Unit tangent bundle

Abstract: If M is a differentiable ra-dimensional manifold and V a linear connection for M, then the 2 rc-dimensional manifold TM, which is the total space of the tangent bündle of M, admits an almost complex structure /, naturally determined by V *). (I learned of this almost complex structure, which occurs e. g. in the theory of partial differential equations on Riemannian manifolds, frorn Professor W. Ambrose. I wish to thank him very much for the stimulating conversations which I have had with him on that topic.) We shall give here a computation of the Eckmann-Frölicher torsion tensor for this almost complex structure /, which implies the following result: / is integrable if and only if the linear connection has vanishing torsion and curvature). An appendix is devoted to some questions on the geometry of the tangent bündle TM which arise in connection with the construction of / and which can be answered easily by methods similar to those which we have used in order to compute the EckmannFrölicher torsion tensor of /. We list here only two of these results: If g is a Riemann metric for M and V its Levi-Civita connection, then TM admits a canonical hermitian metric hg with respect to the almost complex structure / on TM, which is determined (see above) by VWe prove, (confer Appendix (iii)) that hg is kählerian. If V is any linear connection for M, then the distribution of the \"horizontal subspaces\" on TM is invariant under the action of the multiplicative group R* of non vanishing real numbers on TM. We prove (confer Appendix (iv)) that if oppositely an n-dimensional distribution on TM is given, which is invariant under the action of the group JR* on TM and which contains no nonzero vertical\" vector, then this distribution

Abstract: Various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined, and the results applied to time-independent Lagrangian dynamics. It is shown that a certain type (1, 1) tensor field which is part of the intrinsic geometry of a tangent bundle, being a tensorial equivalent of the projection map of tangent vectors, plays a role in Lagrangian theory scarcely less important than that of the canonical one-form on a cotangent bundle in Hamiltonian theory. Recent results in Lagrangian theory are interpreted from this new viewpoint.

Abstract: 2. Generalities [2; l]. If xGA, vi, v2(E.Xx, let (vu v2) denote the inner product that defines the Riemannian metric. If a: [0, l]—»A is a curve, let a':t-^a'(t) denote the tangent vector field to a. If v: H»(i)GI,(i) is a vector-field along a, let Av denote the covariant derivative of v along a [l]. Let us recall how it is defined by Elie Cartan's method of orthonormal moving frames: Suppose U is an open set of A and w, (léi,j, k • ■ • 5¡w = dim X, summation convention) 1-differential forms in U with