We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from S 1 to the plane modulo the group of diffeomorphisms of S 1 , acting as reparameterizations. In particular we investigate the metric for a constant A > 0: where κc is the curvature of the curve c and h, k are normal vector fields to c. The term Aκ 2 is a sort of geometric Tikhonov regularization because, for A = 0, the geodesic distance between any 2 distinct curves is 0, while for A > 0 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 ≥ 0, while for curves with high curvature or perturbations of high frequency, the curvatures are ≤ 0. Date: November 21, 2005.
Michor and Mumford REF studied the Riemann curvature tensor on the space of smooth planar curves and gave an explicit formula for the sectional curvature.
Riemannian geometries on spaces of plane curves
J. Eur. Math. Soc. (JEMS
Mathematics