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

Abstract

We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained Euler-Lagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.

Variations of the total mixed scalar curvature of a distribution

“…It can be decomposed into three independent parts, two of them being symmetric with respect to interchanging the given distribution and its orthogonal complement. Those two equations are also the same as the ones obtained for more restrictive, adapted variations considered in [14]. It is worth noting that the variation formulas for geometric quantities, that we obtain along the way to the Euler-Lagrange equation, can be of use also for many other functionals.…”

“…Remark 1.4. For bothḡ ⊥ -andḡ ⊤ -variations we have Vol(Ω,ḡ t ) = Vol(Ω, g) for all t for which g t is defined, see [14]. Obviously, every g ⊥ -variation, that preserves the volume of Ω,…”

“…This section recalls necessary definitions of some functions and tensors, see [14], and introduces several new notions related to geometry of pseudo-Riemannian almost-product manifolds.…”

“…where {e λ } is a full orthonormal basis of T M and ǫ λ = g(e λ , e λ ) ∈ {−1, 1}. We also use symmetric (0, 2)-tensors Φ h and Φ T defined as in [14], i.e., for any symmetric (0, 2)-tensor S we have…”

“…It is easy to see that all the above tensors defined with the use of an adapted orthonormal frame in fact do not depend on the choice of such frame. Remark 1.1 (see [14]). If g is definite on D then Φ h = 0 if and only if one of the following point-wise conditions holds:…”