Book ChapterDOI
Connections and M-Tensors on the Tangent Bundle TM
Yung-Chow Wong,Kam-Ping Mok +1 more
- pp 157-172
TLDR
In this article, the authors discuss the M-tensor and three types of connections on TM, highlight their properties, and explore the relationship between them, and present the proofs of two decomposition theorems, one of which is a sharpened version of a theorem of Grifone.Abstract:
Publisher Summary After the concept of tangent bundle was discovered, it occurred to few differential geometers, including Professor E. T. Davies, that the proper setting for the Finsler metrics and general paths on a smooth manifold M is the tangent bundle TM and not M itself. This chapter discusses the concept of M-tensor and three types of connections on TM, highlights their properties, and explores the relationship between them. Some of the results help to define the relationship between several related known concepts in the differential geometry of TM, such as the system of general paths of Douglas, the nonlinear connections of Barthel and Yano and Ishihara, and the nonhomogeneous connection of Grifone, while others are generalizations of known results. The chapter describes the structure of the tangent bundle TM and the slit tangent bundle STM. It explores the M-tensors and three types of connections on TM and STM. The chapter highlights a (1, l)-connection on TM as horizontal distribution on TM and discusses the relationship between a vector field on TM and the horizontal distribution associated with a (1, l)-connection. It presents the proofs of two decomposition theorems, one of which is a sharpened version of a theorem of Grifone.read more
Citations
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Journal ArticleDOI
On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds
TL;DR: In this article, it was shown that if the tangent bundle TM of a Riemannian manifold (M,g) is endowed with the Sasaki metric gs, then the flatness property on TM is inherited by the base manifold.
Journal ArticleDOI
Derivations of differential forms along the tangent bundle projection
TL;DR: In this article, the authors studied the calculus of forms along the tangent bundle projection τ, initiated in a previous paper with the same title, and provided a list of commutators of important derivations and special attention paid to degree zero derivations having a Leibnitz-type duality property.
References
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Journal ArticleDOI
On the differential geometry of tangent bundles of riemannian manifolds ii
TL;DR: In this paper, a Riemannian metric on the tangent sphere-bundles of the manifold T{M] was introduced, and the geodesic flow on it was considered.
Journal Article
On the Geometry of the Tangent Bundle.
TL;DR: In this paper, the Eckmann-Frölicher tensor of the tangent bündle of a manifold is computed, which implies that the manifold is integrable if and only if the linear connection has vanishing torsion and curvature.
Journal ArticleDOI
Structure presque tangente et connexions II
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/legal.php) are defined, i.e., the copie ou impression de ce fichier doit contenir la présente mention de copyright.