It is important to study asymptotic behavior of complete manifold without the assumption of pointwise Ricci curvature bound. A volume growth and curvature decay result was obtained in [4] for various classes of complete, noncompact, Bach-flat metrics in dimension 4. Some similar results were also claimed in [1]. In this note we consider a more general case, that is, the Bach tensor may not nece...

A. FollowingGeroch, Traschen,Mars andSenovilla,we considerLorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.e., coordinate-free) framework. First, we determine the minimal assumpti...

We introduce an evolution equation which deforms metrics on 3-manifolds with sectional curvature of one sign. Given a closed 3-manifold with an initial metric with negative sectional curvature, we conjecture that this flow will exist for all time and converge to a hyperbolic metric after a normalization. We shall establish a monotonicity formula in support of this conjecture. Note that in contr...

In this paper, some properties of m−projective curvature tensor in Kenmotsu manifolds are studied.

This paper presents a novel approach towards two-dimensional (2D) image structures modeling. To obtain more degrees of freedom, a 2D image signal is embedded into a certain geometric algebra. Coupling methods of differential geometry, tensor algebra, monogenic signal and quadrature filter, we can design a general model for 2D structures as the monogenic extension of a curvature tensor. Based on...

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 the...

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor Ag associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σk(Ag), 1 ≤ k ≤ n} of the eigenvalues of Ag with respect to g; we call σk(Ag) the k-th Schouten curva...

We introduce a new notion of a curvature of a superconnection, different from the one obtained by a purely algebraic analogy with the curvature of a linear connection. The naturalness of this new notion of a curvature of a superconnection comes from the study of the singularities of smooth sections of vector bundles (Catastrophe Theory). We demonstrate that the classical examples of obstruction...