Cross Curvature Flow on Locally Homogenous Three-manifolds (i)
نویسنده
چکیده
Chow and Hamilton introduced the cross curvature flow on closed 3manifolds with negative or positive sectional curvature. In this paper, we study the negative cross curvature flow in the case of locally homogenous metrics on 3manifolds. In each case, we describe the long time behavior of the solutions of the corresponding ODE system.
منابع مشابه
Cross Curvature Flow on Locally Homogeneous Three-manifolds (ii)
In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cross curvature flow on locally homogeneous 3-manifolds. We show that, typically...
متن کاملBackward Ricci Flow on Locally Homogeneous Three-manifolds
In this paper, we study the backward Ricci flow on locally homogeneous 3-manifolds. We describe the long time behavior and show that, typically and after a proper re-scaling, there is convergence to a sub-Riemannian geometry. A similar behavior was observed by the authors in the case of the cross curvature flow.
متن کاملThe Backward Behavior of the Ricci and Cross Curvature Flows on Sl(2,r)
This paper is concerned with properties of maximal solutions of the Ricci and cross curvature flows on locally homogeneous three-manifolds of type SL2(R). We prove that, generically, a maximal solution originates at a sub-Riemannian geometry of Heisenberg type. This solves a problem left open in earlier work by two of the authors.
متن کاملCommutative curvature operators over four-dimensional generalized symmetric spaces
Commutative properties of four-dimensional generalized symmetric pseudo-Riemannian manifolds were considered. Specially, in this paper, we studied Skew-Tsankov and Jacobi-Tsankov conditions in 4-dimensional pseudo-Riemannian generalized symmetric manifolds.
متن کاملConformal mappings preserving the Einstein tensor of Weyl manifolds
In this paper, we obtain a necessary and sufficient condition for a conformal mapping between two Weyl manifolds to preserve Einstein tensor. Then we prove that some basic curvature tensors of $W_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. Also, we obtained the relation between the scalar curvatures of the Weyl manifolds r...
متن کامل