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