Fundamental Theorem for Submanifolds in General Ambient Spaces

Abstract

In this paper, we extend the fundamental theorem for submanifolds to general ambient spaces by viewing it as a higher codimensional Cartan-Ambrose-Hicks theorem. The key ingredient in obtaining this is a generalization of development of curves in the positive codimensional case. One advantage of our results is that it also provide a geometric construction of the isometric immersion when the isometric immersion exists.

Some of the results are strengthen and the paper are rewritten into arXiv:1911.13242 and 1911.07493