It is well known 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 [Kowalski, J. Reine Angew. Math. 250 (1971) 124–129]. This motivates us to the general question if the flatness and also other simple geometrical properties remain “hereditary” if we replace gs by the most general Riemannian “g-natural metric” on TM (see [Kowalski and Sekizawa, Bull. Tokyo Gakugei Univ. (4) 40 (1988) 1–29; Abbassi and Sarih, Arch. Math. (Brno), submitted for publication]). In this direction, we prove that if (TM,G) is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then (M,g) possesses the same property, respectively. We also give explicit examples of g-natural metrics of arbitrary constant scalar curvature on TM.

On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds

The study of the calculus of forms along the tangent bundle projection τ, initiated in a previous paper with the same title, is continued. The idea is to complete the basic ingredients of the theory up to a point where enough tools will be available for developing new applications in the study of second-order dynamical systems. A list of commutators of important derivations is worked out and special attention is paid to degree zero derivations having a Leibnitz-type duality property. Various ways of associating tensor fields along τ to corresponding objects on TM are investigated. When the connection coming from a given second-order system is used in this process, two important concepts present themselves: one is a degree zero derivation called the dynamical covariant derivative; the other one is a type (1, 1) tensor field along τ, called the Jacobi endomorphism. It is illustrated how these concepts play a crucial role in describing many of the interesting geometrical features of a given dynamical system, which have been dealt with in the literature.

https://cage.ugent.be/zwc/geomech/tm/pdf/list60.pdf

Derivations of differential forms along the tangent bundle projection

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.

Kam-Ping Mok

Papers

Connections and M-Tensors on the Tangent Bundle TM