A Rigidity Theorem for the Hemi-sphere

Put in another way, for a compact manifold with boundary, if we know that the boundary is S(intrinsic geometry on the boundary) and totally geodesic (extrinsic geometry) then we recognize the manifold as the hemisphere S+, provided Ric ≥ (n− 1) g. To put this result in a context, we first recall the following Theorem 2. Let (M, g) be a compact Riemannian manifold with boundary and scalar curvature R ≥ 0. If the boundary is isometric to S and has mean curvature n − 1, then (M, g) is isometric to the unit ball Bn ⊂ R. (If n > 7 we need to assume that M is spin.)