We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.

We define a manifold M where objects c ∈ M are curves, which we parameterize as c : S → lR (n ≥ 2, S is the circle). We study geometries on the manifold of curves, provided by Sobolev–type Riemannian metrics H . These metrics have been shown to regularize gradient flows used in Computer Vision applications, see [13, 16] and references therein. We provide some basic results of H metrics; and, fo...

In Theorem 1, we generalize some results of Szabó [Sz1, Sz2] for Berwald metrics that are not necessarily strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F . As an application we show (Corollary 3) that every Berwald projectively flat metric is a Minkowski metric; this statement is a “Berwald” version of Hilbert’s 4th problem...

In this note, we announce some results showing unexpected similarities between the moduli spaces of constant curvature metrics on 2-manifolds (the Riemann moduli space) and moduli spaces of Einstein metrics on 4manifolds. Let J? denote the moduli space of Einstein metrics of volume 1 on a compact, orientable 4-manifold M. If J£\ denotes the space of smooth Riemannian metrics of volume 1 on M, e...

In this paper, the segmentation problem is formulated as a problem of segmenting a Riemannian manifold. The image domain is endowed with an anisotropic metric and its segmentation is obtained by thresholding the second eigenvector of the Laplace-Beltrami operator on the Riemannian manifold so defined. The formulation is an analytic analog of a recently proposed approach to segmentation based on...

One of the key properties of the length of a curve is its lower semicontinuity : if a sequence of curves γi converges to a curve γ, then length(γ) ≤ lim inf length(γi). Here the weakest type of pointwise convergence suffices. There are higher-dimensional analogs of this semicontinuity for Riemannian (and even Finsler) metrics. For instance, the Besicovitch inequality (see, e.g., [1] and [4]) im...