DDVV-type inequality for skew-symmetric matrices and Simons-type inequality for Riemannian submersions

TL;DR

This paper establishes a new optimal inequality for skew-symmetric matrices and applies it to derive a Simons-type inequality for Riemannian submersions, revealing a duality between submanifold geometry and Riemannian submersions.

Contribution

It introduces a DDVV-type inequality for skew-symmetric matrices and connects it to Riemannian submersion geometry, highlighting a duality between different geometric structures.

Findings

Derived a DDVV-type inequality for skew-symmetric matrices.

Established a Simons-type inequality for Riemannian submersions.

Revealed duality between submanifold geometry and Riemannian submersions.

Abstract

In this paper, we will first derive a DDVV-type optimal inequality for real skew-symmetric matrices, then we apply it to establish a Simons-type integral inequality for Riemannian submersions with totally geodesic fibres and Yang-Mills horizontal distributions. In this way, we show phenomenons of duality between Submanifold geometry and Riemannnian submersion, particularly between second fundamental form of a submanifold and integrability tensor of a Riemannian submersion.