Riemann curvature tensor

About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.

Physics of Finance

Abstract: We give a brief introduction to the Gauge Theory of Arbitrage. Treating a calculation of Net Present Values (NPV) and currencies exchanges as a parallel transport in some fibre bundle, we give geometrical interpretation of the interest rate, exchange rates and prices of securities as a proper connection components. This allows us to map the theory of capital market onto the theory of quantized gauge field interacted with a money flow field. The gauge transformations of the matter field correspond to a dilatation of security units which effect is eliminated by a gauge transformation of the connection. The curvature tensor for the connection consists of the excess returns to the risk-free interest rate for the local arbitrage operation. Free quantum gauge theory is equivalent to the assumption about the log-normal walks of assets prices. In general case the consideration maps the capital market onto lattice QED.

A Curvature-Tensor-Based Perceptual Quality Metric for 3D Triangular Meshes

Abstract: Perceptual quality assessment of 3D triangular meshes is crucial for a variety of applications. In this paper, we present a new objective metric for assessing the visual difference between a reference triangular mesh and its distorted version produced by lossy operations such as noise addition, simplification, compression and watermarking. The proposed metric is based on the measurement of a distance between curvature tensors of the two meshes under comparison. Our algorithm uses not only tensor eigenvalues (i.e., curvature amplitudes) but also tensor eigenvectors (i.e., principal curvature directions) to derive a perceptually-oriented tensor distance. The proposed metric also accounts for the visual masking effect of the human visual system, through a roughness-based weighting of the local tensor distance. A final score that reflects the visual difference between two meshes is obtained via a Minkowski pooling of the weighted local tensor distances over the mesh surface. We validate the performance of our algorithm on four subjectively-rated mesh visual quality databases, and compare the proposed method with state-of-the-art objective metrics. Experimental results show that our approach achieves high correlation between objective scores and subjective assessments.

Prescribing the behaviour of geodesics in negative curvature

Abstract: Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon described by Hall in special arithmetic situations and by Schmidt-Sheingorn for hyperbolic surfaces is in fact only a negative curvature property.