On a geometric black hole of a compact manifold

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.