On isometric immersions of sub-Riemannian manifolds

TL;DR

This paper investigates curvature invariants in sub-Riemannian manifolds, establishing geometric inequalities for submanifolds with orthogonal distributions, extending understanding of their curvature properties and relationships.

Contribution

It introduces new curvature invariants related to mutual curvature of orthogonal subspaces and proves inequalities for submanifolds, advancing geometric analysis in sub-Riemannian geometry.

Findings

Derived inequalities for submanifolds with orthogonal distributions.

Established relations between mutual curvature and scalar curvature.

Extended integral formulas to sub-Riemannian contexts.

Abstract

We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove geometrical inequalities for a sub-Riemannian submanifold. As applications, inequalities are proved for submanifolds with mutually orthogonal distributions that include scalar and mutual curvature. For compact submanifolds, inequalities are obtained that are supported by known integral formulas for almost-product manifolds.