Extrinsic curvatures of distributions of arbitrary codimension

TL;DR

This paper introduces a generalized approach to define higher order mean curvatures for distributions of any codimension, aligning with previous definitions, and explores their properties and associated vector fields.

Contribution

It extends the concept of mean curvatures to arbitrary codimension distributions using the generalized Newton transformation, confirming consistency with prior definitions.

Findings

Higher order mean curvatures are well-defined for arbitrary codimension distributions.

The divergence of higher order mean curvature vector fields is computed for specific distributions.

Total extrinsic mean curvatures are derived from divergence calculations.

Abstract

In this article, using the generalized Newton transformation, we define higher order mean curvatures of distributions of arbitrary codimension and we show that they agree with the ones from Brito and Naveira (Ann. Global Anal. Geom. 18, 371-383 (2000)). We also introduce higher order mean curvature vector fields and we compute their divergence for certain distributions and using this we obtain total extrinsic mean curvatures.