The space $LV$ of free loops on a manifold $V$ inherits an action of the circle group $\T$. When $V$ has an almost-complex structure, the tangent bundle of the free loopspace, pulled back to a certain infinite cyclic cover $\LV$, has an equivariant decomposition as a completion of $\bT V \otimes (\oplus \C(k))$, where $\bT V$ is an equivariant bundle on the cover, nonequivariantly isomorphic to the pullback of $TV$ along evaluation at the basepoint (and $\oplus \C(k)$ denotes an algebra of Laurent polynomials). On a flat manifold, this analogue of Fourier analysis is classical.
Homology, Homotopy and Applications, Vol. 3(2), 2001, pp. 407-415
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