Integral formulas for a metric-affine manifold with two complementary orthogonal distributions

TL;DR

This paper derives integral formulas for metric-affine manifolds with two orthogonal distributions, linking curvature and tensor invariants to manifold splitting and obstructions to distributions.

Contribution

It introduces new integral formulas involving Ricci and scalar curvatures, aiding in understanding manifold decompositions and distribution existence.

Findings

Formulas relate curvature invariants to manifold splitting

Obstructions to distributions and foliations identified

Applications include submersions and twisted products

Abstract

We obtain integral formulas for a metric-affine space equipped with two complementary orthogonal distributions. The integrand depends on the Ricci and mixed scalar curvatures and invariants of the second fundamental forms and integrability tensors of the distributions. The formulas under some conditions yield splitting of manifolds (including submersions and twisted products) and provide geometrical obstructions for existence of distributions and foliations (or compact leaves of them).