Isometric immersions into products of many space forms: an introductory study

TL;DR

This paper introduces a foundational theory for submanifolds immersed in products of multiple space forms, extending existing tensors, the Bonnet theorem, and codimension reduction results to more complex product spaces.

Contribution

It generalizes key tensors, the Bonnet theorem, and codimension reduction for submanifolds into products of many space forms, expanding the theoretical framework.

Findings

Generalization of tensors $ extbf{R}$, $ extbf{S}$, $ extbf{T}$ for multiple space forms

Extension of the Bonnet theorem to product spaces

Generalization of the codimension reduction theorem

Abstract

This article begins the theory of submanifolds into products of 2 or more space forms. The tensors $\mathbf{R}$, $\mathbf{S}$ and $\mathbf{T}$ defined by Lira, Tojeiro and Vit\'orio at \cite{LTV} and the Bonnet theorem proved by them are generalized for the product of many space forms. Besides, some examples given by Mendon\c{c}a and Tojeiro at \cite{MT} and the reduction of codimension theorem proved by them are also generalized.