Second order tangent bundles of infinite dimensional manifolds

Details Second order tangent bundles of infinite dimensional manifolds Second order tangent bundles of infinite dimensional manifolds

2003-10-29

arxiv.org/abs/math/0310455v1

Mathematics

Differential Geometry

8 pages

Scientific paper

10.1016/j.geomphys.2004.02.005

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 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 Fr\'{e}chet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite-dimensional dynamical systems.

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