A notion of metric invariance entropy is constructed with respect to a conditionally invariant measure for control systems in discrete time. It is shown that the metric invariance entropy is invariant under conjugacies, the power rule holds, and the (topological) invariance entropy provides an upper bound. November 20, 2014 Key words. invariance entropy, conditionally invariant measures, quasi-...

In the present paper a global conformal invariant Y of a closed initial data set is constructed. A spacelike hypersurface Σ in a Lorentzian spacetime naturally inherits from the spacetime metric a differentiation De, the so-called real Sen connection, which turns out to be determined completely by the initial data hab and χab induced on Σ, and coincides, in the case of vanishing second fundamen...

In this article we consider the possible sets of distances in Polish metric spaces. By a Polish metric space we mean a pair (X, d), where X is a Polish space (a separable, completely-metrizable space) and d is a complete, compatible metric for X. We will consider two aspects. First, we will characterize which sets of reals can be the set of distances in a Polish metric space. We will also obtai...

This paper focuses on the study of open curves in a manifold M , and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions M = Imm([0,1], M) by pullback of a metric on the tangent bundle TM derived from the Sasaki...

Abstract We study left-invariant conformal Killing 2- or 3-forms on simply connected 2-step nilpotent Riemannian Lie groups. show that if the center of group is dimension greater than equal to 4, then every such form automatically coclosed (i.e. it a form). In addition, we prove only groups with at most 3 and admitting non-coclosed are following: The Heisenberg their trivial 1-dimensional exten...

The object of this paper is to study invariant submanifolds M of Kenmotsu manifolds M̃ admitting a quarter symmetric metric connection and to show that M admits quarter symmetric metric connection. Further it is proved that the second fundamental forms σ and σ with respect to LeviCivita connection and quarter symmetric metric connection coincide. Also it is shown that if the second fundamental f...

<p style='text-indent:20px;'>The goal of this paper is the study integrability geodesic flow on <inline-formula><tex-math id="M1">\begin{document}$ k $\end{document}</tex-math></inline-formula>-step nilpotent Lie groups, = 2, 3, when equipped with a left-invariant metric. Liouville proved in low dimensions. Moreover, it shown that complete families first integrals ...

In mathematics and engineering, a manifold is topological space that locally resembles Euclidean near each point. Defining the best metric for these manifolds have several engineering science implications from controls to optimization generalized inner product applications of Gram Matrices appear in applications. These smooth geometric can be formalized by Lie Groups their Algebras on its infin...