Displaying similar documents to “On natural metrics on tangent bundles of Riemannian manifolds”

Invariance of g -natural metrics on linear frame bundles

Oldřich Kowalski, Masami Sekizawa (2008)

Archivum Mathematicum

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In this paper we prove that each g -natural metric on a linear frame bundle L M over a Riemannian manifold ( M , g ) is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define g -natural metrics on the orthonormal frame bundle O M and we prove the same invariance result as above for O M . Hence we see that, over a space ( M , g ) of constant sectional curvature, the bundle O M with an arbitrary g -natural metric G ˜ is locally homogeneous.

The natural transformations between T-th order prolongation of tangent and cotangent bundles over Riemannian manifolds

Mariusz Plaszczyk (2015)

Annales UMCS, Mathematica

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If (M,g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T* M given by v → g(v,−) between the tangent TM and the cotangent T* M bundles of M. In the present note first we generalize this isomorphism to the one JrTM → JrTM between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT M of cotangent TM bundles of M. Further we describe all base preserving vector bundle maps DM(g) : JrTM → JrT* M depending...

On the completeness of total spaces of horizontally conformal submersions

Mohamed Tahar Kadaoui Abbassi, Ibrahim Lakrini (2021)

Communications in Mathematics

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In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for...

The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds

Mariusz Plaszczyk (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T*M given by v → g(v, –) between the tangent TM and the cotangent T*M bundles of M. In the present note first we generalize this isomorphism to the one JrTM → JrT*M between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT*M of cotangent T*M bundles of M. Further we describe all base preserving vector bundle maps DM(g) : JrTM → JrT*M depending...

On the geometry of frame bundles

Kamil Niedziałomski (2012)

Archivum Mathematicum

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Let ( M , g ) be a Riemannian manifold, L ( M ) its frame bundle. We construct new examples of Riemannian metrics, which are obtained from Riemannian metrics on the tangent bundle T M . We compute the Levi–Civita connection and curvatures of these metrics.