Second order tangent bundles of infinite dimensional manifolds

Abstract

Abstract. The second order tangent bundle T 2 M of a smooth manifold M consists of the equivalent classes of curves on M that agree up to their acceleration. It is known [1] that in the case of a finite n-dimensional manifold M , T 2 M becomes a vector bundle over M if and only if M is endowed with a linear connection. Here we extend this result to M modeled on an arbitrarily chosen Banach space and more generally to those Fréchet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite-dimensional dynamical systems.

A generalized second-order frame bundle for Fréchet manifolds

“…Using the above methodology, we have defined in [6] a Fréchet vector bundle structure on the second order tangent bundle (T 2 M, π 2 , M ), which consists of all equivalence classes of curves in M that agree up to their acceleration. To be more specific, if …”

“…Following the results obtained in [6], if we assume further that M is endowed with a linear connection…”

“…In a previous paper ( [6]) we studied the second order tangent bundle T 2 M of a smooth infinite dimensional manifold M , i.e. the bundle of curves of M that agree up to their acceleration.…”

Remarks on the behaviour of higher-order derivations on the gluing of differential spaces

“…Much can also be found in [3], [5], [21], [23]. For historical reasons, of course, the original paper [27] should be consulted, and also [28] presents an interesting discussion.…”

Isomorphism classes for Banach vector bundle structures of second tangents

“…Dodson and Galanis ( [4]) extended that study to the infinite dimensional case, namely to tangent bundles of order two over Banach and Fréchet manifolds (see also Section 1). In all these cases, existence of a vector bundle structure on T 2 M relies heavily on the choice of a linear connection ∇ on the base manifold M ; the trivializations are directly defined by the Christoffel symbols of ∇.…”