A flat affine manifold is said to Hessian if it is endowed with a Riemannian metric whose local expression has the form where is a -function and is an affine local coordinate system. Let be a Hessian manifold. We show that if is homogeneous, the universal covering manifold of is a convex domain in 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 .
Une variété d’une connexion affine plate est dite hessienne si elle est munie d’une métrique riemannienne qui s’exprime localement où est une fonction et est un système de coordonnées locales affines. Soit une variété hessienne. On montre que si est homogène, le revêtement universel de est un domaine convexe dans 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 .
@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}, zbl = {0424.53023}, mrnumber = {597019}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.794/} }
Shima, Hirohiko. Homogeneous hessian manifolds. Annales de l'Institut Fourier, Volume 30 (1980) no. 3, pp. 91-128. doi : 10.5802/aif.794. http://www.numdam.org/articles/10.5802/aif.794/
[1] Homogeneous Kähler manifolds, in “Geometry of Homogeneous Bounded Domains”, Centro Int. Math. Estivo, 3 Ciclo, Urbino, Italy, 1967, 3-87. | Zbl
, and ,[2] On the geometry of the tangent bundles, J. Reine Angew. Math., 210 (1962), 73-88. | MR | Zbl
,[3] Faithful representations of Lie groups I, Math. Japon., 1 (1948), 1-13. | Zbl
,[4] Doctorat de 3e cycle “Radical et groupe formel d'une algèbre symétrique à gauche” novembre 1975, Grenoble.
,[5] Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. of Tokyo, IA 24 (1977), 129-135. | MR | Zbl
,[6] Domaines bornés homogènes et orbites de groupes de transformations affines, Bull. Soc. Math. France, 89 (1961), 515-533. | Numdam | MR | Zbl
,[7] Variétés localement plates et convexité, Osaka J. Math., 2 (1965), 285-290. | MR | Zbl
,[8] On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math., 13 (1976), 213-229. | MR | Zbl
,[9] Symmetric spaces with invariant locally Hessian structures, J. Math. Soc. Japan, 29 (1977), 581-589. | MR | Zbl
,[10] Compact locally Hessian manifolds, Osaka J. Math., 15 (1978) 509-513. | MR | Zbl
,[11] Une notion d'hyperbolicité sur les variétés localement plates, C.R. Acad. Sci. Paris, 266 (1968), 622-624. | MR | Zbl
,[12] The Morozov-Borel theorem for real Lie groups, Soviet Math. Dokl., 2 (1961), 1416-1419. | MR | Zbl
,[13] The theory of convex homogeneous cones, Trans. Moscow Math. Soc., 12 (1963), 340-403. | MR | Zbl
,[14] Kaehlerian manifolds admitting a transitive solvable automorphism group, Math. Sb., 75 (116) (1967), 333-351. | Zbl
and ,[15] A theorem concerning the semi-simple Lie groups, Tohoku Math. J., 43 (Part II) (1937), 81-84. | JFM | Zbl
,Cited by Sources: