×

Generalized spherical functions on projectively flat manifolds. (English) Zbl 0837.53022

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.

MSC:

53B10 Projective connections
53C35 Differential geometry of symmetric spaces

Citations:

Zbl 0833.58042
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Berger, P. Gauduchon, and E. Mazet. Le spectre d’une variété Riemannienne. Lecture Notes in Mathematics 194. Springer-Verlag, Berlin, 1971. · Zbl 0223.53034
[2] L. P. Eisenhart. Non-Riemannian Geometry, volume VIII of Colloquium Publications. AMS, 1927. · JFM 52.0721.02
[3] S. Kobayashi and K. Nomizu. Foundations of differential geometry, volume I. John Wiley & Sons, New York, 1963. · Zbl 0119.37502
[4] K. Nomizu and U. Pinkall. On a certain class of homogeneous projectively flat manifolds. Tôhoku Math. J., 39:407–427, 1987. · Zbl 0641.53053 · doi:10.2748/tmj/1178228287
[5] K. Nomizu and U. Pinkall. On the geometry of affine immersions. Math. Z., 195:165–178, 1987. · Zbl 0629.53012 · doi:10.1007/BF01166455
[6] K. Nomizu and U. Simon. Notes on conjugate connections. In F. Dillen and L. Verstraelen, editors, Geometry and Topology of Submanifolds, IV. Proc. Conf. Diff. Geom. Vision, pages 152-172, Leuven (Belgium), 1991. World Scientific, Singapore, 1992. · Zbl 0840.53018
[7] M. Obata. Certain conditions for a Riemannian manifolds to be isometric with a sphere. J. Math. Soc. Japan, 14:333–340, 1962. · Zbl 0115.39302 · doi:10.2969/jmsj/01430333
[8] V. Oliker and U. Simon. Codazzi tensors and equations of Monge-Ampére type on compact manifolds of constant sectional curvature. J. reine angew. Math., 342:35–65, 1983. · Zbl 0502.53038
[9] U. Pinkall, A. Schwenk-Schellschmidt, and U. Simon. Geometric methods for solving Codazzi and Monge-Ampère equations. Math. Annalen, 298:89–100, 1994. · Zbl 0934.53029 · doi:10.1007/BF01459727
[10] J. A. Schouten. Ricci-Calculus. Springer-Verlag, Berlin, 2nd edition, 1954. · Zbl 0057.37803
[11] U. Simon. Transformation techniques for partial differential equations on projectively flat manifolds. Results in Mathematics, this volume, 1994.
[12] U. Simon, A. Schwenk-Schellschmidt, and H. Viesel. Introduction to the affine differential geometry of hypersurfaces. Lecture Notes. Science University of Tokyo, 1991. [Distribution TU Berlin, ISBN 3 7983 1529 9]. · Zbl 0780.53002
[13] K. Tandai. Riemannian manifolds admitting more than n 1 linearly independent solutions of % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $\(\backslash\)nabla 2\(\backslash\)rho+c 2\(\backslash\)rho g=0$ . Hokkaido Math. J., 1:12–15, 1972. · Zbl 0252.53016 · doi:10.14492/hokmj/1381759031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.