Some notes on vector fields in the tangent bundle

2021

Let (M, g) be a Riemannian manifold, g and g be respectively the corresponding Sasaki metric and the horizontal lift on the tangent bundle TM . In this paper we study the harmonicity of g-natural metrics with respect to g and then with respect to g. We compute the tension fields in terms of the Ricci curvature and we describe the harmonicity conditions accordind to whether the basis metric g is Einstein or not. In the case of a Ricci flat manifold (M, g) we obtain that the horizontal lift g is an harmonic metric w.r.t. the Sasaki metric g, and conversely that g is harmonic w.r.t. g on TM . Futhermore for a connected non-Einstein manifold (M, g), we prove that there exists no nontrivial Riemannian g-natural metric on TM that is harmonic w.r.t the horizontal lift g.

Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 2018

Let (M; g) be a n-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by G f and called the vertical rescaled metric on the tangent bundle T M. We calculate its Levi-Civita connection and Riemannian curvature tensor. We study the geometry of (T M; G f) and several important results are obtained on curvature, Einstein structure, scalar and sectional curvatures.

Differential Geometry and its Applications, 1993

The study of the calculus of forms along the tangent bundle projection r, 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 r 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,l) tensor field along r, 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. Keyzuords: Derivations, forms along a map, second-order equations, nonlinear connections. MS classification: 58A10, 53CO5, 70D05.

The main aim of this survey paper is to write a detailed and unified presentation of some of the best known results on the geometry of tangent bundles of Riemannian manifolds.

2008

Let ( M n; g ) be a Riemannian manifold and T ( M n) its tangent bundle with diagonal lift connection and adapted almost paracomplex structure. We determine the innitesimal paraholomorphica projective transformation on T ( M n). Furthermore, if T ( M n) admits a non-affine innitesimal paraholomorphica projective transformation, then M n and T ( M n) are locally at. M.S.C. 2000: 53C20, 53C15.

International Electronic Journal of Geometry

Let (M n , g) be a Riemannian manifold and T M n the total space of its tangent bundle. In this paper, we determine the infinitesimal fiber-preserving holomorphically projective (IFHP) transformations on T M n with respect to the Levi-Civita connection of the deformed complete lift metricG f = g C + (f g) V , where f is a nonzero differentiable function on M n and g C and g V are the complete lift and the vertical lift of g on T M n , respectively. Morevore, we prove that every IFHP transformation on (T M n ,G f) is reduced to an affine and induces an infinitesimal affine transformation on (M n , g).

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz HORIZONT.AL LIFTS OF TENSOR FIELDS AND CONNECTIONS TO TИE TANGENT ВДNDLE OF HIGHER ORDER Jacek Gancarzewicz and Modesto Salgдdo 0. Introduction Let T^M = {i r Y I f :(-£++£)-• M of class 0°°} be the tangent bundle of order r t where M is a manifold of dimension n. We denote by Ttt^M • M > ^(i^f) s yCo-) the bundle projection. Let T be a connection of order r on M-. that is* P is a connection in the principal fibre bundle F r M of frames of order r. Since T^H is an associated bundle with. F r M > this connection defines a distribution H on T r M» called the horizontal distribution,, such that T(T r M) 8 VClr^M) ® H where VCT 1 *!) = ker d-rr is the distribution of vertical vectors on T r M. The restriction ^irJH is an isomorphism of H onto ^-.(^M and we can define the horizontal lift X of a vector field X from M to T^Vl by the formula д»(p>-(VІV" 1(X-ЧP)> In this paper we will discuss horizontal lifts of tensor fields from M to TFM* This paper has six sections. In Section 1 we recall results of A. Morimoto [?]» [tl] about lifts of tensor fields to the bundle T r M. This paper is in final form and no version of it will be submitted for publication elsewhere.

Bulletin of the Transilvania University of Brasov Series III Mathematics and Computer Science

In this paper, we have concentrated on a group action on the tangent bundle of some smooth/differentiable manifolds which has been built from a regular Lie group action on such smooth/differentiable manifolds. Interestingly, elements of orbit space yield smooth sections of the tangent bundle having beautiful algebraic properties. Moreover, each of those smooth sections behaves nicely as a left-invariant vector field with respect to Lie group action by $G$. We have explained here a simple isomorphism between the set of such smooth sections and each tangent space of that smooth/differentiable manifold. Also we have discussed more about $F$-relatedness and have introduced vector field relatedness by notations $rel_{\mathfrak{X}(M)}(F), rel_{Diff(M)}(X)$, etc. which are sets based on both vector field related diffeomorphisms and diffeomorphism related vector fields. We have presented consequences based on the algebraic structure on $rel_{\mathfrak{X}(M)}(F), rel_{Diff(M)}(X)$, etc. sets...

Journal of Geometry and Physics, 1987

We show that the cotangent bundle T * TM of the tangent bundle of any differentiable manifold M carries an integrable almost tangent structure which is generated by a natural lifting procedure from the canonical almost tangent structure (vertical endomorphism) of TM. Using this almost tangent structure we show that T * TM is diffeomorphic to a tangent bundle, namely TT * M. This provides a new and geometrically instructive proof of a result of Tulczyjew, which has applications in Lagrangian and Hamiltonian dynamics and in field theory. The requisite general definitions and results concerning liftings of geometric objects from a manifold to its cotangent bundle are given. As an application, we shed new light on the meaning of so-called adjoint symmetries of second-order differential equations.

Journal of Mathematical Analysis and Applications, 2008

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometers using some Riemannian and pseudo-Riemannian lift metrics on TM. Here we consider the Riemannian or pseudo-Riemannian lift metric G on TM which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on (TM, G).