Abstract
Local and global properties of the first order spherical functions are generalized to projectively flat manifolds.
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M. Berger, P. Gauduchon, and E. Mazet. Le spectre d’une variété Riemannienne. Lecture Notes in Mathematics 194. Springer-Verlag, Berlin, 1971.
L. P. Eisenhart. Non-Riemannian Geometry, volume VIII of Colloquium Publications. AMS, 1927.
S. Kobayashi and K. Nomizu. Foundations of differential geometry, volume I. John Wiley & Sons, New York, 1963.
K. Nomizu and U. Pinkall. On a certain class of homogeneous projectively flat manifolds. Tôhoku Math. J., 39:407–427, 1987.
K. Nomizu and U. Pinkall. On the geometry of affine immersions. Math. Z., 195:165–178, 1987.
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.
M. Obata. Certain conditions for a Riemannian manifolds to be isometric with a sphere. J. Math. Soc. Japan, 14:333–340, 1962.
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.
U. Pinkall, A. Schwenk-Schellschmidt, and U. Simon. Geometric methods for solving Codazzi and Monge-Ampère equations. Math. Annalen, 298:89–100, 1994.
J. A. Schouten. Ricci-Calculus. Springer-Verlag, Berlin, 2nd edition, 1954.
U. Simon. Transformation techniques for partial differential equations on projectively flat manifolds. Results in Mathematics, this volume, 1994.
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].
K. Tandai. Riemannian manifolds admitting more than n − 1 linearly independent solutions of \(\nabla 2\rho+c 2\rho g=0\). Hokkaido Math. J., 1:12–15, 1972.
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Dedicated to Katsumi Nomizu
Partially supported by the exchange-program UNC-TUB and by NSF grants DMS-9204942 and 9409037
Partially supported by the DFG-project Si 163/4-1
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Gardner, R.B., Kriele, M. & Simon, U. Generalized spherical functions on projectively flat manifolds. Results. Math. 27, 41–50 (1995). https://doi.org/10.1007/BF03322268
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DOI: https://doi.org/10.1007/BF03322268