A Note on a Conjecture of Schroeder and Strake

We prove some rigidity results for compact manifolds with boundary. In particular for a compact Riemannian manifold with nonnegative Ricci curvature and simply connected mean convex boundary, it is shown that if the sectional curvature vanishes on the boundary, then the metric must be ‡at. In [Schroeder and Strake 1989, Theorem 1], Schroeder and Strake proved the following rigidity theorem. Let (M; g) be a compact Riemannian manifold with convex boundary and nonnegative Ricci curvature. Assume that the sectional curvature is identically zero in some neighborhood U of @M and that one of the following conditions holds: @M is simply connected dim @M is even and @M is strictly convex at some point p 2 @M . Then M is ‡at. As remarked in [Schroeder and Strake 1989], the condition that the metric is ‡at in a whole neighborhood of @M is very strong. They conjectured that it suf…ces to only assume that the sectional curvature vanishes on @M and proved this in the special case of a convex metric ball. The problem was studied by Xia in [Xia 1997, Xia 2002] who con…rmed the conjecture under various additional conditions: like the boundary has constant mean curvature or constant scalar curvature, or the second fundamental form satis…es some pinching condition etc. We refer to [Xia 1997, Xia 2002] for the precise statements. Here we present some results related to the conjecture. Theorem 1. Let M be a smooth compact connected Riemannian manifold with boundary and nonnegative Ricci curvature. If every component of @M is simply connected and has nonnegative mean curvature and the sectional curvature of M vanishes on @M , then M is ‡at and @M has only one component. Therefore when @M is simply connected the conjecture of Schroeder and Strake is true. Moreover one only needs @M to be mean convex instead of convex. We remark that the conclusion that @M has only one component follows from theorems in [Ichida 1981, Kasue 1983]. Below we will present a di¤erent argument for it based on the Reilly’s formula ([Reilly 1977]). To continue the discussion we need to …x some notations. We will often write h ; i for the metric onM and denote the connection as D. For convenience we write 1991 Mathematics Subject Classi…cation. 53C24, 53C21.