On curvatures of linear frame bundles with naturally lifted metrics.
Kowalski, O., Sekizawa, M. (2005)
Rendiconti del Seminario Matematico
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Displaying similar documents to “Curvatures of the diagonal lift from an affine manifold to the linear frame bundle”
On curvatures of linear frame bundles with naturally lifted metrics.
On the geometry of frame bundles
Kamil Niedziałomski (2012)
Archivum Mathematicum
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Let be a Riemannian manifold, its frame bundle. We construct new examples of Riemannian metrics, which are obtained from Riemannian metrics on the tangent bundle . 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 -natural metrics on linear frame bundles
Oldřich Kowalski, Masami Sekizawa (2008)
Archivum Mathematicum
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In this paper we prove that each -natural metric on a linear frame bundle over a Riemannian manifold is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define -natural metrics on the orthonormal frame bundle and we prove the same invariance result as above for . Hence we see that, over a space of constant sectional curvature, the bundle with an arbitrary -natural metric is locally homogeneous.
Differential geometric structures of stable state feedback systems with dual connections
Atsumi Ohara, Shun-ichi Amari (1994)
Kybernetika
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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...