In this paper we characterize the existence of Riemannian covering maps from a complete simply connected Riemannian manifold $(M,g)$ onto a complete Riemannian manifold $(\widehat{M},\widehat{g})$ in terms of developing geodesic triangles of $M$ onto $\widehat{M}$. More precisely, we show that if $A_0:T{{|}_{x_0}M\to T|}_{{\widehat{x}}_0}\widehat{M}$ is some isometric map between the tangent spaces and if for any two geodesic triangles $\gamma$, $\omega$ of $M$ based at $x_0$ the development through $A_0$ of the composite path $\gamma ·\omega$ onto $\widehat{M}$ results in a closed path based at ${\widehat{x}}_0$, then there exists a Riemannian covering map...

The space of the torsion (0,3)-tensors of the linear connections on almost contact manifolds with B-metric is decomposed in 15 orthogonal and invariant subspaces with respect to the action of the structure group. Three known connections, preserving the structure, are characterized regarding this classification.

The space of the closures of leaves of a Riemannian foliation is a nice topological space, a stratified singular space which can be topologically embedded in ${ℝ}^k$ for k sufficiently large. In the case of Orbit Like Foliations (OLF) the smooth structure induced by the embedding and the smooth structure defined by basic functions is the same. We study geometric structures adapted to the foliation and present conditions which assure that the given structure descends to the leaf closure space. In Section...

This article deals with vector valued differential forms on $C^{\infty }$-manifolds. As a generalization of the exterior product, we introduce an operator that combines $Hom({⨂}^s\left(W\right),Z)$-valued forms with $Hom({⨂}^s\left(V\right),W)$-valued forms. We discuss the main properties of this operator such as (multi)linearity, associativity and its behavior under pullbacks, push-outs, exterior differentiation of forms, etc. Finally we present applications for Lie groups and fiber bundles.

In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities (I). The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a...

Let $ℳ=(M,{𝒪}_{ℳ})$ be a smooth supermanifold with connection $\nabla$ and Batchelor model ${𝒪}_{ℳ}\cong {\Gamma }_{\Lambda E^{*}}$. From $(ℳ,\nabla )$ we construct a connection on the total space of the vector bundle $E\to M$. This reduction of $\nabla$ is well-defined independently of the isomorphism ${𝒪}_{ℳ}\cong {\Gamma }_{\Lambda E^{*}}$. It erases information, but however it turns out that the natural identification of supercurves in $ℳ$ (as maps from ${ℝ}^{1|1}$ to $ℳ$) with curves in $E$ restricts to a 1 to 1 correspondence on geodesics. This bijection is induced by a natural identification of initial conditions for geodesics...

This paper studies the relationship between the sections and the Chern or Pontrjagin classes of a vector bundle by the theory of connection. Our results are natural generalizations of the Gauss-Bonnet Theorem.

It is proven that every flat connection or covariant derivative ∇ on a left A-module M (with respect to the universal differential calculus) induces a right A-module structure on M so that ∇ is a bimodule connection on M or M is a flat differentiable bimodule. Similarly a flat hom-connection on a right A-module M induces a compatible left A-action.