Kähler Structures on Spaces of Framed Curves Tom Needham The Ohio State University

APS March Meeting Knotting in Filaments and Fields March 15, 2017

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Motivation: Distance on Spaces of Curves Question What is the correct notion of distance between a pair of curves? Should be invariant under rotations and choice of parameterization. Rough Idea Use geodesic distance in the ∞-dimensional space of curves w.r.t. some Riemannian metric (think L2 inner product). Visualized as an optimal homotopy between the curves.

A priori, finding the geodesic requires solving a nonlinear PDE. Can avoid numerical methods using geometry of framed loop space. 2 / 10

Framed Loops Definitions A framed loop is a parameterized closed space curve with a choice of normal vector field. A relatively framed loop is an equivalence class of framed loops up to S 1 -action of twisting all normal vectors by fixed angle.

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Framed loop space is the ∞-dimensional manifold M := {relatively framed loops}/Sim(R3 ) where Sim(R3 ) = {translation, rotation, scaling}. 3 / 10

Kähler Structure on Framed Loop Space Theorem (N) Each component of framed loop space M is isometric to a dense open subset of Gr2 (C ∞ (S 1 , C)) = {2-planes in complex vector space C ∞ (S 1 , C)} with L2 Riemannian metric. Grassmannian is a Kähler manifold with its L2 Hermitian structure ⇒ M is a Kähler manifold. Structures on M ◦ Riemannian metric: generalization of elastic metrics from computer vision (Mio-Srivastava-Joshi). ◦ Complex structure/symplectic form: related to Marsden-Weinstein structures on loop spaces from fluid dynamics. Corollary Planar loop space Imm(S 1 , R2 )/Sim(R2 ) embeds as a totally geodesic, Lagrangian submanifold of M. ⇒ Geodesics between framed loops and plane curves are explicit! 4 / 10

The Action of Diff+ (S 1 ) Diff+ (S 1 ) acts on M (and planar loop space) by reparameterization. The action is Hamiltonian and by isometries. Want distance between curves to be invariant to parameterization—i.e., want induced distance in M/Diff+ (S 1 ). Practically: ◦ Search over Diff+ (S 1 )-fiber for optimal parameterizations. ◦ Compute geodesic distance in total space M. ◦ Optimal parameterizations found via dynamic programming.

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Comparing Geodesics Geodesics between horse shapes in the tot. geod. submanifold Imm(S 1 , R2 )/Sim(R2 ) ⊂ M.

Geodesic between arclength-parameterized curves. Dist ≈ 0.58.

Starting curve is arclength-param., ending curve has optimal param. Dist ≈ 0.22. 6 / 10

Geodesics in Framed Loop Space Geodesics between tight trefoil and circle in M.

Both curves arclength param. with Frenet frames.

Trefoil arclength param. with Frenet frame. Circle has optimal parameterization. Framing is optimized over C ∞ (S 1 , S 1 )/S 1 -fibers. 7 / 10

Protein Registration and Clustering Similar constructions hold for spaces of open curves. The spaces are isometric to ∞-dim’l projective spaces. 1. 1A0H 2. 1A2C 3. 1A46 4. 1A4W 5. 1A5G 6. 1A5I 7. 1 A61 8. 1A7Y 9. 1A7Z 10. 1A85 1

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1A5G to 1A85 geodesic distance ~ 1.32

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1A5G to 1A61 geodesic distance ~ 0.27

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Related Work Theorem (Younes-Michor-Shah-Mumford) The planar loop space Imm(S 1 , R2 )/Sim(R2 ) is isometric to an ∞-dimensional real Grassmannian. Theorem (N) The frame twisting action of C ∞ (S 1 , S 1 )/S 1 is Hamiltonian. Symplectic reduction of M by this action is isomorphic to isometric immersion space {γ ∈ Imm(S 1 , R3 ) | kγ 0 (t)k = 1}/Sim(R3 ), studied by Millson and Zombro. Theorem (Howard-Manon-Millson, Hausmann-Knutson) ×2n

Gr2 (Cn ) −−→ {relatively framed n-gons in R3 }/Sim(R3 ) G = U(1)n /U(1) acts on Gr2 (Cn ) Gr2 (Cn ) G ≈ {n-gons w/ fixed edgelengths}/Sim(R3 ). 9 / 10

Thanks for Listening! Future Work ◦ Riemannian shape analysis with respect to other metrics ◦ Applications to protein clustering, surface matching, animation ◦ Framed loop spaces of general 3-manifolds References ◦ Kähler structures on spaces of framed curves. Arxiv 1701.03183 ◦ Grassmannian geometry of framed curve spaces. Phd Dissertation. ◦ Elastic shape analysis for framed space curves. In preparation.

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Knot Clustering Clustering tight knot shapes up to 7 crossings by geodesic distance with optimized parameterizations and framings.

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Knot Clustering

41 to 63 . Geod dist ≈ 0.25.

41 to 62 . Geod dist ≈ 0.43.