Dual connections and affine geometry
type: Item
instance of: scholarly article
title: Dual connections and affine geometry
zbMATH Open document ID: 0696.53005
DOI: 10.1007/BF02570725
author: Takashi Kurose
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174116
review text: Two torsion free affine connections \(\nabla\) and \({\bar \nabla}\) of a pseudo-Riemannian manifold \((M,g)\), \(\dim M=n\) are called dual if \(Xg(Y,Z)=g(\nabla_ XY,Z)+g(Y,{\bar \nabla}_ XZ)\) for any vector fields X, Y, Z. Such an \((M,g,\nabla,{\bar \nabla})\) is called a statistical manifold. It is proved that if an \((M,g,\nabla,{\bar \nabla})\) has constant curvature (defined in the paper), then there exist affine immersions \((x,\xi)\) of \((M,\nabla)\) and \((\bar x,{\bar \xi})\) of \((M,{\bar \nabla})\) into an affine \({\mathbb{R}}^{n+1}\) such that their second fundamental forms are equal to g; and conversely. Corollary 1 gives a local description for explicit calculation. In Corollary 2 the case of positive definite g is discussed and a theorem of S.-T. Yau concerning complete affine hyperspheres and affine mean curvature is involved.
reviewed by: Lajos Tamássy
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53B05
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 4140706
zbMATH Keywords: affine connections
zbMATH Keywords: pseudo-Riemannian manifold
zbMATH Keywords: affine immersions
zbMATH Keywords: second fundamental forms
zbMATH Keywords: complete affine hyperspheres
zbMATH Keywords: affine mean curvature
MaRDI profile type: Publication
cites work: Differential-geometrical methods in statistics
cites work: Complete affine hypersurfaces. Part I. The completeness of affine metrics
cites work: Compact locally Hessian manifolds
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description: scientific article; zbMATH DE number 4140706