8 - Geometry of second order tangent bundle

Summary

The second order tangent bundle T²B of a smooth manifold B consists of the equivalence classes of curves in B that agree up to their acceleration, and arises in a natural way in several problems of theoretical physics and differential geometry (cf., for instance, [DG05], [DR82]). However, the vector bundle structure on T²B is not as straightforward as that of the ordinary (viz. first order) tangent bundle TB of B; in fact, it relies on the choice of a linear connection on B.

Aiming at the reader's convenience, in §§ 8.1.1–8.1.3 we specialize to the ordinary tangent bundle a few facts from the theory of linear connections on vector bundles, exhibited in Chapter 1. In § 8.2 we proceed to the details of the structure of T²B, for a Banach manifold B. Our next target is to find out the extent of the dependence of the vector bundle structure of T²B on the choice of the linear connection on B. This naturally leads us to the notion of second order differentials (§ 8.3). With their help, we prove (in § 8.4) that related (or conjugate) connections induce—up to isomorphism—the same vector bundle structure on T²B.

The last two sections are devoted to the projective limits of second order tangent bundles (§ 8.5), and the generalized second order frame bundle (§ 8.6). Note that, whereas the tangent vectors of curves and velocities of particles naturally form vector bundles, their derivatives, which yield curvatures and accelerations, do not. In order to cover this gap, second order vector bundle structures are constructed for projective limits of Banach modelled manifolds.

The (first order) tangent bundle in brief

We transcribe a few features of the general theory of linear connections to the particular case of the tangent bundle of a Banach manifold. This will pave the way to the main topics of the present chapter.

Linear connections on manifolds

Let B be a Banach manifold with atlas {(U_α, ϕ_α)}_α∈I.