Riemannian Geometries on Spaces of Plane Curves

  title={Riemannian Geometries on Spaces of Plane Curves},
  author={Peter W. Michor and David Mumford},
  journal={Journal of the European Mathematical Society},
  • P. Michor, D. Mumford
  • Published 19 December 2003
  • Mathematics
  • Journal of the European Mathematical Society
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… 

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