Cut Locus of Submanifolds: A Geometric and Topological Viewpoint

Prasad, Sachchidanand (2022) Cut Locus of Submanifolds: A Geometric and Topological Viewpoint. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Sachchidanand Prasad (17RS038)) 17RS038.pdf - Submitted Version Restricted to Repository staff only Download (2MB)

Abstract

Associated to every closed, embedded submanifold N of a connected Riemannian manifold M, there is the distance function dN which measures the distance of a point in M from N. We analyze the square of this function and show that it is Morse-Bott on the complement of the cut locus Cu(N) of N, provided M is complete. Moreover, the gradient flow lines provide a deformation retraction of M − Cu(N) to N. If M is a closed manifold, then we prove that the Thom space of the normal bundle of N is homeomorphic to M/Cu(N). We also discuss several interesting results which are either applications of these or related observations regarding the theory of cut locus. These results include, but are not limited to, a computation of the local homology of singular matrices, a classification of the homotopy type of the cut locus of a homology sphere inside a sphere, a deformation of the indefinite unitary group U(p, q) to U(p)×U(q) and a geometric deformation of GL(n,R) to O(n,R) which is different from the Gram-Schmidt retraction. If a compact Lie group G acts on a Riemannian manifold M freely then M/G is a manifold. In addition, if the action is isometric, then the metric of M induces a metric on M/G. We show that if N is a G-invariant submanifold of M, then the cut locus Cu(N) is G-invariant, and Cu(N)/G = Cu(N/G) in M/G. An application of this result to complex projective hypersurfaces has been provided.

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