" $A = \{ u \} + P$ (direct sum of vector spaces,
\item"(E.2)" $u \ \cdot u = u, \ u \neq 0$,
\item"(E.3)" $u \cdot P \subset P, \ P \cdot u = \{ 0 \}$,
\item"(E.4)" $p \cdot q = \Phi(p,q) u$ for $p,q \in P$,
where $\Phi$ is a symmetric bilinear form on $P$.
\endroster
\enddefinition
\vskip 30pt
\proclaim{Proposition}
Let $\Om$ be a homogeneous domain in $\R^n$ containing 0, on which an
affine Lie group $G$ acts simply transitively.
Suppose that the left-symmetric algebra $A$ of $\Om$ at 0 is
elementary, i.e., $A = \{ u \} + P$ satisfies the above conditions
from (E.1) to (E.4).
Then we have
$$\Om = \{ t u + p \mid t - {1 \over 2}\Phi(p,p) > -1 \text{ for }
t \in \R, p \in P \}.$$
In particular, if $Phi$ is a positive definite symmetric bilinear form
on $P$, then
$\Om$ is the interior of a paraboloid.
\endproclaim
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
\ref\key H2
\by \ \ H. Shima
\paper Homogeneous convex domains of negative sectional curvature
\jour J. Differential Geometry
\vol 12
\yr 1977
\pages 327 -- 332
\endref
}
\vskip 30pt
\bigtype
\proclaim{Theorem}
An affine homogeneous proper convex domain $\Om$
has negative sectional curvature with respect to $D\alpha$
if and only if
$\Om$ is the interior of a parabloid :
$$y\sp0 - {\textstyle\frac 1{2}}\{(y\sp 1)\sp 2+\cdots+(y\sp {n-1})\sp 2\} > -1,$$
where $\{y\sp 0,y\sp 1,\cdots,y\sp {n-1}\}$ is an affine coordinate system of $V$.
\endproclaim
\vskip 30pt
\definition{Definition}
A clan $A$ is called {\it elementary}
if $A$ satisfies the following conditions:
\roster
\item"(E.1)" $A = \{ u \} + P$ (direct sum of vector spaces,
\item"(E.2)" $u \ \cdot u = u, \ u \neq 0$,
\item"(E.3)" $u \cdot p = {1 \over 2} p, \ p \cdot u = \{ 0 \} \text{ for } p \in P $,
\item"(E.4)" $p \cdot q = \Phi(p,q) u$ for $p,q \in P$,
where $\Phi$ is a positive definite symmetric bilinear form on $P$.
\endroster
\enddefinition
\vskip 10pt
\proclaim{Theorem}
Let $A$ be a clan.
Then the following conditions are equivalent:
\roster
\item"(i)" the setional curvature $\Cal K < 0.$
\item"(ii)" $A$ is an elementary clan.
\endroster
\endproclaim
%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%
Let $\Om$ be an affine homogeneous proper convex domain in $(\R^n, D)$.
Let $G$ be a connected triangular affine Lie group which acts simply transitively on $\Om$,
Then $(\R^n,\cdot)$ becomes a left-symmetric algebra
and let $L_x(y) = x \cdot y $.
Define $\alpha_0(x) = \tr L_x$ for $x \in \R^n = T_0\Om$
and let $\lan x,y \ran = \alpha_0(x \cdot y).$
Since $A$ is a LSA, we have
$$\lan xy, z \ran - \lan yx, z \ran = \lan x, yz \ran - \lan y, xz \ran.$$
The algebra $A$ together with the linear function $\alpha_0$
is said to be a {\it clan} corresponding to $\Om$.
\vskip 10pt
\proclaim{Proposition}
The Riemannian connection $\na$ for $D\alpha$ is given by
$$\na_x y = {1 \over 2} (L_x - {}^t\! L_x) y,$$
i.e., $\na_x$ is the skew symmetric part of $L_x$.
\endproclaim
\vskip 10pt
\proclaim{Proposition}
Let $S_x$ be the symmetric part of $L_x$, i.e.,
$S_x y = {1 \over 2} (L_x + {}^t\! L_x) y.$
Then we have
\roster
\item"(i)" $S_x y = S_y x,$ i.e., $\na_x$ is the skew symmetric part of $L_x$.
\item"(ii)" the curvaturetensor $R$ is given by
$$R(x,y) = - [S_x,S_y],$$
\item"(iii)" the sectional curvature $\Cal K$ is given by
$$\Cal K(x,y) = {{{\vert S_x y \vert}^2 - \lan S_x x, S_y y \ran}
\over {{\vert x\vert}^2{\vert y\vert}^2 - {\lan x, y \ran}^2 }}.$$
\endroster
\endproclaim
\vskip 10pt
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
\ref\key H3
\by \ \ H. Shima
\paper Compact locally Hessian manifolds
\jour Osaka J. Math.
\vol 15
\yr 1978
\pages 509 -- 513
\endref
}
\vskip 30pt
\medtype
Let $M$ be an oriented manifold with a locally flat connection $D$
and with a Riemannian metric g such that
in a neighborhood of every point,
g is of the form $\pa^2 \phi /\pa x^i \pa x^j$
where $\phi$ is a $C^\infty$ function
and where $x^i$ are local coordinate system satisfying $D(dx^i)=0.$
The connection $D$ and the Riemannain volume $\nu$ associated to g
determine on $M$ a differentiable form $\beta$ of degree 2 ;
locally,
if $\nu = K dx^1 \wedge ... \wedge dx^n,$
$\beta = \sum \beta_{i j} dx^i dx^j$,
where $\beta_{i j}= \pa^2 \log (K)/ \pa x^i \pa x^j.$
\vskip 10pt
The author prove that
if $M$ is compact, then
\roster
\item"(i)" $\int_M (\sum \beta_{ii}) \nu \geq 0,$
\item"(ii)" if this integral is zero, then g is locally flat.
\endroster
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
\ref\key H4
\by \ \ H. Shima
\paper Homogeneous (locally) Hessian manifolds
\jour Ann. Inst. Fourier, Grenoble
\vol 30, No. 3
\yr 1980
\pages 91 -- 128
\endref
}
\vskip 30pt
\medtype
A differentiable manifold $M$ provided
with a locally flat linear connection $D$ and a Riemannian metric g
is said to be Hessian(in fact, {\bf locally Hessian}) (and g is called a Hessian metric)
if g has the form $\text{g} = D^2 \phi$, where $\phi$ is a function,
in a neighbourhood of any point of $M$.
The tangent bundle $TM$ of a differentiable manifold $M$ provided
with a locally flat linear connection and a Riemannian metric g
admits in a natural way a Hermitian metric $\text{g}^T$,
and g is a Hessian metric on $M$
if and only if
$\text{g}^T$ is a Kahlerian metric on $TM$.
Thus the geometry of Hessian manifolds is related with that of certain Kahlerian manifolds.
A diffeomorphism of a Hessian manifold onto itself is called an automorphism
if it preserves both the locally flat linear connection and the Hessian metric.
A Hessian manifold $M$ is said to be homogeneous
if the group of its automorphisms $\on{Aut}(M)$ acts transitively on $M$.
\vskip 10pt
The main result of the paper under review is the following theorem.
If M is a connected homogeneous Hessian manifold
then the domain of definition $E_x$ for the exponential mapping $exp_x$
at a point $x \in M$ given by the locally flat linear connection
is an affine homogeneous convex domain in the tangent space $T_xM$ of $M$ at $x$ ;
moreover,
$E_x$ is the universal covering manifold of $M$ with covering projection
$exp_x : E_x \ra M$.
Several corollaries are deduced from the theorem.
Here are some of them:
(1) If the connected Lie subgroup $G$ of $\on{Aut}(M)$
acts transitively on a Hessian manifold $M$
and if the isotropy subgroup of $G$ at a point in $M$ is discrete,
then $G$ is a solvable Lie group.
(2) If a connected homogeneous Hessian manifold $M$
admits a transitive reductive Lie subgroup of $\on {Aut}(M)$,
then the universal covering manifold of $M$ is a direct product of a Euclidean space
and an affine homogeneous convex self-dual cone not containing any full straight line.
(3) A compact connected homogeneous Hessian manifold is a Euclidean torus.
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
\ref\key H5
\by \ \ H. Shima
\paper Hessian manifolds and convexity
\jour in manifolds, and Lie groups, papers in honor of Y. Matsushima,
Progress in mathematics
\vol 14
\yr 1981
\pages 385 -- 392
\endref
}
\vskip 30pt
\bigtype
Define a cotangent valued 1-form $\text{g}^0$ by
$\text{g}^0(X)(Y) = \text{g}(X,Y)$ for $X,Y \in \frak X(M)$.
$\ul d $ : exterior differential for tensor bundle valued forms on $M$.
\vskip 20pt
\proclaim{Theorem}
Let $M$ be a connected Hessian manifold with Hessian metric {\text\rm g}.
suppose $\text{\rm g}^0$ is $\ul d$-exact, that is,
$M$ admits a closed 1-form $\alpha$ such that
$\text{\rm g}^0 = \ul d \alpha$.
If there exists a subgroup $G$ of affine transformations of $M$
such that $M/G$ is quasi-compact and that
$G$ leaves $\alpha$ invariant,
then
the universal covering manifold of $M$ is a proper convex domain
in a real affine space.
\endproclaim
\vskip 20pt
Let $M$ be a Hessian manifold with flat affine connection $D$ and
a Hessian metric g.
Define $f \in \on{Aut}(M)$ if $f \in \on{Diff}(M)$,
$f^*D = D, f^*\text{g} = \text{g}$.
\proclaim{Theprem}
Let $M$ be a connected Hessian manifold.
If there exists a subgroup $G$ of $\on{Aut}(M)$ such that
$M/G$ is quasi-compact,
then
the universal covering manifold of $M$ is a convex domain
in a real affine space.
\endproclaim
\vskip 20pt
\proclaim{Corollary}
Let $M$ be a connected Hessian manifold.
If $M$ admits a transitive automorphism group or is compcat
then
the universal covering manifold of $M$ is a convex domain
in a real affine space.
\endproclaim
\vskip 10pt
\noindent
{\bf Remark} ([Yagi, 1981]) showed in "On Hessian structures on an affine manifold" :
Flat affine manifold whose universal covering manifold
is a convex domain in a real affine space
does not necessarily admit a Hessian metric.
%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%
Let $\Om$ be a convex domain in a real affine space $\R^n$
and $\Gamma$ a properly discontinuous subgroup of
affine transformation group of $\R^n$
acting freely on $\Om$.
Then the quotient space $\Om/\Gamma$ admits in a natural way
a flat affine structure induced from that of $\R^n$.
Let $\Cal A$ denote the vector space of all affine functions on $\R^n$.
Then $\Gamma$ acts on $\Cal A$ by $g \cdot f = f \circ g^{-1}$,
where $g \in \Gamma, f \in \Cal A$
and this makes $\Cal A$ into a $\Gamma$-module.
Let $H^r(\Gamma, \Cal A)$ be the $r$-th cohomology group of $\Gamma$
with coefficients in $\Cal A$.
\vskip 30pt
\proclaim{Theorem}
Suppose that $\Om/\Gamma$ admits a Hessian metric $\text{\rm g}$.
\roster
\item"(i)" If $H^1(\Gamma, \Cal A) = 0$,
there exists a primitive of $\text{\rm g}$ globally defined on $\Om/\Gamma$.
\item"(ii)" If $\Om/\Gamma$ is compact,
then we have $H^1(\Gamma, \Cal A) \neq 0$.
\endroster
\endproclaim
\vskip 30pt
Let $\Cal P$ denote the vector space of all parallel
(with respect to the natural flat affine connection on $\R^n$)
1-forms on $\R^n$.
Then $\Gamma$ acts on $\Cal P$ by
$g \cdot \omega = (g^{-1})^* \omega$,
where $g \in \Gamma, \omega \in \Cal P$
and this action makes $\Cal P$ into a $\Gamma$-module.
Let $H^r(\Gamma, \Cal P)$ be the $r$-th cohomology group of $\Gamma$
with coefficients in $\Cal P$.
\vskip 10pt
\proclaim{Theorem}
Suppose that $\Om/\Gamma$ admits a Hessian metric $\text{\rm g}$.
If $H^1(\Gamma, \Cal P) = 0$,
the cotangent bundle valued 1-form $\text{\rm g}^0$
corresponding to $\text{\rm g}$ is $\ul d$-exact.
Assuming further that $\Om/\Gamma$ is compact,
$\Om$ is a proper convex domain.
\endproclaim
%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
{
\ref\key H6
\by \ \ H. Shima
\paper A differential geometric characterization of homogeneous self-dual cones
\jour Tsukuba J. Math.
\vol 6, No. 1
\yr 1982
\pages 79 -- 88
\endref
}}
\vskip 30pt
\medtype
Let $\Omega$ be an affine homogeneous convex domain
in an n- dimensional real vector space $V^n$ not containing any full straight line.
Then, by using the characteristic function $\phi$ of $\Omega$,
we can define an invariant Riemannian metric g on $\Omega$ in the following way:
$$\text{g} = \sum_{i,j=1}^n {\pa^2 \log \phi \over \pa x^i \pa x^j} d x^i d x^j,$$
where $(x^1, ... ,x^n)$ denotes an affine coordinate system on $V^n$.
\vskip 10pt
Let $\na$ and $R$ be the Riemannian connection and the Riemannian curvature tensor for g, respectively.
Denoting by $\Gamma^i_{jk}$ the components of $\na$ with respect to $(x^1, ... ,x^n)$,
we can see that $\Gamma^i_{jk}$ defines a tensor field on $\Omega.$
We denote this tensor field by the same letter $\na.$
\vskip 10pt
In this article,
the author gives characterizations of self-dual cones among affine homogeneous convex domains as follows:
(1) A homogeneous convex cone $\Omega$ not containing any full straight line is a self-dual cone
if and only if $R$ is parallel with respect to $\na.$
(2) A homogeneous convex domain $\Omega$ not containing any full straight line is a self-dual cone
if and only if $\na$ is parallel with respect to $\na.$
\vskip 10pt
The result (1) above gives an affirmative answer to a problem posed by Y. Matsushima in 1965.
The same result as in (1) was also obtained by T. Tsuji independently
[Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), 185 - 187].
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
{
\ref\key H7
\by \ \ H. Shima
\paper Vanishing theorems for compact Hessian manifolds
\jour Ann. Inst. Fourier, Grenoble
\vol 36, 3
\yr 1986
\pages 183 -- 205
\endref
}}
\vskip 30pt
\bigtype
([Serre, 1955] duality Theorem)
\proclaim{Theorem}
Let $M$ be a compact oriented flat affine manifold of dimension $n$.
Then we have
$$H\sp {p,q}(F) \cong H\sp {n-p,n-q}((K\otimes F)\sp *),$$
where $K$ is the canonical line bundle over $M$ and
$(K\otimes F)\sp *$ denotes the dual bundle of $K\otimes F$.
\endproclaim
\vskip 30pt
(KodairaNakano's vanishing Theorem)
\proclaim{Theorem}
Let $M$ be a compact oriented Hessian manifold.
Denote by $K$ the canonical line bundle over $M$.
Let $F$ be a locally constant line bundle over $M$.
\roster
\item"(i)" If $2F + K$ is positive, then
$$H\sp {p,q}(F) = 0 \ \text{ for } p + q > n.$$
\item"(ii)" If $2F+ K$ is negative, then
$$ H\sp {p,q}(F) = 0 \ \text{ for } p + q < n.$$
\endroster
\endproclaim
%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%
([Koszul, 1968])
\proclaim{Theorem}
Let $M$ be a compact oriented hyperbolic affine manifold.
Then we have
$$H\sp {p,q}(1) = 0 \ \text{ for } p, q > 0,$$
where $1$ is the trivial line bundle over $M$.
\endproclaim
\vskip 30pt
Let $F$ be a locally constant line bundle over $M$.
Choose an open covering $\{ U_\la \}$ of $M$
such that the local triviality holds on each $U_\la$.
Denote $\{ f_{\la \mu} \}$the constant transition functions
with respect to $\{ U_\la \}$.
A fiber metric $ a = \{a_\la \}$ on $F$
is a collection of positive $C^\infty$-functions $a_\la$ on $U_\la$
such that
$$a_\mu = f_{\la\mu}^2 a_\la.$$
Using this we can define a globally defined closed 1-form
$-D \log a_\la$ and $-D^2 \log a_\la$.
\vskip 10pt
\definition{Definition}
A locally constant line bundle $F$ is said to be
{\it positive} (resp. {\it negative})
if
the globally defined 2-form $(-D\sp 2\log a\sb \lambda)$ is positive definite
[resp. negative definite].
\enddefinition
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
\ref\key H8
\by \ \ H. Shima
\paper Hessian manifolds of constant Hessian sectional curvature
\jour J. Math. Soc. Japan
\vol 47, No. 4
\yr 1995
\pages 735 -- 753
\endref
}
\vskip 30pt
\bigtype
Let $(M,D,\text{g})$ a Hessian manifold,
$\{x^1, \cdots, x^n\}$ be an affine coordinate with respect to $D$.
Let $\na$ be the Levi-Civita connection for g.
Let $\gamma (X,Y) = \na_XY - D_XY$,
and $S = D\gamma$.
Let $\nu$ be the volume element for g,
and $D_X\nu = \alpha(X) \nu$,
$\beta = D \alpha$.
Then $S^i_{jkl} = {\pa \gamma^i_{jl} \over \pa x^k}$.
Let $\Cal S$ be an endomorphism of the space of covariant
symmetric tensor fields of degree 2 defined by
$$\Cal (\xi)^{ik} = {S^i_j}^k_l \xi^{jl}.$$
\definition{Definition}
For a non-zero contravariant symmetric tensor $\xi_x$
of degree 2 at $x$ we set
$$h(\xi_x) = \lan \Cal S (\xi_x), \xi_x \ran
\over \lan \xi_x, \xi_x\ran,$$
and call it the {\it Hessian sectional curvature}
in the direction $\xi_x$,
where $\lan, \ran$ is the inner product by g.
\enddefinition
\vskip 10pt
\definition{Definition}
A Hessian manifold $(M,D,\text{g})$ is said to be a {\it space of
constant sectional curvature c}
if $h(\xi_x)$ is constant $c$ for all contravariant symmetric
tensor $\xi_x$ at $x$ and for all point $x$ in $M$.
\enddefinition
\vskip 10pt
\proclaim{Theorem 1}
Let $(M,D,\text{\rm g})$ be a Hessian manifold of $dim \geq 2$.
if the Hessian sectional curvature $h(\xi_x)$ depends only on $x$,
then $(M,D,\text{\rm g})$is of constant Hessian sectional curvature $c$
if and only if
$$S_{ijkl} = {c \over 2}(\text{\rm g}_{ij} \text{\rm g}_{kl}
+ \text{\rm g}_{il} \text{\rm g}_{kj}).$$
\endproclaim
\vskip 10pt
\proclaim{Corollary 2}
If a Hessian manifold $(M,D,\text{\rm g})$ is of constant Hessian sectional curvature $c$,
then
the Riemannian manifold $(M, \text{\rm g})$
is a space of constant sectional curvature $-c \over 4$.
\endproclaim
\vskip 10pt
\proclaim{Corollary 3}
Let $(M,D,\text{\rm g})$ be a simply connected Hessian manifold
of constant Hessian sectional curvature $c$.
If the Riemannian metric $\text{\rm g}$ is complete,
then $c \geq 0$.
\endproclaim
\vskip 10pt
\definition{Definition}
A Hessian manifold $(M,D,\text{\rm g})$ is said to be
{\it Hessian-Einstein}
if $\beta = \la \text{\rm g}$.
\enddefinition
\vskip 10pt
\proclaim{Corollary 4}
A Hessian manifold of constant Hessian sectional curvature $c$,
is Hessian-Einstein ;
$$\beta = \{ {(n+1) c \over 2} \} \text{\rm g}.$$
\endproclaim
\vskip 10pt
Define a tensor field $W$ by
$$W^i_{jkl} = S^i_{jkl} - {1 \over {n+1}}(\delta^i_j\beta_{kl}
+ \delta^i_l\beta_{kj}).$$
\vskip 10pt
\proclaim{Theorem 5}
$(M,D,\text{\rm g})$ is of constant Hessian sectional curvature
if and only if
$W = 0$.
\endproclaim
\vskip 10pt
\proclaim{Theorem 6}
We have
$$\tr\Cal S^2 \geq {2 \over {n(n+1)}}(\tr \beta)^2,$$
where $\tr \beta = \beta^i_i$.
The equality holds
if and only if
$(M,D,\text{\rm g})$ is of constant Hessian sectional curvature
\endproclaim
\vskip 20pt
2. Constructions of Hessian manifold of constant Hessian sectional
curvature.
4. Characterizations of Hessian manifolds of constant Hessian
sectional curvature by affine Chern classes.
\vskip 10pt
\proclaim{Theorem 10}
Let $(M,D,\text{\rm g})$ be a Hessian-Einstein manifold of dimension $n$.
Then we have
$$\{ -n c_1^2 + 2(n+1)c_2)c_2 \} \wedge \text{\rm g}^{n-2} \geq 0.$$
The equality holds
if and only if
$(M,D,\text{\rm g})$ is of constant Hessian sectional curvature.
\endproclaim
\vskip 10pt
\proclaim{Theorem 11}
Let $(M,D,\text{\rm g})$ be a Hessian-Einstein manifold of dimension $n$.
Then we have
$$\{ -(n-2) c_1^2 + 2(n-1)c_2)c_2 \} \wedge \text{\rm g}^{n-2} \geq 0.$$
The equality holds
if and only if
$(M,D,\text{\rm g})$ is of constant Hessian sectional curvature.
\endproclaim
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
{
\ref\key H9
\by \ H. Shima, K. Yagi
\paper Geometry of Hessian manifolds
\jour Differential Geom. Appl.
\vol 7, No. 3
\yr 1997
\pages 277--290
\endref
}}
\vskip 30pt
\bigtype
\proclaim{Theorem 2.1}
If $M$ is a simply connected Hessian manifold
whose metric $\text{\rm g}$ is complete,
then
the affine development of $M$ is an affine-diffeomorphism
whose image is a convex domain in $\E^n$.
\endproclaim
\vskip 30pt
\proclaim{Theorem 4.2}
The following conditions are equivalent :
\roster
\item"(i)" $M$ admits a $D$-parallel volume element ;
\item"(ii)" $\na = D$ ;
\item"(iii)" $\alpha = 0$ ;
\item"(iv)" $\beta = 0$.
\endroster
\endproclaim
\vskip 10pt
\proclaim{Corollary 4.3}
Suppose the Hessian metric is rational.
If $D$ is complete, then $\na = D$.
\endproclaim
%%%%%%%%%%%\vskip 20pt
\vfill\newpage
%%%%%%%%%%%%
Let $M$ is an oriented compact Hessian manifold
with Hessian structure $(D,\text{g})$.
Let $\{ x_1, \cdots, x^n \}$ be a local affine coordinate
with respect to $D$, i.e., $D dx^i = 0$.
Let $\na$ be the levi-Civita connection for g.
Denote by $\nu$ the volume element determined by g.
Define a closed 1-form $\alpha$ and
a symmetric bilinear form $\beta$ by
$$D_X \nu = \alpha(X) \nu,$$
$$\beta = D \alpha.$$
\vskip 10pt
For a flat vector bundle $F$ over $M$,
denote $\Gamma(F)$ the space of all smooth sections of $F$.
Set
$$\Om^{p,q} = \Gamma((\Lambda^pT^*M) \otimes (\Lambda^qT^*M)).$$
We call $\varphi$ in $\Om^{p,q}$ a $(p,q)$-form
and express $\varphi$ locally as follow ;
$$\varphi = c \sum_{I, \ol J} \varphi_I dx^I \otimes dx^{\ol J},$$
where $I =( i_1, \cdots, i_p), J = (j_1, \cdots, j_q),$
$dx^I = dx^{i_1} \wedge \cdots \wedge dx^{i_p}$.
Define an operator $\pa_D : \Om^{p,q} \ra \Om^{p+1,q}$ by
$$\pa_D \varphi
= c \sum_{I, \ol J} \{ d \varphi_I \wedge dx^I \} \otimes dx^{\ol J}.$$
Then $\{ \sum_p\Om^{p,q}, \pa_D \}$ is a complex
which is similar to the Dolbeault complex for a complex manifold.
\vskip 10pt
\definition{Definition}
We define certain covariant derivatives $D^\prime_X$ and $\ol D_X^\prime$
of $\Gamma(TM \otimes TM)$ in the direction of $X \in \Gamma(TM)$ by
$$D^\prime_X = 2 \gamma_X \otimes I + D_X,$$
$$\ol D_X^\prime = 2 I \otimes \gamma_x + D_X,$$
where $\gamma_X = \na_X - D_X$ and $I$ is the identity map.
\enddefinition
%%%%%%%%%%%%\vskip 20pt
\vfill\newpage
%%%%%%%%%%%%%%
Let $\fg$ be the abelian Lie algebra
of all $D$-parallel vector fields on $M$
By Corollary 4.5 and $(H_6)$, the restriction of the Hessian metric g on $\fg$
is an inner product on $\fg$.
Let $\{ X_1, \cdots, X_r \}$ be an orthonomal basis of $\fg$
with respect to g ; $\text{g}(X_i,X_j) = \delta_{ij}$.
Again by Corollary 4.5, the dual 1-forms
$\{ \omega^1, \cdots, \omega^r \}$ defined by
$$\omega^i(Y) = \text{\rm g}(X_i,Y) \ \text{ for } Y \in \Gamma^(TM)$$
are $D$-parallel.
Set $\fg_p = \{ X_p \in T_pM \mid X \in \fg \}$
and denote by $\Cal D_p$ the orthogonal complement of $\fg_p$
with respect to g.
Since $\Cal D_p = \{ v \in T_pM \mid \omega^i(v) = 0 \text{ for all} i \}$
and $d \omega^i = 0$,
the assignment $\Cal D :\ p \mapsto \Cal D_p$ is an integrable distribution.
\vskip 10pt
\proclaim{Theorem 5.1}
Let $G$ be the abelian Lie group generated by $\fg$
and $N_p$ the maximal integal submanifold of $\Cal D$ through $p$.
Then we have
\roster
\item"(i)" Each $G$-orbit $G \cdot q$ is totally geodesic with respect to
$D, D^\prime$ and $\na$.
The restrictions of $(D,\text{\rm g})$ and $(D^\prime, \text{\rm g})$ on $G \cdot q$
are Hessian structures and $D = D^\prime = \na$ on $G \cdot q$.
\item"(ii)" $N_p$ is totally geodesic with respect to $D, D^\prime$ and $\na$
and the restrictions of $(D, \text{\rm g})$ and $(D^\prime, \text{\rm g})$ on $N_p$
are Hessian structures.
\item"(iii)" $M = G \cdot N_p$.
\endroster
\endproclaim
\vskip 10pt
\proclaim{Corollary 5.2}
Suppose $N_p$ is closed in $M$.
Let $H$ be a closed subgroup of $G$ which consists of all elements
in $G$ preserving $N_p$.
Then $G/H$ is a torus and
$$M = G \times_H N_p \ra G/H$$
is a fiber bundle.
\endproclaim
%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
{
\ref\key I1
\by \ J. H. Hao, H. Shima
\paper Level surfaces of non-degenerate functions in $\R^{n+1}$
\jour Geometriae Dedicata
\vol 50, No. 2
\yr 1994
\pages 193 -- 204
\endref
}}
\vskip 30pt
\bigtype
\proclaim{Thorem1} Let $\tilde E$ denote the affine normal on a level surface
$M$ of $\phi$ with respect to the volume element $d x^1 \wedge \cdots d\wedge
x^{n+1}$ on $\Om$. Then we have $$ \tilde E = Z + \mu \tilde E,$$ w
here $\mu = \vert { {\det ( \tilde {\text{\rm g}_{ij} })} \over {(\tilde E \phi)^{n+1}} }
\vert^{1 \over {n+2}}$, and $Z \in \frak X (M)$ is given by
$$ \tilde{ \text{\rm g} } (X,Z) = ( d ( \mu \tilde E \phi ) )(X) \text{ for } X \in \frak X (M).$$
Moreover, we have
$$ d \log \vert \mu \tilde E \phi \vert = {1 \over {n+1}} ( \tau + 2 \tilde \alpha) \text{ on } \frak X (M).$$
\endproclaim
\vskip 20pt
\proclaim{Corollary 2}
$\tilde E$ is the direction of the affine normal $\ol E$
if and only if
$$\tilde \alpha(X) = - {1 \over 2} \tau(X) \text{ for } X \in \frak X (M).$$
\endproclaim
\vskip 20pt
\proclaim{Corollary 3}
Suppose $\tau = 0$.
$\tilde E$ is the direction of the affine normal $\ol E$
if and only if
$$\tilde \alpha(X) = f d \phi .$$
\endproclaim
\vskip 10pt
\proclaim{Theorem 4} The following
$$ S = I, \ \ \tau = 0, \ \ \la = 1, \ \ D h = 0$$
hold for all level surfaces of $\phi$
if and only if
$\phi$ is a non-degenerate polynomial of degree 2.
In this case $\tilde E$ is in the direction of the affine normal $\ol E$.
\endproclaim
\vskip 10pt
\proclaim{Theorem 5} The following
$$ S = 0, \ \ \tau = 0, \ \ \la = -1$$
hold for all level surfaces of $\phi$
if and only if,
with an appropriate choice of affine coordinate systems,
$\phi$ is expressed as
$$ \phi = k \log( x^{n+1} - F(x^1, \cdots, x^n)),$$
where $F$ is a smooth function on a domain in $\R^n$ with
$\det ({\pa^2F \over \pa x^i \pa x^j}) \neq 0$
and $k$ is a non-zero constant.
\endproclaim
\vskip 10pt
\proclaim{Corollary 6}
Let $F$ be a smooth convex function defined on $\R^n$, i.e.,
$({\pa^2F \over \pa x^i \pa x^j})$ is positive definite on $\R^n$.
For the non-degenerate function
$ \phi = \log( x^{n+1} - F(x^1, \cdots, x^n))$
defined on $x^{n+1} > F(x^1, \cdots, x^n)$,
$\tilde E$ is in the direction of the affine normal $\ol E$
if and only if
$F$ is a convex polynomial of degree 2.
\endproclaim
\vskip 10pt
\proclaim{Theorem 7} The following
$$ S = - I, \ \ \tau = 0, \ \ \la = - 1$$
hold for all level surfaces of $\phi$
if and only if
$d \phi$ is invariant under a certain 1-parameter group of dilations.
\endproclaim
\vskip 10pt
\proclaim{Corollary 8}
Let $\Om$ be an open proper convex cone
and $\psi$ be the characteristic function of $\Om$.
For a non-degenerate function $\phi = \log \psi$, we have
$$ S = - I, \ \ \tau = 0, \ \ \la = - 1.$$
\endproclaim
\vskip 10pt
\proclaim{Corollary 9}
Let $\Om$ be an open proper convex domain.
Suppose $\Om$ is homogeneous under the action of a group $G$
of affine automorphism group of $\Om$.
For the non-degenerate function $\phi = \log \psi$, we have
$$ \tau = 0, \ \ \la = - 1 \ \ \tilde \alpha = d \phi,$$
and $\tilde E$ is in the direction of the affine normal $\ol E$.
Moreover, $\Om$ is a cone
if and only if
$S = - I$.
\endproclaim
%%%%%%%%%%%%%%
\vskip 20pt
Let $\tilde D$ be the canonical flat affine connection on $\R^{n+1}$,
and let $\{ x^1, \cdots x^{n+1} \}$ be an affine coordinate system on $\R^{n+1}$.
A real valued smooth function $\phi$ defined on a domain $\Om$ in $\R^{n+1}$
is said to be {\it non-degenerate}
if it satisfies the following conditions:
\roster
\item"(i)" $d \phi_x \neq 0$ for $x \in \Om$,
\item"(ii)" the Hessian $\tilde D d \phi = \sum {\pa^2 \phi \over \pa x^i \pa x^j} d x^i d x^j$
of $\phi$ is non-degenerate on $\Om$.
\endroster
From now on $\phi$ si always assumed to be a non-degenerate function on $\Om$.
The Hessian $\tilde D d \phi$ of $\phi$ gives a pseudo-Riemannian meric on $\Om$
and is denoted by $\tilde {\text{g}}$.
Define a canonical vector field $\tilde E$ on $\Om$ by
$$ \tilde{\text{g}} (\tilde X, \tilde E) = (d \phi)(\tilde X) \ \text{ for } \tilde X \in \frak X(\Om).$$
Assume that $d \phi(\tilde E)$ does not identically vanish in $\Om$.
Let $M$ be a level surface of $\phi$. Denote $\frak X(M)$ the set of all
tangent vector fields on $M$. A vector field $X$ along $M$ is tangent to $M$ if
and only if $(d \phi)(X) = 0$.
\vskip 10pt
Whenever $\tilde E$ is transversal to $M$,
we can define the induced affine connection $D$ and the affine fundamental form $h$ by
$$ \tilde D_X Y = D_XY + h(X,Y)\tilde E \text{ for } X, Y \in \frak X(M).$$
We also define the shape operator $S$ and the transversal connection form $\tau$ on $M$ by
$$ \tilde D_X \tilde E = S(X) + \tau(X)\tilde E \text{ for } X \in \frak X(M).$$
\vskip 10pt
\proclaim{Lemma 1}
The covariant tensor field of degree 3 defined by
$$ (\tilde D_{\tilde X} \tilde{ \text{\rm g} }) (\tilde Y, \tilde Z))
\text{ for } \tilde X, \tilde Y, \tilde Z \in \frak X(\Om),$$
is symmetric with respect to $\tilde X, \tilde Y, \tilde Z$.
\endproclaim
\vskip 10pt
\proclaim{Lemma 2}
We have
\roster
\item"(i)" $h(X,Y) = - (\tilde E \phi)^{-1} \tilde{\text{\rm g}}(X,Y)$,
\item"(ii)" $\tau(X) = (d \log \vert \tilde E \phi \vert)(X) \text{ for } X, Y \in \frak X(M)$.
\endroster
\endproclaim
\vskip 10pt
\proclaim{Lemma 3}
The following conditions are equivalent :
\roster
\item"(i)" $\tau = 0$.
\item"(ii)" $\tilde D_{\tilde E}\tilde E = \la \tilde E$.
\endroster
In this case, we have
$$d(\tilde E \phi) = (\la + 1)d \phi,$$
in particular, $\tilde E\phi$ is a constant on $\Om$
if and only if
$\la = -1$.
\endproclaim
\vskip 10pt
We define a closed 1-form$\tilde{\alpha}$ on $\Om$ by
$$ \tilde{ \alpha} = d \log \vert \det (\tilde{ {\rm g}_{ij} }) \vert^{1 \over 2}.$$
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
{
\ref\key I2
\by \ H. Shima
\paper Harmonicity of grandient mapping of level surfaces in
a real affine spaces
\jour Geometriae Dedicata
\vol 56, No. 2
\yr 1995
\pages 177 -- 184
\endref
}}
\vskip 30pt
\bigtype
\proclaim{Theorem 1}
Let $\tilde D^\prime$ be a flat affine connection on $\Om$
locally defined by
$$ \iota_* (\tilde D^\prime_{\tilde X} (\tilde Y)) = \tilde D^*_{\tilde X}(\tilde Y)
\text{ for } \tilde X, \tilde Y \in \frak X(\Om),$$
where the right-hand $\tilde D^*$ of the above formula is the covariant
differentiation along $\iota$ induced by the flat affine connection
$\tilde D^*$ on $\E^*_{n+1}$.
Then we have
\roster
\item"(i)" $\tilde D^\prime = 2 \tilde \na - \tilde D$,
\item"(ii)" $\tilde X \tilde{\text{\rm g} }(\tilde Y, \tilde Z)
= \tilde{\text{\rm g} }(\tilde D_{\tilde X}\tilde Y, \tilde Z)
+ \tilde{\text{\rm g} }(\tilde Y, {{\tilde D}^\prime}_{\tilde X} \tilde Z)$
for $\tilde X, \tilde Y, \tilde Z \in \frak X(\Om)$.
\endroster
\endproclaim
The connection $\tilde D^\prime$ is called the dual(conjugate) affine connection
of $\tilde D$ with respect to $\tilde{\text{g}}$.
%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%
Let $\tilde D$ be the canonical flat affine connection on $\E^{n+1}$,
and let $\{ x^1, \cdots x^{n+1} \}$ be an affine coordinate system on $\E^{n+1}$.
A domain $\Om$ in $\E^{n+1}$ admitting a function $\phi$ on $\Om$
is said to be a {\it Hessian domain}
if the Hessian $\tilde D d \phi = \sum {\pa^2 \phi \over \pa x^i \pa x^j} d x^i d x^j$
of $\phi$ is non-degenerate,
and is denoted by $(\Om,\tilde D, \tilde{\text{g} } = \tilde D d \phi)$.
Let ${\E_{n+1}}^*$ be the dual affine space of $\E^{n+1}$
with the dual affine coordinate system $\{ x_1^*, \cdots, x_{n+1}^* \}$
and let $\tilde D^*$ be the canonical flat affine connection
given by $\tilde D^* d x_i^* = 0$.
Consider an immersion $\iota$ of a Hessian domain $(\Om, \tilde D, \tilde D d \phi)$
into $\E_{n+1}^*$ defined by
$$ x_i^* \circ \iota = { \pa \phi \over \pa x^i}.$$
Since $\iota$ is locally invertible,
we define locally the Legendre transformation $\phi^*$ of $\phi$ by
$$ \phi^* = \sum^{n+1}_{i=1} (x^i \circ \iota^{-1}) {x_i}^* - \phi \circ \iota^{-1}.$$
Then we have
$$ {\pa \phi^* \over \pa x_i^*} = x^i \circ \iota^{-1},$$
$${\pa^2 \phi^* \over \pa x_i^* \pa x_j^*} = \tilde{\text{g}}^{ij} \circ \iota^{-1},$$
where $({\tilde{\text{g}}}^{ij})= ({\tilde{\text{g}}}_{ij})^{-1}$
and ${\tilde{\text{g}}}_{ij} = {\pa^2 \phi \over \pa x^i \pa x^j}$.
Suppose $\iota$ is invertible on a domain $\Om_0$ in $\Om$.
Then $\iota$ maps a Hessian domain $(\Om_0, \tilde D, \tilde D d \phi)$
to a Hessian domain $(\Om_0^* = \iota (\Om_0), \tilde D^*, \tilde D^* d \phi^*)$
and $\iota$ is locally involutive, i.e.,
$\iota^{-1} = \iota$ on $\Om_0^*$.
%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%
Let recall the Laplacian $\Delta \iota$ of $\iota$ relative to $(\tilde{\text{g}}, \tilde D^*):$
$$\Delta \iota = \sum_{ij}{ \tilde{\text{g}} }^{ij}
\{ \tilde D^*_{\pa \over \pa x^i}\iota_* ( {\pa \over \pa x^j} )
- \iota_*( \tilde \na_{\pa \over \pa x^i} {\pa \over \pa x^j} ) \}.$$
By calculation, then we have
$$ \Delta \iota = \tilde \alpha.$$
\vskip 10pt
\proclaim{Theorem 2}
Let $\phi$ be a convex function on $\E^{n+1}$.
Then the following conditions are equivalent :
\roster
\item"(i)" $\iota$ is harmonic, i.e., $\Delta \iota = 0$;
\item"(ii)" $\tilde \alpha = 0$;
\item"(iii)" $\phi$ is a polynomial of degree 2.
\endroster
\endproclaim
\vskip 20pt
Assume that a function $\phi$ on $\Om$ satisfies
\roster
\item"(i)" $\tilde D d \phi$ is non-degenerate ;
\item"(ii)" $d \phi(x) \neq 0$ for $x \in \Om$ ;
\item"(iii)" $d \phi(\tilde E)$ does not vanish at any point in $\Om$,
\endroster
where $\tilde E$ is a canonical vector field $\tilde E$ on $\Om$ defined by
$$ \tilde{\text{g}} (\tilde X, \tilde E) = (d \phi)(\tilde X) \ \text{ for } \tilde X \in \frak X(\Om).$$
Let $M$ be a level surface of $\phi$. Denote $\frak X(M)$ the set of all
tangent vector fields on $M$. A vector field $X$ along $M$ is tangent to $M$ if
and only if $(d \phi)(X) = 0$.
Whenever $\tilde E$ is transversal to $M$,
we can define the induced affine connection $D$ and the affine fundamental form $h$ by
$$ \tilde D_X Y = D_XY + h(X,Y)\tilde E \text{ for } X, Y \in \frak X(M).$$
$$ \tilde D_X \tilde E = S(X) + \tau(X)\tilde E \text{ for } X \in \frak X(M).$$
Let $\iota$ be the restriction of $\tilde \iota$ (= $\iota$ on $\Om$ ain the above) to $M$
and g be the restriction of $\tilde{g}$ to $M$ with
$\na$ the Levi-Civita connection for g.
Let $\{X_1, \cdots, X^n \}$ be a local basis on $M$.
Then
$$\Delta \iota = \sum_{ij}{{\text{g}} }^{ij}
\{ \tilde D^*_{X_i}\iota_* (X_j)- \iota_*(\na_{X_i} X_j ) \},$$
where $(\text{g}^{ij}) = (\text{g}(X_i,X_j))^{-1}$.
\vskip 10pt
\proclaim{Theorem 3}
We have
\roster
\item"(i)" $\Delta\iota(X) = \tilde \alpha (\tilde X) + {1 \over 2} \tau(X)$ for $X \in \frak X(M)$,
\item"(ii)" $\Delta\iota(\tilde E) = - \tr S$.
\endroster
\endproclaim
\vskip 10pt
\proclaim{Corollary 4}
The following conditions are equivalent :
\roster
\item"(i)" $\iota$ is harmonic, i.e., $\Delta\iota = 0$ ;
\item"(ii)" $\tr S = 0$ and $\tilde \alpha = - {1 \over 2}\tau$ on $M$ ;
\item"(iii)"$\tr S = 0$ and $\tilde E$ is in the direction of the affine normal.
\endroster
\endproclaim
\vskip 20pt
The affine Bernstein problem conjectured by Chern can be restated :
Let $F$ be a smooth convex function defined on $\E^n$.
If the conormal mapping of the graph $x^{n+1} = F(x^1, \cdots, x^n)$
is harmonic, then
the graph is an elliptic paraboloid.
\vskip 10pt
\proclaim{Corollary 5}
Let $F$ be a smooth convex function defined on $\E^n$,
and let $\Om$ be the domain in $\E^{n+1}$
lying above the graph $x^{n+1} = F(x^1, \cdots, x^n)$.
For a level surface $M$ of the function
$$ \phi = \log( x^{n+1} - F(x^1, \cdots, x^n))\text{ on } \Om, $$
the mapping $\iota$, defined at first,
is harmonic
if and only if
$M$ is elliptic paraboloid.
\endproclaim
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%
\bigsm{
{
\ref\key P1
\by \ H. Shima
\paper Homogeneous spaces with invariant projectively flat affine connections
\jour Trans. Amer. Math. Soc.
\vol 351, No. 12
\yr 1999
\pages 4713 -- 4726
\endref
}}
\vskip 30pt
\bigtype
Let $D$ be an affine connection on an $n$-dimensional manifold.
Assume that $D$ is torsion free and Ricci-symmetric.
$D$ is said to be {\it projectively flat}
if $D$ is locally projectively equivalent to a flat affine connection.
\vskip 3pt
\centerline{
$T(X,Y) = T^D(X,Y) = D_XY - D_YX - [X,Y],$}
\vskip 3pt
\centerline{
$R(X,Y)Z = R^D(X,Y)Z
= D_XD_YZ - D_YD_XZ - D_{[X,Y]}Z.$}
\vskip 3pt
\noindent
If $T = 0$ and $R$ is identically 0 on $M$, we say that
$\na$ is a {\it flat} affine connection.
We always assume the torsion tensor is zero.
\definition{Definition 1}
An affine connection $D$ with zero torsion is said to be
(locally) {\it equiaffine} if it admits locally a parallel volume element,
that is,
a nonvanishing $n$-form $\omega$ such that
$D \omega = 0$.
\enddefinition
\vskip 10pt
\definition{Definition 2}
The Ricci tensor of an affine connection $D$ is defined by
$$Ric(Y,Z) = \text{ \rm{ trace of the linear map } }
X \mapsto R(X,Y)Z \tag 1 $$
for any vector fields $X,Y,Z$.
\enddefinition
\vskip 10pt
\proclaim{Proposition 3}
A torsion free affine connection $D$ has symmetric Ricci tensor
if and only if it admits locally a parallel volume element
if and only if $D$ is equiaffine.
\endproclaim
For a proof, see [NP] or [NS, p14].
\vskip 10pt
\definition{Definition 4}
Two torsion free (locally) equiaffine connections $D$ and $\ol D$
are {\it projectively equivalent} if there is a 1-form $\rho$ such that
$$\ol{D}_XY = D_XY + \rho(X)Y + \rho(Y)X \tag 2 $$
for all tangent vector fields $X$ and $Y$.
In this case, $\rho$ is necessarily closed, i.e,
$\rho = d \log \la$ for some positive function $\la$ (see [NS, p38]).
\enddefinition
\vskip 10pt
\proclaim{Proposition 5}
Let $D$ be a torsion free affine connection on a manifold $M$.
Given a volume element $\omega$ on $M$,
there is a unique torsion free affine connection $\ol D$ on $M$
such that
\roster
\item"(i)" $\ol D$ is projectively equivalent to $D$ ;
\item"(ii)" $\ol D \omega = 0$, i.e.,
$\omega$ is parallel relative to $\ol D$.
\endroster
\endproclaim
For a proof, see [NP] or [Ei, p104].
\vskip 10pt
\definition{Definition 6}
An affine connection $D$ with zero torsion and symmetric Ricci tensor
is said to be {\it projectively flat} if in a neighborhood of each point,
$D$ is projectively equivalent to a flat affine connection.
\enddefinition
\proclaim{Proposition 7}
A torsion free equiaffine connection $D$ is projectively flat
if and only if
\roster
\item"(i)" the Weyl projective curvature tensor $W$ defined by
$$W(X,Y)Z = R(X,Y)Z - \{ \gamma(Y,Z)X - \gamma(X,Z)Y\}$$
is identically zero ; and
\item"(ii)" $\gamma$ satisfies Codazzi equation :
$$(D_{\!X}\gamma)(Y,Z) = (D_{\!Y}\gamma)(X,Z),$$
where $\gamma = {1 \over \text{ n-1}}Ric$ for all vector fields $X, Y, Z$.
\endroster
Furthermore, if dim $\geq 3$ then (ii) follows from (i).
For dim $=2$, (i) is automatically satisfied.
\endproclaim
\vskip 10pt
\remark{\bf Remark 8}
(i) For proofs see [Ei, p95].
\noindent
(ii) It is known that the Levi-Civita connection of a nondegenerate metric
$\text{g}$ is projectively flat if and only if
$\text{g}$ has constant sectional curvature (see [NP, p411]).
\endremark
\vskip 10pt
\proclaim{Proposition 9}
Let {\rm g} be a metric tensor with Levi-Civita connection $D$
with respect to $\text{g}$. If $M$ has constant sectional curvature $k$,
then
$$R(X,Y)Z = k \{\text{\rm g}(Y,Z)X - \text{\rm g}(X,Z)Y \}.$$
\endproclaim
\noindent
The converse is also true.
For a proof see [O'ne, p80] or [KN, p203].
We say that an affine connection $D$ on $M$ is {\it metric}
if $D \text{g} = 0$.
That is,
$ 0 = (D_X \text{g})(Y,Z)$
$ = X\text{g}(Y,Z) - \text{g}(D_XY, Z)$
$- \text{g}(Y,D_XZ)$.
If $x,y,z$ in $\fg$ then
with $xy = D_X Y$, this is equivalent to
$0 = \text{g}(xy,z) + \text{g}(y,xz)$ and also
$\la_x + \la_x^\prime = 0$ under the nondegenerate $\text{g}$.
\vskip 10pt
\definition{Definition 10}
A hypersurface $M$ embedded in $\R^{n+1}$ is said to be a
{\it centro-affine hypersurface}
if the position vector $x$ (from 0) for each point $x \in M$ is transversal
to the tangent plane of $M$ at $x$.
\enddefinition
%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%
Let $G$ be a simply connected Lie group and
$K$ a connected closed subgroup of $G$.
Assume that $G$ acts effectively on $G/K$.
Denote $\fg$ and $\frak k$ the Lie algebras of $G$ and $K$, resp.
Enlarge $\tilde \fg = \fg \oplus \R E$,
with $[\tilde \fg , E] = \{0 \}$.
\vskip 10pt
\proclaim{Theorem}
A simply connected effective homogeneous space $G/K$
admits a $G$-invariant projectively flat affine connection
if and only if
$\tilde \fg$ has an affine representation $(\tilde f, \tilde q)$
on a real affine space $\tilde V$,
that is,
\roster
\item"(i)" $\tilde f : \tilde\fg \ra \fgl(\tilde V)$
is a Lie algebra homomorphism,
\item"(ii)" $\tilde q$ is a linear mapping from $\tilde \fg$ to $\tilde V$
such that
$$\tilde q ([\tilde X, \tilde Y]) = \tilde f(\tilde X)\tilde q(\tilde Y)
- \tilde f(\tilde Y)\tilde q(\tilde X),$$
\endroster
with the following properties,
\roster
\item"(iii)" $\text{\rm dim}\tilde V = \text{\rm dim}G/K + 1$,
\item"(iv)" $\tilde q$ is surjective and the kernel id $\frak k$,
\item"(v)" $\tilde f(E)$ is the identity mapping $I_{\tilde V}$ of $\tilde V$
and $\tilde q(E) \neq 0.$
\endroster
\endproclaim
\vskip 10pt
\proclaim{Corollary}
Let $G/K$ be a simply connected homogeneous space. Then
the following conditions are equivalent.
\roster
\item"(i)" $G/K$ admits an invariant projectively flat affine connection.
\item"(ii)" $G/K$ has an equivalent centro-affine hypersurface immersion
into a real affine space.
\endroster
\endproclaim
\vskip 10pt
\proclaim{Theorem} $($[Koszul, Vinberg]$)$
There is a natural one-to-one correspondence between
\roster
\item"(i)" $n$-dimensional simply connected Lie groups
with left invariant flat affine connections
up to affine diffeomorphism ;
\item"(ii)" $n$-dimensional left symmetric algebra over $\R$
up to algebraic isomorphism.
\endroster
\endproclaim
\vskip 10pt
\proclaim{Theorem}
There is a natural one-to-one correspondence between
\roster
\item"(i)" $n$-dimensional simply connected Lie groups
with left invariant projectively flat affine connections
up to equivariant projective diffeomorphism ;
\item"(ii)" $(n+1)$-dimensional left symmetric algebra over $\R$
with unit up to algebraic isomorphism.
\endroster
\endproclaim
\vskip 20pt
\proclaim{Theorem}
Let $G/K$ be an effective symmetric pair
where $G$ is semisimple.
Suppose that the space $G/K$ admits a $G$-invarinat projectively flat affine connection.
Then there exists a central-simple Jordan algebra $\tilde V$
with unit $e$ such that
\roster
\item"(i)" $\tilde V = V \oplus \R e$ (direct sum as vector spaces).
\item"(ii)" Let $\frak m(V) = \{ L_u \mid u \in V \}$ and
let $\frak k (V)$ be the vector space spanned by
$[L_u,L_v]$ where $u,v \in V$.
Then $\fg (V) = \frak k(V) + \frak m(V)$ is a Lie algebra and
is isomorphic to the Lie algebra $\fg = \frak k + \frak m$ of $G$
including decompositions.
\endroster
\endproclaim
Conversely, we have
\vskip 10pt
\proclaim{Theorem}
Let $\tilde V$ be a central-simple Jordan algebra with unit $e$.
We set $V = \{ v \in \tilde V \mid \tr L_v= 0 \}$.
Let $\frak m(V)= \{ L_v \mid v \in V \}$
and let $\frak k(V)$ be the vector space spanned by $[L_u,L_v]$
for $u,v\in V$.
Then $\frak k(V)$ and $\fg (V) = \frak k(V) + \frak m(V)$
are linear Lie algebras.
Let $G(V)$ and $K(V)$ are linear Lie groups generated by $\fg(V)$ and $\frak k(V)$, respectively.
Then $(G(V), K(V))$ is a symmetric pair,
where $G(V)$ is a semi-simple Lie group,
and $G(V)/K(V)$ admits a $G(V)$-invariant projectively flat affine connection.
\endproclaim
\vskip 10pt
Let $(G,K)$ be an effective symmetric pair where $G$ is semi-simple
and let $\fg = \frak k + \frak m$ be the canonical decomposition,
that is,
$$[\frak k, \frak k] \subset \frak k,
[\frak k, \frak m] \subset \frak m,
[\frak m, \frak m] \subset \frak k.$$
Suppose that $G/k$ admits a $G$-invariant projectively flat affine connection.
Enlarg $\fg$ so that
$\tilde \fg = \fg \oplus \R E$,
with $[\tilde \fg , E] = \{0 \}$,
and set $\tilde {\frak k} = \frak k$, $\tilde {\frak m} = \frak m + \R E$.
Then
$$[\tilde{\frak k}, \tilde{\frak m}] \subset \tilde{\frak m},
[\tilde{\frak m}, \tilde{\frak m}] \subset \tilde{\frak k}.$$
By above Theorem,
there exists an affine representation
$(\tilde f, \tilde q) : \tilde \fg \ra \faff(\tilde V)$,
where $V = T_0 G/K$, $\tilde V$ is a real affine space with dim$\tilde V$ = dim$V$ + 1.
The restriction of $\tilde q$ to $\tilde \frak m$ being an siomorphism,
for each $u \in \tilde V$
there exists a unique element $X_u \tilde \frak m$ such that
$\tilde q (X_u) = u$.
We put
$$ L_u = \tilde f(X_u),$$
and define a multiplication law in $\tilde V$ by
$$ u \cdot v = L_u(v).$$
Then the algebra $\tilde V$ is commutative and
has unit $e = \tilde q (E)$.
\vskip 10pt
An algebra $\tilde V$ over $\R$ is said to {\it Jordan algebra}
if for all $u,v$ in $\tilde V$,
$$ u \cdot v = v \cdot u,$$
$$ u \cdot (u^2 \cdot v) = u^2 \cdot (u \cdot v).$$
The {\it center} $Z(\tilde V)$ of a Jordan algebra $\tilde V$
is by definition([Braun, Koecher])
$$ Z(\tilde V) = \{ u \in \tilde V \mid (u,v,w) = 0 \}.$$
A Jordan algebra $\tilde V$ with unit $e$ is said to be
{\it central-simple}
if $\tilde V$ is simple and $Z(\tilde V) = \R e.$
%%%%%%%%%%%%%%%%%%%%%
\vfill\newpage
%%%%%%%%%%%%%%%%%%%%%
\vskip 1cm
\Refs
\vskip 0.5cm
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\endref
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\by H. Shima
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\paper Homogeneous convex domains
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\jour J. Differential Geometry
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\by J. H. Hao, H. Shima
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\paper Homogeneous spaces with invariant projectively flat affine connections
\jour Trans. Amer. Math. Soc.
\vol 351, No. 12
\yr 1999
\pages 4713 -- 4726
\endref
\endRefs
\enddocument