Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHomogeneity of Lorentzian three-manifolds with recurrent curvature
163
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
k-Curvature homogeneous three-dimensional Walker metrics are described for k=0,1,2. This allows a complete description of locally homogeneous three-dimensional Walker metrics, showing that there exist exactly three isometry classes of such manifolds. As an application one obtains a complete description of all locally homogeneous Lorentzian manifolds with recurrent curvature. Moreover, potential functions are constructed in all the locally homogeneous manifolds resulting in steady gradient Ricci and Cotton solitons.
rate research
Read More
A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the behavior of Gaussian curvature at a non-degenerate lightlike point of a mixed type surface. To characterize the boundedness of Gaussian curvature at a non-degenerate lightlike points, we introduce several fundamental invariants along non-degenerate lightlike points, such as the lightlike singular curvature and the lightlike normal curvature. Moreover, using the results by Pelletier and Steller, we obtain the Gauss-Bonnet type formula for mixed type surfaces with bounded Gaussian curvature.
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.
We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanzurova are genuine generalizations of the ordinary notion of k-curvature homogeneity. The homothety group plays an essential role in the analysis. We give a complete classification of homothety homogeneous manifolds which are not homogeneous and which are not VSI and show that such manifolds are cohomogeneity one. We also give a complete description of the local geometry if the homothety character defines a split extension.
We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanv{z}urova are genuine generalizations of the ordinary notion of $k$-curvature homogeneity. The homothety group plays an essential role in the analysis.
For a homotopically energy-minimizing map $u: N^3to S^1$ on a compact, oriented $3$-manifold $N$ with boundary, we establish an identity relating the average Euler characteristic of the level sets $u^{-1}{theta}$ to the scalar curvature of $N$ and the mean curvature of the boundary $partial N$. As an application, we obtain some natural geometric estimates for the Thurston norm on $3$-manifolds with boundary, generalizing results of Kronheimer-Mrowka and the second named author from the closed setting. By combining these techniques with results from minimal surface theory, we obtain moreover a characterization of the Thurston norm via scalar curvature and the harmonic norm for general closed, oriented three-manifolds, extending Kronheimer and Mrowkas characterization for irreducible manifolds to arbitrary topologies.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
