Riemannian geometries on spaces of plane curves

Details Riemannian geometries on spaces of plane curves Riemannian geometries on spaces of plane curves

2003-12-19

arxiv.org/abs/math/0312384v3

J. Eur. Math. Soc. (JEMS) 8 (2006), 1-48

Mathematics

Differential Geometry

amslatex, 45 pagex, 8 figures, typos corrected

Scientific paper

We study some Riemannian metrics on the space of regular smooth 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 $A> 0$: $$ G^A_c(h,k) := \int_{S^1}(1+A\ka_c(\th)^2)< h(\th),k(\th) > |c'(\th)| d\th $$ where $\ka_c$ is the curvature of the curve $c$ and $h,k$ are normal vector fields to $c$. The term $A\ka^2$ 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 $\ge 0$, while for curves with high curvature or perturbations of high frequency, the curvatures are $\le 0$.

Affiliated with

Also associated with

No associations

Rating

Riemannian geometries on spaces of plane curves does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Riemannian geometries on spaces of plane curves, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Riemannian geometries on spaces of plane curves will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-198898