Fundamental Theorem for Submanifolds in General Ambient Spaces

2021/09/08 by Chengjie Yu, Yu, Chengjie
Engineering · #Advanced Numerical Analysis Techniques

paper · pdf · doi:10.48550/arxiv.2109.03474

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.

Citations

• A Bonnet theorem for isometric immersions into products of space forms
• Isometric immersions into products of space forms

Related

• Extensions of isometric immersions and Cartan-Ambrose-Hicks theorem based on submanifolds
• C^1,α isometric extensions
• A new obstruction to minimal isometric immersions into a real space form
• Developments of curves with respect to symmetric tensors and existence of isometric immersions with prescribed second fundamental form
• Extrinsic Characterisations of Immersions