Investigation of conformally killing vector fields on 5-dimensional 2-symmetric lorentzian manifolds

Conformally Killing fields play an important role in the theory of Ricci solitons and also generate an important class of locally conformally homogeneous (pseudo) Riemannian manifolds. In the Riemannian case, V. V. Slavsky and E.D. Rodionov proved that such spaces are either conformally flat or conformally equivalent to locally homogeneous Riemannian manifolds. In the pseudo-Riemannian case, the question of their structure remains open. Pseudo-Riemannian symmetric spaces of order k, where k 2, play an important role in research in pseudo-Riemannian geometry. Currently, they have been investigated in cases k=2,3 by D.V. Alekseevsky, A.S. Galaev and others. For arbitrary k, non-trivial examples of such spaces are known: generalized Kachen - Wallach manifolds. In the case of small dimensions, these spaces and Killing vector fields on them were studied by D.N. Oskorbin, E.D. Rodionov, and I.V. Ernst with the helpof systems of computer mathematics. In this paper, using the Sagemath SCM, we investigate conformally Killing vector fields on five-dimensional indecomposable 2- symmetric Lorentzian manifolds, and construct an algorithm for their computation.

Carnot metrics, dynamics and local rigidity

This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.

Stability of Einstein metrics on symmetric spaces of compact type

We prove the linear stability with respect to the Einstein-Hilbert action of the symmetric spaces $${\text {SU}}(n)$$ SU ( n ) , $$n\ge 3$$ n ≥ 3 , and $$E_6/F_4$$ E 6 / F 4 . Combined with earlier results, this resolves the stability problem for irreducible symmetric spaces of compact type.

The Injectivity Theorem on a Non-Compact Kähler Manifold

In this paper, we establish an injectivity theorem on a weakly pseudoconvex Kähler manifold X with negative sectional curvature. For this purpose, we develop the harmonic theory in this circumstance. The negative sectional curvature condition is usually satisfied by the manifolds with hyperbolicity, such as symmetric spaces, bounded symmetric domains in Cn, hyperconvex bounded domains, and so on.

Completeness of b−Metric Spaces and Best Proximity Points of Nonself Quasi-Contractions

The aims of this article are twofold. One is to prove some results regarding the existence of best proximity points of multivalued non-self quasi-contractions of b−metric spaces (which are symmetric spaces) and the other is to obtain a characterization of completeness of b−metric spaces via the existence of best proximity points of non-self quasi-contractions. Further, we pose some questions related to the findings in the paper. An example is provided to illustrate the main result. The results obtained herein improve some well known results in the literature.

Black holes

We discuss event horizons and black holes. First Birkhoff’s theorem is derived, and we consider the general nature of spherically symmetric spaces. Then the concepts of null surface, Killing horizon and event horizon are defined and related to one another. Cosmic censorship is briefly discussed. The Schwarzshild horizon is discussed in detail. The divergence or otherwise of redshift, acceleration, speed and proper time is obtained for infalling observers and for Schwarzschild observers. Eddington-Finkelstein coordinates are introduced and used to discuss gravitational collapse. The growth of the horizon is noted, and the causality structure is briefly considered via an introduction to the conformal (Penrose-Carter) diagram. The maximal extension is then presented, with the Kruskal-Szekeres coordinates and associated diagram. Wormholes are briefly discussed. The chapter finishes with a survey of astronomical evidence for black holes.