We prove Obata’s rigidity theorem for metric measure spaces that satisfy a Riemannian curvaturedimension condition. Additionally,we show that a lower bound K for the generalizedHessian of a sufficiently regular function u holds if and only if u is K-convex. A corollary is also a rigidity result for higher order eigenvalues.
This article presents the theory of focal locus applied to the hypersurfaces in the projective space which are (finitely) covered by linear spaces and such that the tangent space is constant along these spaces.
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