Developments of curves with respect to symmetric tensors and existence of isometric immersions with prescribed second fundamental form

TL;DR

This paper introduces a new geometric method for constructing isometric immersions with prescribed second fundamental forms using developments of curves relative to symmetric tensors.

Contribution

It presents a novel approach to prove the existence of isometric immersions with specific second fundamental forms through geometric curve developments.

Findings

Established a method for constructing isometric immersions with prescribed second fundamental forms.

Proved the existence of such immersions in general ambient spaces.

Provided a geometric construction technique for these immersions.

Abstract

In this paper, we introduce the notion of developments of curves with respect to symmetric tensors and use it to prove the existence of isometric immersions into a general ambient space with prescribed second fundamental form. Our method provides a geometric construction of such an isometric immersion.