Extrinsic Curvature of Semiconvex Subspaces in Alexandrov Geometry
نویسندگان
چکیده
In Alexandrov spaces of curvature bounded either above (CBA) or below (CBB), we obtain extrinsic curvature bounds on subspaces associated with semiconcave functions. For CBA spaces, we obtain new intrinsic curvature bounds on subspaces. For CBB spaces whose boundary is extrinsically curved, we strengthen Perelman’s concavity theorem for distance from the boundary, deriving corollaries on sharp diameter bounds, contractibility, and rigidity.
منابع مشابه
Gauss Equation and Injectivity Radii for Subspaces in Spaces of Curvature Bounded Above
A Gauss Equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K. That is, a subspace of extrinsic curvature ≤ A, defined by a cubic inequality on the difference of arc and chord, has intrinsic curvature ≤ K +A. Sharp bounds on injectivity radii of subspaces, new even in the Riemannian case, are derived.
متن کاملOn the k-nullity foliations in Finsler geometry
Here, a Finsler manifold $(M,F)$ is considered with corresponding curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of $M$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant, then the distribution is involutive...
متن کاملDirac Operators on Hypersurfaces of Manifolds with Negative Scalar Curvature
We give a sharp extrinsic lower bound for the first eigenvalues of the intrinsic Dirac operator of certain hypersurfaces bounding a compact domain in a spin manifold of negative scalar curvature. Limiting-cases are characterized by the existence, on the domain, of imaginary Killing spinors. Some geometrical applications, as an Alexandrov type theorem, are given. Mathematics Subject Classificati...
متن کاملExtrinsic sphere and totally umbilical submanifolds in Finsler spaces
Based on a definition for circle in Finsler space, recently proposed by one of the present authors and Z. Shen, a natural definition of extrinsic sphere in Finsler geometry is given and it is shown that a connected submanifold of a Finsler manifold is totally umbilical and has non-zero parallel mean curvature vector field, if and only if its circles coincide with circles of the ambient...
متن کاملMinkowski Formulae and Alexandrov Theorems in Spacetime
The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit “hidden symmetry” from conformal KillingYano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimensiontwo submanifold with constant normalized null expansion (null mean curva...
متن کامل