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Some Aspects of Differential Topology of Subcartesian Spaces
Version 1
: Received: 20 August 2024 / Approved: 20 August 2024 / Online: 22 August 2024 (03:12:09 CEST)
How to cite: Chen, L.; Xia, Q. Some Aspects of Differential Topology of Subcartesian Spaces. Preprints 2024, 2024081490. https://doi.org/10.20944/preprints202408.1490.v1 Chen, L.; Xia, Q. Some Aspects of Differential Topology of Subcartesian Spaces. Preprints 2024, 2024081490. https://doi.org/10.20944/preprints202408.1490.v1
Abstract
In this paper we investigate the differential topological properties of a subclass of singular space--subcarteisan spaces. First, we get further result on partition of unity for differential spaces. Second, we establish the tubular neighborhood theorem for subcartesian spaces with constant structural dimension. Third, we generalize the concept of Morse functions on smooth manifolds to differential spaces. For subcartesian space with constant structural dimension, we provide examples of Morse functions; With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, we prove that smooth bounded functions on this space can be approximated by Morse functions; We study the infinitesimal stability of Morse functions on subcartesian spaces. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.
Keywords
Singular space; Subcartesian space; Partition of unity; Tubular neighborhood; Morse function; Differential topology
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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