Variational problem on a metric-affine almost product manifold

TL;DR

This paper investigates a variational problem on manifolds with multiple distributions, generalizing Einstein metrics, and identifies conditions for critical metrics involving umbilical distributions and specific geometric structures.

Contribution

It introduces a new variational functional on manifolds with multiple distributions, extending Einstein metrics, and analyzes critical points involving metric-contorsion pairs.

Findings

Critical metrics make all distributions totally umbilical.

Examples include twisted products and 3-Sasaki manifolds.

Obstructions are identified in certain geometric contexts.

Abstract

We study a variational problem on a smooth manifold with a decomposition of the tangent bundle into $k>2$ subbundles (distributions), namely, we consider the integrated sum of their mixed scalar curvatures as a functional of adapted pseudo-Riemannian metric (keeping the pairwise orthogonality of the distributions) and contorsion tensor, defining a linear connection. This functional allows us to generalize the class of Einstein metrics in the following sense: if all of the distributions are one-dimensional, then it coincides with the geometrical part of the Einstein-Hilbert action restricted to adapted metrics. We prove that metrics in pairs metric-contorsion critical for our functional make all of the distributions totally umbilical. We obtain examples and obstructions to existence of those critical pairs in some special cases: twisted products with statistical connections; semi-symmetric connections and 3-Sasaki manifolds with metric-compatible connections.