In this paper first it is proved that if ξ is a nontrivial closed conformal vector field on an n-dimensional compact Riemannian manifold (M, g) with constant scalar curvature S satisfying S ≤ λ1(n − 1), λ1 being first nonzero eigenvalue of the Laplacian operator ∆ on M and Ricci curvature in direction of a certain vector field is non-negative, then M is isometric to the n-sphere S(c), where S =...

We estimate the lower bound of the first non-zero eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature in terms of the diameter and the lower bound of Ricci curvature and give an affirmative answer to the conjecture of H. C. Yang.

We prove in the Tucker-Wang approach to non-Riemannian Gravity that a general homogeneous Lagrangian density in the general connection with order of homogeneity of at least two gives no contribution to the generalised Einstein equations. Using this result other important cases are also considered. 04.20.-q, 04.40.-b, 04.50.+h, 04.62.+v [email protected]

Gromov-Hausdorff convergence is an important tool in comparison Riemannian geometry. Given a sequence of Riemannian manifolds of dimension n with Ricci curvature bounded from below, Gromov’s precompactness theorem says that a subsequence will converge in the pointed Gromov-Hausdorff topology to a length space [G-99, Section 5A]. If the sequence has bounded sectional curvature, then the limit wi...

We introduce a Riemannian metric with non positive curvature in the (infinite dimensional) manifold Σ∞ of positive invertible operators of a Hilbert space H, which are scalar perturbations of Hilbert-Schmidt operators. The (minimal) geodesics and the geodesic distance are computed. It is shown that this metric, which is complete, generalizes the well known non positive metric for positive defin...

Let M be a differentiable manifold. We say that a tensor field g defined on M is non-regular if g is in some local L space or if g is continuous. In this work we define a mollifier smoothing gε of g that has the following feature: If g is a Riemannian metric of class C, then the Levi-Civita connection and the Riemannian curvature tensor of gε converges to the Levi-Civita connection and to the R...