Variations of the total mixed scalar curvature of a distribution

Abstract

We examine the total mixed scalar curvature of a smooth manifold endowed with a distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution and use this key and technical result to find the critical points of this action. Together with the arbitrary variations of the metric, we consider also variations that preserve the volume of the manifold or partially preserve the metric (e.g., on the distribution). For each of those cases, we obtain the Euler-Lagrange equation and its several solutions. Examples of critical metrics that we find are related to various fields of geometry such as contact and 3-Sasakian manifolds, geodesic Riemannian flows, codimension-one foliations, and distributions of interesting geometric properties (e.g., totally umbilical and minimal).

Einstein-Hilbert type action on spacetimes

“…Section 2 starts with variation formulas for quantities on a manifold with a distribution. Using them, we deduce Euler-Lagrange equations of (0.1), which generalize results in [2,8] and use them to build the mixed Ricci tensor Ric D obeying (0.2). In Proposition 2.2 we observe that the replacement of S mix by Scal D in (0.1) leads to the same Euler-Lagrange equations (0.2).…”

“…In this paper, we explore (0.1) for any spacetime, the obtained mixed gravitational field equations generalize the result of [1]. In fact, we work in arbitrary number of dimensions of a pseudo-Riemannian manifold endowed with a distribution, and also generalize certain results of [2,8], where the particular case of variations of metric (called adapted variations) has been examined. In this setting, we present the Euler-Lagrange equations for (0.1) in the classical form of the Einstein field equations:…”

“…The section reviews necessary definitions of tensors, some of them were introduced in [8,11]. We consider a connected manifold M n+p with a pseudo-Riemannian metric g and a non-degenerate distribution D of rank dim…”

“…Given D, its g t -orthogonal complement D ⊥ t depends on t; hence, the g t -projections onto D (denoted by ⊤ ) and onto D ⊥ 0 = D ⊥ (denoted by ⊥ ) also depend on t. If (g t ) preserve orthogonality of D and D ⊥ then projections ⊤ and ⊥ do not depend on t, and we obtain adapted variations, see [2,8]. Among adapted variations, those preserving metric on D will be called g ⊥ -variations.…”

The Mixed Scalar Curvature of a Twisted Product Riemannian Manifolds and Projective Submersions

“…. , E n } on TM adapted to V and H, respectively (see also [8] (p. 117); [9] (p. 23) and [10][11][12][13]). It is easy to see that this expression is independent of the chosen adapted frames.…”