A polarized version of Bochner's identity is used to obtain symmetry results for an overdetermined boundary value problem on constant curvature manifolds. The identity is also used to prove a Rellich identity for Dirichlet eigenvalues of the Laplacian. Finally, a reflection argument which was developed by Alexandrov is used to obtain symmetry results for overdetermined boundary value problems on constant curvature manifolds.1980 Mathematics Subject Classification (1985 Revision): 53C20. l IntroductionIn this paper we consider some overdetermined boundary value problems for the Laplacian on spaces of constant curvature. Suppose M is a manifold of constant curvature and Ω c M is a domain with C 2 boundary. Let u e C 2 ( ) be a function such that Au=ffor some given radially Symmetrie function / and suppose in addition u satisfies the boundary conditions w = 0 on δ Ω and (d/dn)(u) = k (constant) on 3Ω. We want to then show that Ω is a metric ball in M. The given radially Symmetrie function,/, and the manifold M dictate the result one obtains s well s the method used. An example due to C. Berenstein and M. Karlowitz, [3], will help illustrate this point.On the Standard sphere, S" 1 , they construct a domain Ω c S" with smooth boundary and a smooth function u such that Au = -i on Ω, w = 0 on 5 Ω and (djdn)(u) = k on du. Furthermore the domain Ω is not radially Symmetrie. On the other hand if Ω is contained in the hemisphere and we consider the equation Au=f where/ = cos r and r is the geodesic distance from a fixed point or/ = -l, then in fact Ω is radially Symmetrie. Moreover, the methods used to prove these two results are quite different. The first technique we use involves an integral formula obtained from a polarized version of Bochner's identity. The second technique is a reflection argument due to Alexandrov [1].

Abstract. In this paper we define, in two equivalent ways, the Schwarzian derivative of a map between complex manifolds equipped with complex projective connections. Also, a new, coordinate-free definition of complex projective connections is given. We show how the Schwarzian derivative is related to the projective structure of the manifolds, to projective linear transformations, and to complex geodesics.

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