Scalar curvature

About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.

The Riemannian geometry of the configuration space of gauge theories

Abstract: We state some new results about the configuration space of pure Yang-Mills theory. These results come from the study of the kinetic energy term of the Lagrangian of the theory. This term defines a riemannian metric on the space of non-equivalent gauge potentials. We develop a riemannian calculus on the configuration space, compute the riemannian connection, the curvature tensor, and solve for the geodesics, etc. We show that the Gribov ambiguity is more than an artefact of the choice of a gauge condition, and is related to the existence of conjugate points on the geodesics, and is thus an intrinsic feature of the theory.

The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature

Abstract: With an immersion x of a Riemannian n-manifold M into a Euclidean Nspace E there is associated the Gauss map, which assigns to a point p of M the n-plane through the origin of E and parallel to the tangent plane of x(M) at x(p), and is a map of M into the Grassmann manifold G Λ f # = O(N)/O(n)χO(N-n). An isometric immersion of M into a Euclidean iV-sphere S can be viewed as one into a Euclidean (N + l)-sρace E**, and therefore the Gauss map associated with such an immersion can be determined in the ordinary sense. However, for the Gauss map to reflect the properties of the immersion into a sphere, instead of into the Euclidean space, it seems desirable to modify the definition of the Gauss map appropriately. To this end we consider the set Q of all the great n-spheres in 5\", which is naturally identified with the Grassmann manifold of (« -f l)-planes through the center of S in E^ , since such {n + l)-planes determine unique great n-spheres and conversely. In this paper by the Gauss map of an immersion x into 5* is meant a map of M into the Grassmann manifold Gn+ltK+1 which assigns to each point p of M the great H-sphere tangent to x(M) at *(/?), or the (n + l)-plane spanned by the tangent space of x(M) at x(p) and the normal to S at x(p) in E. More generally, with an immersion x of M into a simply-connected complete N-sρace V of constant curvature there is associated a map which assigns to each point p of M the totally geodesic w-subspace tangent to x(M) at*(p). Such a map is called the (generalized) Gauss map. Thus the Gauss map in our generalized sense is a map: M —• Q, where Q stands for the space of all the totally geodesic H-subspaces in V. The purpose of the present paper will be first to obtain a relationship among the Ricci form of the immersed manifold and the second and third fundamental forms of the immersion, and then to give a geometric interpretation of the

Spherical and Hyperbolic Embeddings of Data

Abstract: Many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects. For many types of data, these dissimilarities are not euclidean (i.e., they do not represent the distances between points in a euclidean space), and therefore cannot be isometrically embedded in a euclidean space. Examples include shape-dissimilarities, graph distances and mesh geodesic distances. In this paper, we provide a means of embedding such non-euclidean data onto surfaces of constant curvature. We aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data. The space can be either of positive curvature (spherical) or of negative curvature (hyperbolic). We give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data. Our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation. For objects which do not reside exactly on the manifold, we develop a optimisation-based procedure for approximate embedding on a hyperspherical manifold. We use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space. This process is efficient enough to allow us to embed data sets of several thousand objects. We apply our method to a variety of data including time warping functions, shape similarities, graph similarity and gesture similarity data. In each case the embedding maintains the local structure of the data while placing the points in a metric space.

Newton's method on Riemannian manifolds: covariant alpha theory

Abstract: In this paper, Smale's α theory is generalized to the context of intrinsic Newton iteration on geodesically complete analytic Riemannian and Hermitian manifolds. Results are valid for analytic mappings from a manifold to a linear space of the same dimension, or for analytic vector fields on the manifold. The invariant γ is defined by means of high-order covariant derivatives. Bounds on the size of the basin of quadratic convergence are given. If the ambient manifold has negative sectional curvature, those bounds depend on the curvature. A criterion of quadratic convergence for Newton iteration from the information available at a point is also given.