Generalized spherical functions on projectively flat manifolds (Q1895215)

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.