Rigidity theorems on hemispheres in non-positive space forms

Rigidity Theorems on Hemispheres in Non–positive Space Forms

We study the curvature condition which uniquely characterizes the hemisphere. In particular, we prove the Min–Oo conjecture for hypersurfaces in Euclidean space and hyperbolic space.

Rigidity Results for Hypersurfaces in Space Forms

Rigidity of Minimal Submanifolds in Space Forms

(1.1) Let c be a real number. Represent by A4"(c) a n-dimensional space form of curvature c. Let M N be a N-dimensional connected Riemannian manifold. The question that served as the starting point for this paper was to find simple conditions on the metric of M so that, if f:M"~ffl"+P(c), p>= 1, is an isometric minimal immersion then f is rigid in the following sense: given another minimal imme...

Local Rigidity of Surfaces in Space Forms

We prove that isometric embeddings of closed, embedded surfaces in R are locally rigid, i.e. they admit no non-trivial local isometric deformations, answering a question in classical differential geometry. The same result holds for abstract surfaces embedded in constant curvature 3-manifolds, provided a mild condition on the fundamental group holds.

On Family Rigidity Theorems