A sharp upper bound on the first S invariant eigenvalue of the Laplacian for S invariant metrics on S is used to find obstructions to the existence of S equivariant isometric embeddings of such metrics in (R, can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in (R, can). This leads to ge...

We study the existence of projectable G-invariant Einstein metrics on the total space of G-equivariant fibrations M = G/L → G/K, for a compact connected semisimple Lie group G. We obtain necessary conditions for the existence of such Einstein metrics in terms of appropriate Casimir operators, which is a generalization of the result by Wang and Ziller about Einstein normal metrics. We describe b...

Abstract. Let Dn be the generalized unit disk of degree n. In this paper, we find Riemannian metrics on the Siegel-Jacobi disk Dn × C (m,n) which are invariant under the natural action of the Jacobi group explicitly and also compute the Laplacians of these invariant metrics explicitly. These are expressed in terms of the trace form. We give a brief remark on the theory of harmonic analysis on t...

We develop techniques for classifying the nonnegatively curved left-invariant metrics on a compact Lie group G. We prove rigidity theorems for general G and a partial classification for G = SO(4). Our approach is to reduce the general question to an infinitesimal version; namely, to classify the directions one can move away from a fixed bi-invariant metric such that curvature variation formulas...

Image segmentation is a critical step in building a computer vision algorithm that is able to distinguish between separate objects in an image scene. Image segmentation is based on two fundamentally intertwined components: pixel comparison and pixel grouping. In the pixel comparison step, pixels are determined to be similar or different from each other. In pixel grouping, those pixels which are...

We simplify the Hitchin's description of SU (2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system. The SU (2) invariant self-dual Einstein metrics were studied in a number of papers [1, 2, 3]. The local classification of the metrics of this type was given in extensive paper by Hitchin [3]. However, the final form of the metric coefficie...

We combine the results of Bartal [Proc. 37th FOCS, 1996, pp. 184–193] on probabilistic approximation of metric spaces by tree metrics, with those of Klein, Plotkin and Rao [Proc. 25th STOC, 1993, pp. 682–690] on decompositions of graphs without small Ks,s minors (such as planar graphs) to show that metrics induced by such graphs (with unit lengths on the edges) can be probabilistically approxim...

A Riemannian manifold (M, ρ) is called Einstein if the metric ρ satisfies the condition Ric(ρ) = c · ρ for some constant c. This paper is devoted to the investigation of G-invariant Einstein metrics with additional symmetries, on some homogeneous spaces G/H of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds SO(n)/SO(l), and on the symplecti...

in this paper, we study a class of finsler metrics which contains the class of p-reducible andgeneral relatively isotropic landsberg metrics, as special cases. we prove that on a compact finsler manifold,this class of metrics is nothing other than randers metrics. finally, we study this class of finsler metrics withscalar flag curvature and find a condition under which these metrics reduce to r...