Homogeneous hessian manifolds

Hirohiko Shima

Annales de l'institut Fourier (1980)

  • Volume: 30, Issue: 3, page 91-128
  • ISSN: 0373-0956

Abstract

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A flat affine manifold is said to Hessian if it is endowed with a Riemannian metric whose local expression has the form g i j = ∂ 2 Φ ∂ x i ∂ x j where Φ is a C ∞ -function and { x 1 , ... , x n } is an affine local coordinate system. Let M be a Hessian manifold. We show that if M is homogeneous, the universal covering manifold of M is a convex domain in R n and admits a uniquely determined fibering, whose base space is a homogeneous convex domain not containing any full straight line, and whose fiber is an affine subspace of R n .

How to cite

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Shima, Hirohiko. "Homogeneous hessian manifolds." Annales de l'institut Fourier 30.3 (1980): 91-128. <http://eudml.org/doc/74467>.

@article{Shima1980,
abstract = {A flat affine manifold is said to Hessian if it is endowed with a Riemannian metric whose local expression has the form $g_\{ij\}=\{\partial ^2\Phi \over \partial x^i\partial x^j\}$ where $\Phi $ is a $C^\infty $-function and $\lbrace x^1,\ldots ,x^n\rbrace $ is an affine local coordinate system. Let $M$ be a Hessian manifold. We show that if $M$ is homogeneous, the universal covering manifold of $M$ is a convex domain in $\{\bf R\}^n$ and admits a uniquely determined fibering, whose base space is a homogeneous convex domain not containing any full straight line, and whose fiber is an affine subspace of $\{\bf R\}^n$.},
author = {Shima, Hirohiko},
journal = {Annales de l'institut Fourier},
keywords = {homogeneous hessian manifolds; affine manifold; universal covering manifold; base space},
language = {eng},
number = {3},
pages = {91-128},
publisher = {Association des Annales de l'Institut Fourier},
title = {Homogeneous hessian manifolds},
url = {http://eudml.org/doc/74467},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Shima, Hirohiko
TI - Homogeneous hessian manifolds
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 3
SP - 91
EP - 128
AB - A flat affine manifold is said to Hessian if it is endowed with a Riemannian metric whose local expression has the form $g_{ij}={\partial ^2\Phi \over \partial x^i\partial x^j}$ where $\Phi $ is a $C^\infty $-function and $\lbrace x^1,\ldots ,x^n\rbrace $ is an affine local coordinate system. Let $M$ be a Hessian manifold. We show that if $M$ is homogeneous, the universal covering manifold of $M$ is a convex domain in ${\bf R}^n$ and admits a uniquely determined fibering, whose base space is a homogeneous convex domain not containing any full straight line, and whose fiber is an affine subspace of ${\bf R}^n$.
LA - eng
KW - homogeneous hessian manifolds; affine manifold; universal covering manifold; base space
UR - http://eudml.org/doc/74467
ER -

References

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  11. [11] J. VEY, Une notion d'hyperbolicité sur les variétés localement plates, C.R. Acad. Sci. Paris, 266 (1968), 622-624. Zbl0155.30602MR38 #5131
  12. [12] E.B. VINBERG, The Morozov-Borel theorem for real Lie groups, Soviet Math. Dokl., 2 (1961), 1416-1419. Zbl0112.02505MR26 #252
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