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.

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

Let $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity. First let G be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$ or $\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$ , and let $\Gamma $ be an irreducible cocompact lattice in G. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type. Secondly we show that if $M\stackrel {\rho _0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho _0$ with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property

By [5] it is known that a geodesic γ in an abstract reflection space X (in the sense of Loos, without any assumption of differential structure) canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we modify these arguments in order to prove an analog of this result stating that, if X contains an embedded hyperbolic plane 𝓗 ⊂ X, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL2(ℝ). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact split type Y lying in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac–Moody symmetric space G/K for an algebraically simply connected two-spherical split Kac–Moody group G, as defined in [5], satisfies a universal property similar to the universal property that the group G satisfies itself.

Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions

The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush–Kuhn–Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash’s critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces.