Fundamental Theorem for Submanifolds in General Ambient Spaces

TL;DR

This paper generalizes the fundamental theorem for submanifolds to broader ambient spaces by framing it as a higher codimensional Cartan-Ambrose-Hicks theorem, offering a geometric construction of isometric immersions.

Contribution

It introduces a new perspective by viewing the theorem as a higher codimensional Cartan-Ambrose-Hicks theorem and generalizes curve development for positive codimension cases.

Findings

Extended the fundamental theorem to general ambient spaces.

Provided a geometric construction method for isometric immersions.

Generalized development of curves in higher codimension.

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.