Displaying similar documents to “Curvatures of the diagonal lift from an affine manifold to the linear frame bundle”

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.

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...

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...

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.

Affine analogues of the Sasaki-Shchepetilov connection

Maria Robaszewska (2016)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections...