The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric space comes from...

In this work we study the issue of geodesic extendibility on complete and locally compact metric length spaces. We focus geometric structure space $(\Sigma (X),d_H)$ balls endowed with Hausdorff distance give an explicit isometry between closed half-space $ X\times \mathbb{R}_{\ge 0}$ a taxicab metric. Among applications establish group $\mbox{Iso} (X,d)$ (\Sigma when $(X,d)$ is Hadamard space.

The Riemannian metric on the manifold of positive de nite matrices is de ned by a kernel function in the formK D(H;K) = P i;j ( i; j) TrPiHPjK when P i iPi 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! G(D) is a ...

The present paper discusses the problem of least-squares over the real symplectic group of matrices Sp(2n,R). The least-squares problem may be extended from flat spaces to curved spaces by the notion of geodesic distance. The resulting non-linear minimization problem on manifold may be tackled by means of a gradient-descent algorithm tailored to the geometry of the space at hand. In turn, gradi...

Abstract Given a compact orientable surface with finitely many punctures Σ, let S(Σ) be the set of isotopy classes of essential unoriented simple closed curves in Σ. We determine a complete set of relations for a function from S(Σ) to R to be the geodesic length function of a hyperbolic metric with geodesic boundary and cusp ends on Σ. As a consequence, the Teichmüller space of hyperbolic metri...

Spheres and hemispheres allow for their interpretation as quantum states spaces quite similar as it is known about projective spaces. Spheres describe systems with two levels of equal degeneracy. The geometric key is in the relation between transition probability and geodesics. There are isometric embeddings as geodesic submanifolds into the space of density operators assuming the latter is equ...

In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality d(x, y) ≤ σ(d(x, z) +d(z, y)) for some constant σ ≥ 1, rather than the usual triangle inequality. Such a space is called a nearmetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzelà theorem) still hold in nearmetric...

In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality d(x, y) ≤ σ(d(x, z) +d(z, y)) for some constant σ ≥ 1, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzelà theorem) still hold in quasimetr...

For any ε > 0, we construct an explicit smooth Riemannian metric on the sphere Sn, n ≥ 3, that is within ε of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is ε-dense in the unit tangent bundle. Moreover, for any ε > 0, we construct a smooth Riemannian metric on S, n ≥ 3, that is within ε of the round metric and has a geodesic for which the complemen...

We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In this article we compare our approach with the diffeomorphic matching framework. In the latter approach a deformation is prescribed on the ambient space, which then drags along an embedded surface. In...