On a geometric black hole of a compact manifold

TL;DR

The paper introduces the concept of a geometric black hole in a compact manifold, showing how it can be used to simplify the deformation of tensor fields while preserving the manifold's structure.

Contribution

It presents a novel geometric construction called a black hole in a manifold, facilitating tensor field deformation with simplified local structure.

Findings

Every compact manifold can be decomposed into a cell and a union of subsimplexes.

A small neighborhood of the black hole can be used to deform tensor fields into simpler forms.

The method applies to various tensor fields and structures on manifolds.

Abstract

Using a smooth triangulation and a Riemannian metric on a compact, connected, closed manifold M of dimension n we have got that every such M can be represented as a union of a n-dimensional cell and a connected union K of some subsimplexes of the triangulation. A sufficiently small closed neighborhood of K is called a geometric black hole. Any smooth tensor field T (or other structure) can be deformed into a continuous and sectionally smooth tensor field T1 where T1 has a very simple construction out of the black hole.