Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExamples of signature (2,2) manifolds with commuting curvature operators
183
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A.
rate research
Read More
We use the modified Riemannian extension of an affine surface to construct Bach flat manifolds. As all these examples are VSI (vanishing scalar invariants), we shall construct scalar invariants which are not of Weyl type to distinguish them. We illustrate this phenomena in the context of homogeneous affine surfaces.
We construct $Q$-curvature operators on $d$-closed $(1,1)$-forms and on $overline{partial}_b$-closed $(0,1)$-forms on five-dimensional pseudohermitian manifolds. These closely related operators give rise to a new formula for the scalar $Q$-curvature. As applications, we give a cohomological characterization of CR five-manifolds which admit a $Q$-flat contact form; and we show that every closed, strictly pseudoconvex CR five-manifold with trivial first real Chern class admits a $Q$-flat contact form provided the $Q$-curvature operator on $overline{partial}_b$-closed $(0,1)$-forms is nonnegative.
In this note we prove that a four-dimensional compact oriented half-confor-mally flat Riemannian manifold $M^4$ is topologically $mathbb{S}^{4}$ or $mathbb{C}mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval $[frac{3sqrt{3}-5}{4},,1].$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the $4$-sphere.
In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.
A connected Riemannian manifold M has constant vector curvature epsilon, denoted by cvc(epsilon), if every tangent vector v in TM lies in a 2-plane with sectional curvature epsilon. By scaling the metric on M, we can always assume that epsilon = -1, 0, or 1. When the sectional curvatures satisfy the additional bound that each sectional curvature is less than or equal to epsilon, or that each sectional curvature is greater than or equal to epsilon, we say that, epsilon, is an extremal curvature. In this paper we study three-manifolds with constant vector curvature. Our main results show that finite volume cvc(epsilon) three-manifolds with extremal curvature epsilon are locally homogenous when epsilon=-1 and admit a local product decomposition when epsilon=0. As an application, we deduce a hyperbolic rank-rigidity theorem.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
