We wish to describe how the hyperbolic geometry of a Riemann surface of genus g y g > 2, leads to a symplectic geometry on Tg, the genus g Teichmüller space, and ~Mg, the moduli space of genus g stable curves. The symplectic structure has three elements: the Weil-Petersson Kahler form, the FenchelNielsen vector fields t+, and the geodesic length functions I*. Weil introduced a Kahler metric for...

Here shape space is either the manifold of simple closed smooth unparameterized curves in R or is the orbifold of immersions from S to R modulo the group of diffeomorphisms of S. We investige several Riemannian metrics on shape space: L-metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two le...

In this article we define and study a notion of asymptotic rank for metric spaces and show in our main theorem that for a large class of spaces, the asymptotic rank is characterized by the growth of the higher filling functions. For a proper, cocompact, simplyconnected geodesic metric space of non-curvature in the sense of Alexandrov the asymptotic rank equals its Euclidean rank.

An H type metric on the space of planar curves is proposed and equation of the geodesic is derived. A numerical example is given to illustrate the differneces between H and H metrics.

In this paper we study geodesic Ptolemy metric spaces X which allow proper and cocompact isometric actions of crystallographic or, more generally, virtual polycyclic groups. We show that X is equivariantly rough isometric to a Euclidean space.

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δneighborhood of the union of the other two sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) ...

We prove that, if a geodesic metric space has Markov type 2 with constant 1, then it is an Alexandrov space of nonnegative curvature. The same technique provides a lower bound of the Markov type 2 constant of a space containing a tripod or a branching point.

The L-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N, g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the L-metric.

The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function φ in the form K D(H,K) = ∑ i,j φ(λi, λj) −1TrPiHPjK when ∑ i λiPi is the spectral decomposition of the foot point D and the Hermitian matrices H,K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping D 7→ ...