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

Abstract

We continue our study of the mixed Einstein–Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler–Lagrange equations (which in the case of space-time are analogous to those in Einstein–Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler–Lagrange equations of the mixed Einstein–Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric connections.

Variational problem on a metric-affine almost product manifold

“…for some λ ∈ R. Following [22] for k = 2, we define auxiliary Casorati type operators T µ : D µ → D µ and self-adjoint (1, 1)-tensors K µ (using the Lie bracket) by…”

“…Variational formulas of terms of Q in (9) obtained in the following lemma are a special case (i.e., for adapted variations of metric) of equations from [22,Lemma 3] for (D µ , D ⊥ µ ). Detailed proof of Lemma 2 for particular choice of T will be given further below in Lemma 6.…”

“…. , D k ) with k = 2 were studied in [5,17,20,21,22,23]. An almost product structure (1) appears naturally in such topics as (multiply) twisted or warped product manifolds, e.g., [13,15]; the theory of webs [12]; almost para-f -manifolds, e.g., [24]; lightlike manifolds, i.e., with degenerate metric of constant rank and index, e.g., [10]; hypersurfaces in space forms with k distinct principal curvatures, see [9].…”

“…where the Ricci tensor and the scalar curvature are replaced by the new Ricci type tensor Ric D and its trace S D . Although Ric D has a complicated expression even for k = 2, see [22], we write it explicitly for some distinguished cases: statistical and semi-symmetric connections, as well as for twisted products. We also find the Euler-Lagrange equation of (2) with a fixed adapted metric for variations of T, which can be decomposed into independent equations -some of which do not contain contorsion tensor, but restrict metrics admitting critical contorsions.…”

“…In Section 3, we study adapted variations of metric (with fixed contorsion tensor) and, similarly to the case of (3) in [18], find the Euler-Lagrange equation of the action (2). In Section 4, we study variations of T and, using results for k = 2 from [22], find the Euler-Lagrange equation of (2) with fixed adapted metric. In Section 5 we apply the results of Sections 3 and 4 to metric connections; in particular, in Section 5.1 we consider semisymmetric connections and in Section 5.2 we examine 3-Sasaki manifolds.…”