This lecture describes the holonomy group for a 4-dimensional Hausdorff, connected and simply connected manifold admitting a Lorentz metric and shows, briefly, some applications to Einstein’s space-time of general relativity. 1. Holonomy Theory on 4-Dimensional Lorentz Manifolds. Let M be a 4-dimensional, smooth, Hausdorff, connected, and simply connected manifold admitting a smooth Lorentz met...

We prove a structure theorem for the isometry group $${\text {Iso}}(M,g)$$ of compact Lorentz manifold, under assumption that closed subgroup has exponential growth. don’t assume anything about identity component , so our results apply discrete groups. infer full classification lattices can act isometrically on manifolds. Moreover, without any growth hypothesis, we Tits alternative subgroups .

In this paper we address several aspects of flat Bogomolnyi-PrasadSommerfeld (BPS) domain walls together with their Lorentz invariant vacua of 4d N = 1 supergravity coupled to a chiral multiplet. The scalar field spans a one-parameter family of 2d Kähler manifolds satisfying a Kähler-Ricci flow equation. We find that BPS equations and the scalar potential deform with respect to the real paramet...

In this paper we address several aspects of flat Bogomolnyi-PrasadSommerfeld (BPS) domain walls together with their Lorentz invariant vacua of 4d N = 1 supergravity coupled to a chiral multiplet. The scalar field spans a one-parameter family of 2d Kähler manifolds satisfying a Kähler-Ricci flow equation. We find that BPS equations and the scalar potential deform with respect to the real paramet...

We show an analogue of the Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds: If a Berwald spacetime nonnegative Ricci curvature satisfies certain completeness and metrizability conditions includes timelike straight line, then it necessarily admits one-dimensional family isometric translations generated by gradient vector field Busemann function. Moreover, our formulation in t...

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous , but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members of the family are geodesically complete, others are not. All have vanishing...