Homogeneous hessian manifolds
Annales de l'Institut Fourier, Tome 30 (1980) no. 3, pp. 91-128.

Une variété d’une connexion affine plate est dite hessienne si elle est munie d’une métrique riemannienne qui s’exprime localement g ij = 2 Φ x i x j Φ est une fonction C et {x 1 ,...,x n } est un système de coordonnées locales affines. Soit M une variété hessienne. On montre que si M est homogène, le revêtement universel de M est un domaine convexe dans R n et admet une fibration uniquement déterminée dont la base est un domaine convexe homogène ne contenant aucune droite et dont le fibré est un sous-espace affine de R n .

A flat affine manifold is said to Hessian if it is endowed with a Riemannian metric whose local expression has the form g ij = 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 .

@article{AIF_1980__30_3_91_0,
     author = {Shima, Hirohiko},
     title = {Homogeneous hessian manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {91--128},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {30},
     number = {3},
     year = {1980},
     doi = {10.5802/aif.794},
     mrnumber = {597019},
     zbl = {0424.53023},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.794/}
}
TY  - JOUR
AU  - Shima, Hirohiko
TI  - Homogeneous hessian manifolds
JO  - Annales de l'Institut Fourier
PY  - 1980
SP  - 91
EP  - 128
VL  - 30
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.794/
DO  - 10.5802/aif.794
LA  - en
ID  - AIF_1980__30_3_91_0
ER  - 
%0 Journal Article
%A Shima, Hirohiko
%T Homogeneous hessian manifolds
%J Annales de l'Institut Fourier
%D 1980
%P 91-128
%V 30
%N 3
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.794/
%R 10.5802/aif.794
%G en
%F AIF_1980__30_3_91_0
Shima, Hirohiko. Homogeneous hessian manifolds. Annales de l'Institut Fourier, Tome 30 (1980) no. 3, pp. 91-128. doi : 10.5802/aif.794. https://aif.centre-mersenne.org/articles/10.5802/aif.794/

[1] S.G. Gindikin, I.I. Pjateckii-Sapiro and E.B. Vinberg, Homogeneous Kähler manifolds, in “Geometry of Homogeneous Bounded Domains”, Centro Int. Math. Estivo, 3 Ciclo, Urbino, Italy, 1967, 3-87. | Zbl

[2] P. Dombrowski, On the geometry of the tangent bundles, J. Reine Angew. Math., 210 (1962), 73-88. | MR | Zbl

[3] M. Goto, Faithful representations of Lie groups I, Math. Japon., 1 (1948), 1-13. | Zbl

[4] J. Helmstetter, Doctorat de 3e cycle “Radical et groupe formel d'une algèbre symétrique à gauche” novembre 1975, Grenoble.

[5] S. Kobayashi, Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. of Tokyo, IA 24 (1977), 129-135. | MR | Zbl

[6] J.L. Koszul, Domaines bornés homogènes et orbites de groupes de transformations affines, Bull. Soc. Math. France, 89 (1961), 515-533. | Numdam | MR | Zbl

[7] J.L. Koszul, Variétés localement plates et convexité, Osaka J. Math., 2 (1965), 285-290. | MR | Zbl

[8] H. Shima, On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math., 13 (1976), 213-229. | MR | Zbl

[9] H. Shima, Symmetric spaces with invariant locally Hessian structures, J. Math. Soc. Japan, 29 (1977), 581-589. | MR | Zbl

[10] H. Shima, Compact locally Hessian manifolds, Osaka J. Math., 15 (1978) 509-513. | MR | Zbl

[11] J. Vey, Une notion d'hyperbolicité sur les variétés localement plates, C.R. Acad. Sci. Paris, 266 (1968), 622-624. | MR | Zbl

[12] E.B. Vinberg, The Morozov-Borel theorem for real Lie groups, Soviet Math. Dokl., 2 (1961), 1416-1419. | MR | Zbl

[13] E.B. Vinberg, The theory of convex homogeneous cones, Trans. Moscow Math. Soc., 12 (1963), 340-403. | MR | Zbl

[14] E.B. Vinberg and S.G. Gindikin, Kaehlerian manifolds admitting a transitive solvable automorphism group, Math. Sb., 75 (116) (1967), 333-351. | Zbl

[15] K. Yoshida, A theorem concerning the semi-simple Lie groups, Tohoku Math. J., 43 (Part II) (1937), 81-84. | JFM | Zbl

Cité par Sources :