Peter W. Michor

Riemannian Geometries on Spaces of Plane Curves

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.

A metric on shape space with explicit geodesics

This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc...) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided

An Overview of the Riemannian Metrics on Spaces of Curves Using the Hamiltonian Approach

Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1 . We investigate several Riemannian metrics on shape space: L 2 -metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on R 2 is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.

Topics in Differential Geometry

Manifolds and vector fields Lie groups and group actions Differential forms and de Rham cohomology Bundles and connections Riemann manifolds Isometric group actions or Riemann

Connections on central bimodules in noncommutative differential geometry

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Sobolev metrics on shape space of surfaces

-manifolds Symplectic and Poisson geometry List of symbols Bibliography Index.

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudo-riemannian structures.

CHOOSING ROOTS OF POLYNOMIALS SMOOTHLY

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The multigraded Nijenhuis-Richardson algebra, its universal property and applications

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Almost Local Metrics on Shape Space of Hypersurfaces in

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