The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds

Abstract

A pseudo-Riemannian manifold endowed with k > 2 orthogonal complementary distributions (called a Riemannian almost multi-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, Dupin hypersurfaces and in studies of the curvature and Einstein equations. In this article, we consider the following two problems on the mixed scalar curvature of a Riemannian almost multi-product manifold with a linear connection: a) integral formulas and applications to splitting of manifolds, b) variation formulas and applications to the mixed Einstein-Hilbert action, and we generalize certain results on the mixed scalar curvature of pseudo-Riemannian almost product manifolds.

The Einstein-Hilbert Type Action on Pseudo-Riemannian Almost-Product Manifolds

“…where the Ricci and scalar curvature are replaced by the mixed Ricci curvature Ric D , see (26), and its trace S D . In [25], we obtained the Euler-Lagrange equations for (4) with fixed g and variable T, see (30a-h), and examined critical contorsion tensors (and corresponding connections) in general and in distinguished classes of (1,2)-tensors. We have shown that T is critical for (4) with fixed g if and only if T obeys certain system of algebraic equations, however, unlike (3b), these equations heavily involve also the pseudo-Riemannian geometry of the distributions.…”

“…In particular, from [25] we know that if T is critical for the action (6), then D and D are totally umbilical with respect to ∇ -and to express this together with other conditions, a pair of equations like (3a,b) is not sufficient. Due to this fact, only in the special case of semi-symmetric connections we present the Euler-Lagrange equation in the form, which directly generalizes (5):…”

“…The article has the Introduction and three other Sections. Section 2 contains background definitions and necessary results from [2,23,24,25], among them the notions of the mixed scalar curvature and the mixed and the partial Ricci tensors are central.…”

“…In Section 3.1, we vary functional (7) with respect to metric g. Compared to its variation with fixed g, which was considered in [25], we obtain additional conditions for general and metric connections. On the other hand, a metric-affine doubly twisted product is critical for (7) if and only if it is critical for the action with fixed g. Similarly, restricting (7) to pairs of metrics and statistical connections also does not give any new Euler-Lagrange equations than those obtained in [25].…”

The weighted mixed curvature of a foliated manifold

“…see [8,10,14], called the mixed scalar curvature, i.e., an averaged mixed sectional curvature. For instance, S mix = Ric(y, y) when n = 1 and a unit vector field y spans D ⊥ locally.…”