Generalized spherical functions on projectively flat manifolds
type: Item
instance of: scholarly article
title: Generalized spherical functions on projectively flat manifolds
zbMATH Open document ID: 0837.53022
DOI: 10.1007/BF03322268
author: Robert B. Gardner
author: Marcus Kriele
author: Udo Simon
published in: Results in Mathematics
publication date: 1996-05-13T00:00:00Z
review text: The paper complements the applications given by the third author [ibid. 27, No. 1--2, 160--187 (1995; Zbl 0833.58042)]. For a given torsion-free and Ricci-symmetric connection \(\nabla\) a function \(f\in C^\infty(M)\) is called by the authors generalized spherical function if \(H(f):= \operatorname{Hess}(f)+ (n+ 1)^{- 1} f\operatorname{Ricc} = 0\), Ricc stands for the Ricci tensor of \(\nabla\). \({\mathcal H}(M, \nabla):= \{f\mid H(f)= 0\}\) denotes the space of solutions. For a projective transformation \(\phi: (M, \nabla)\to (M, \nabla^\#)\) the authors obtain \({\mathcal H}(M, \nabla)\cong {\mathcal H}(M, \nabla^\#)\). The main local theorem of the authors says: For every \(p\in M\) there exists \(U\in M\) such that \(\dim{\mathcal H}(U, \nabla)= n+ 1\) \((n=\dim M)\). They show that for projectively flat \(\nabla\) and \(\nabla^\#\) \({\mathcal H}(M, \nabla)= {\mathcal H}(M, \nabla^\#)\) yields \(\nabla= \nabla^\#\). However, if \(M\) is diffeomorphism to the standard sphere \(S^n\), \(\nabla\) and \(\nabla^\#\) are projectively equivalent and flat and \(\dim\{{\mathcal H}(M, \nabla)\cap {\mathcal H}(M, \nabla^\#)\}\geq 1\), then \(\nabla= \nabla^\#\) globally.
Mathematics Subject Classification ID: 53B10
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 785195
zbMATH Keywords: projectively flat manifolds
zbMATH Keywords: generalized spherical function
reviewed by: Lajos Tamássy
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 785195