Normal Scalar Curvature Inequality on a Class of Austere Submanifolds

TL;DR

This paper introduces sharper inequalities for the normal scalar curvature of austere submanifolds, providing examples and a gap theorem, advancing understanding of their geometric properties.

Contribution

It establishes new DDVV-type inequalities for austere submanifolds and identifies cases where equality holds, along with a related gap theorem.

Findings

A new class of normal scalar curvature inequalities for austere submanifolds.

Existence of examples achieving equality in the inequalities.

A Simons-type gap theorem for closed austere submanifolds in spheres.

Abstract

In this paper, we establish new normal scalar curvature inequalities on a class of austere submanifolds by proving sharper DDVV-type inequalities on associated austere subspaces. We also provide some examples of austere submanifolds in this class and point out one of them achieves the equality in our normal scalar curvature inequality everywhere. As a byproduct, we obtain a Simons-type gap theorem for closed austere submanifolds in unit spheres which belong to that class.