Isotropic affine spheres
type: Item
instance of: scholarly article
title: Isotropic affine spheres
zbMATH Open document ID: 1261.53010
author: Olivier Birembaux
author: Mirjana Djoric
publication date: 2013-01-02T00:00:00Z
review text: The authors apply O'Neill's notion of isotropy to the study of affine hyperspheres contributing a piece to the grand open classification problem of affine hyperspheres. The details are as follows, where hyperspheres will be called spheres for short: In the equiaffine hypersurface theory of Berwald and Blaschke, one considers immersions \( f: M^{n} \to \mathbb R^{n+1} \) where the ambient space \( \mathbb R^{n+1} \) is only equipped with the standard volume element without regard of its Euclidean structure (Klein's geometry of the special affine group). Assuming strict local convexity, one can associate to \( f \) various differential-geometric objects like the affine normal vector field \( \xi \), the affine metric \( h \), and the induced symmetric connection \( \nabla \) on the parameter manifold \( M^{n} \). The metric \( h \) renders \( M^{n} \) a genuine Riemannian manifold (in case \( M^{n} \) is suitably oriented), and the Levi-Civita-connection \( \tilde{\nabla} \) of \( h \) gives rise to the difference tensor field \( K(X,Y) := \nabla_{X}Y-\tilde{\nabla}_{X}Y \), a relative to the classical Darboux tensor or cubic fundamental form. \( f \) is called an affine sphere if all normals go through a fixed point (proper case) or are mutually parallel (improper case). The proper spheres split in two subclasses: `elliptic' resp. `hyperbolic' ones, according to the position of the center on the concave resp. convex side of the sphere. While complete elliptic spheres are well known by the theorem of Blaschke and Deicke in the compact case and its complete extension (they are ellipsoids), and the asymptotic behaviour of complete hyperbolic spheres is clarified by the solution of the Calabi conjecture, it is still open how to classify in general the large variety of equiaffine spheres. See e.g. the book [\textit{A.-M. Li} et al., Global affine differential geometry of hypersurfaces. Berlin: Walter de Gruyter (1993; Zbl 0808.53002)], in particular Chapter 2. A bilinear …
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 6121159
zbMATH Keywords: affine sphere
zbMATH Keywords: Darboux tensor
zbMATH Keywords: cubic form
zbMATH Keywords: isotropy
zbMATH Keywords: symmetric space
zbMATH Keywords: exceptional Lie group
zbMATH Keywords: octonions
reviewed by: Rolf Walter
MaRDI profile type: Publication
full work available at URL: s10114-012-0264-3
OpenAlex ID: W2077264060
cites work: Planar geodesic immersions
cites work: Isotropic submanifolds with parallel second fundamental form in \(P^ m(\)c)
cites work: Isotropic totally real submanifolds
cites work: Q4032082
cites work: Locally strongly convex affine hypersurfaces with parallel cubic form
cites work: Q4846450
cites work: Totally Real Minimal Submanifolds in a Complex Projective Space
cites work: Hyperbolic affine hyperspheres
cites work: Q5565587
cites work: The octonions
DOI: 10.1007/S10114-012-0264-3
published in: Acta Mathematica Sinica, English Series
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description: scientific article; zbMATH DE number 6121159