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

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 .

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 .

@article{AIF_1980__30_3_91_0,
     author = {Shima, Hirohiko},
     title = {Homogeneous hessian manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {30},
     number = {3},
     year = {1980},
     pages = {91-128},
     doi = {10.5802/aif.794},
     mrnumber = {597019},
     zbl = {0424.53023},
     mrnumber = {82a:53054},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1980__30_3_91_0}
}
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/item/AIF_1980__30_3_91_0/

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