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Associated to every closed, embedded submanifold N in 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.
Basu, Somnath 1 ; Prasad, Sachchidanand 1
@article{10_2140_agt_2023_23_4185,
author = {Basu, Somnath and Prasad, Sachchidanand},
title = {A connection between cut locus, {Thom} space and {Morse{\textendash}Bott} functions},
journal = {Algebraic and Geometric Topology},
pages = {4185--4233},
publisher = {mathdoc},
volume = {23},
number = {9},
year = {2023},
doi = {10.2140/agt.2023.23.4185},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.4185/}
}
TY - JOUR AU - Basu, Somnath AU - Prasad, Sachchidanand TI - A connection between cut locus, Thom space and Morse–Bott functions JO - Algebraic and Geometric Topology PY - 2023 SP - 4185 EP - 4233 VL - 23 IS - 9 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.4185/ DO - 10.2140/agt.2023.23.4185 ID - 10_2140_agt_2023_23_4185 ER -
%0 Journal Article %A Basu, Somnath %A Prasad, Sachchidanand %T A connection between cut locus, Thom space and Morse–Bott functions %J Algebraic and Geometric Topology %D 2023 %P 4185-4233 %V 23 %N 9 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2023.23.4185/ %R 10.2140/agt.2023.23.4185 %F 10_2140_agt_2023_23_4185
Basu, Somnath; Prasad, Sachchidanand. A connection between cut locus, Thom space and Morse–Bott functions. Algebraic and Geometric Topology, Tome 23 (2023) no. 9, pp. 4185-4233. doi: 10.2140/agt.2023.23.4185
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