RIEMANNIAN GEOMETRIES ON SPACES OF PLANE CURVES

PETER MICHOR

Abstract: This is joint work with David Mumford. We study some weak Riemannian metrics on spaces of regular smooth curves in the plane, viewed as orbit spaces under the group of diffeomorphisms of the circle acting as reparameterizations. In particular we investigate the metric for a constant A > 0:

G_{c}^{A}(h, k) := ∫_{S^{1}}(1 + Aκ_{c}(θ)^{2})⟨h(θ), k(θ)⟩|c′(θ)| dθ

where κ_{c} is the curvature of the curve c. For A = 0 this induces 0 as geodesic distance function on the orbit space, but for A > 0 is induces a point separating metric. We give some estimates 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.