Riemannian Geometries on Spaces of Plane Curves

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Riemannian Geometries on Spaces of Plane Curves

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Title: Riemannian Geometries on Spaces of Plane Curves
Author: Michor, Peter W.; Mumford, David Bryant

Note: Order does not necessarily reflect citation order of authors.

Citation: Michor, Peter W., and David Bryant Mumford. 2006. Riemannian geometries on spaces of plane curves. Journal of the European Mathematical Society 8(1): 1-48.
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Abstract: We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle, acting as reparameterizations. In particular we investigate the L^2 inner product with respect to 1 plus curvature squared times arclength as the measure along a curve, applied to normal vector field to the curve. The curvature squared term acts as a sort of geometric Tikhonov regularization because, without it, the geodesic distance between any 2 distinct curves is 0, while in our case the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are positive or 0, while for curves with high curvature or perturbations of high frequency, the curvatures are negative.
Published Version: doi:10.4171/JEMS/37
Other Sources: http://www.dam.brown.edu/people/mumford/Papers/DigitizedVisionPapers--forNonCommercialUse/x05a--CurvesMichor.pdf
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3637111
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