A Fundamental Theorem for submanifolds of multiproducts of real space forms

TL;DR

This paper establishes a fundamental theorem for submanifolds in multiproducts of real space forms, including existence results for minimal surfaces and a complex version for the case of 222, 222, highlighting new geometric insights.

Contribution

It extends the Bonnet theorem to submanifolds in multiproducts of real space forms and introduces associated minimal surface families, including a complex structure case for 222.

Findings

Proved a Bonnet theorem for submanifolds in products of real space forms.

Established existence of associated minimal surface families.

Provided a complex version of the theorem for 222.

Abstract

We prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then, we prove the existence of associated families of minimal surfaces in such products. Finally, in the case of $\mathbb{S}^2\times\mathbb{S}^2$, we give a complex version of the main theorem in terms of the two canonical complex structures of $\mathbb{S}^2\times\mathbb{S}^2$.