High-order angles in almost-Riemannian geometry

Ugo Boscain; Mario Sigalotti

Displaying similar documents to “High-order angles in almost-Riemannian geometry”

On almost-Riemannian surfaces

Roberta Ghezzi (2010-2011)

Séminaire de théorie spectrale et géométrie

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An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in general it has rank 1 on a nonempty set which is generically a smooth curve. In this paper we provide a short introduction to 2-dimensional almost-Riemannian geometry highlighting...

A lossless reduction of geodesics on supermanifolds to non-graded differential geometry

Stéphane Garnier, Matthias Kalus (2014)

Archivum Mathematicum

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Let $ℳ = (M, 𝒪_{ℳ})$ be a smooth supermanifold with connection $\nabla$ and Batchelor model $𝒪_{ℳ} ≅ Γ_{Λ E^{*}}$ . 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 $𝒪_{ℳ} ≅ Γ_{Λ E^{*}}$ . 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...

A characterization of isometries between Riemannian manifolds by using development along geodesic triangles

Petri Kokkonen (2012)

Archivum Mathematicum

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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 $(\hat{M}, \hat{g})$ in terms of developing geodesic triangles of $M$ onto $\hat{M}$ . More precisely, we show that if $A_{0} : T {|_{x_{0}} M \to T |}_{{\hat{x}}_{0}} \hat{M}$ is some isometric map between the tangent spaces and if for any two geodesic triangles $γ$ , $ω$ of $M$ based at $x_{0}$ the development through $A_{0}$ of the composite path $γ \cdot ω$ onto $\hat{M}$ results in a closed path based at ${\hat{x}}_{0}$ , then there exists a Riemannian covering...

A class of metrics on tangent bundles of pseudo-Riemannian manifolds

H. M. Dida, A. Ikemakhen (2011)

Archivum Mathematicum

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We provide the tangent bundle $T M$ of pseudo-Riemannian manifold $(M, g)$ with the Sasaki metric $g^{s}$ and the neutral metric $g^{n}$ . First we show that the holonomy group $H^{s}$ of $(T M, g^{s})$ contains the one of $(M, g)$ . What allows us to show that if $(T M, g^{s})$ is indecomposable reducible, then the basis manifold $(M, g)$ is also indecomposable-reducible. We determine completely the holonomy group of $(T M, g^{n})$ according to the one of $(M, g)$ . Secondly we found conditions on the base manifold under which $(T M, g^{s})$ ( respectively $(T M, g^{n})$ ) is Kählerian, locally symmetric...