Frame bundle approach to generalized minimal submanifolds

TL;DR

This paper generalizes the concept of minimal submanifolds to $u$-minimality using a frame bundle approach, analyzing shape operators and providing examples in higher codimension settings.

Contribution

It introduces a new framework for $u$-minimality based on frame bundle analysis, extending classical minimality to arbitrary codimension submanifolds.

Findings

Defined $u$-minimality via variation of $\sigma_u$-symmetric functions

Derived the variation field for $u$-minimality

Provided examples of $u$-minimal submanifolds

Abstract

We extend the notion of $r$-minimality of a submanifold in arbitrary codimension to $u$-minimality for a multi-index $u\in\mathbb{N}^q$, where $q$ is the codimension. This approach is based on the analysis on the frame bundle of orthonormal frames of the normal bundle to a submanifold and vector bundles associated with this bundle. The notion of $u$-minimality comes from the variation of $\sigma_u$-symmetric function obtained from the family of shape operators corresponding to all possible bases of the normal bundle. We obtain the variation field, which gives alternative definition of $u$--minimality. Finally, we give some examples of $u$-minimal submanifolds for some choices of $u$.