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.

Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems

Abstract: where dA -F, on the space of closed curves on the manifold Mn. Here A is a 1-form (i.e., F is an exact 2-form). This functional is a natural generalization of the usual functional of length, and its closed extremals correspond to periodic trajectories of the motion of particles on the Riemannian manifold Mn when the kinetic energy is defined by the metric tensor and the form F defines a magnetic field. Also this functional corresponds to the periodic orbits for other problems of classical mechanics and mathematical physics, as it was shown in [N2], [N3], [NS]. When the Lagrangian function

Statistics on Multivariate Normal Distributions: A Geometric Approach and its Application to Diffusion Tensor MRI

Abstract: This report is dedicated to the statistical analysis of the space of multivariate normal probability density functions. It relies on the differential geometrical properties of the underlying parameter space, endowed with a Riemannian metric, as well as on recent works that led to the generalization of the normal law on non-linear spaces. We will first proceed to the state of the art in section 1, while expressing some quantities related to the structure of the manifold of interest, and then focus on the derivation of closed-form expressions for the mean, covariance matrix, modes of variation and normal law between multivariate normal distributions in section 2. We will also address the derivation of accurate and efficient numerical schemes to estimate the proposed quantities. A major application of the present work is the statistical analysis of diffusion tensor Magnetic Resonance Imaging. We show promising results on synthetic and real data in section 3

Metrics of constant scalar curvature conformal to Riemannian products

Abstract: We consider the conformal class of the Riemannian product go+g, where go is the constant curvature metric on S m and g is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect to the square root of the scalar curvature of g. This is obtained by studying radial solutions of the equation Δu ― λu + λu p = 0 on S m and the number of solutions in terms of λ.

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

Abstract: In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete non-compact complex two dimensional Kahler manifold M of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have maximal volume growth, then M is biholomorphic to C 2. This gives a partial affirmative answer to the well-known conjecture of Yau [41] on uniformization theorem. During the proof, we also verify an interesting gap phenomenon, predicted by Yau in [42], which says that a Kahler manifold as above automatically has quadratic curvature decay at infinity in the average sense.