Kähler Geometry of the Space of Knots
Let M be a smooth, paracompact, oriented manifold of dimension n. We introduce the free loop space LM of smooth maps S 1 → M. We will identify S 1 with ℝ/ℤ and denote by x the parameter in ℝ/ℤ. First, recall from [P-S] how the space LM of smooth loops S 1 → M is made into a smooth manifold, modelled on the topological vector space C ∞(S 1,ℝ n ). This vector space has the structure of an ILH space (inverse limit of Hilbert spaces) in the sense of §1.4.
KeywordsVector Field Vector Bundle Tangent Vector Normal Bundle Coadjoint Orbit
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