The notion of finite type submanifolds was introduced by B. Y. Chen. In this paper the conjectures on scalar curvature of Veronese generating submanifolds in E~σ and the minimal conjecture on Veronese space-like subm...The notion of finite type submanifolds was introduced by B. Y. Chen. In this paper the conjectures on scalar curvature of Veronese generating submanifolds in E~σ and the minimal conjecture on Veronese space-like submanifold Σ and Veronese pseudo-Riemannian submanifold in E_1~σ are proved. We have Σ is minimal in H^5. is minimal in S_1~5, Σ and are of 1-type in E_1~σ.展开更多
A Riemannian manifold （M, g） is called Einstein manifold if its Ricci tensor satisfies r=c.g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifo...A Riemannian manifold （M, g） is called Einstein manifold if its Ricci tensor satisfies r=c.g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.展开更多
The author considers the hyperbolic geometric flow introduced by Kong and Liu. Using the techniques and ideas to deal with the evolution equations along the Ricci flow by Brendle, the author derives the global forms...The author considers the hyperbolic geometric flow introduced by Kong and Liu. Using the techniques and ideas to deal with the evolution equations along the Ricci flow by Brendle, the author derives the global forms of evolution equations for Levi-Civita connection and curvature tensors under the hyperbolic geometric flow. In addition, similarly to the Ricci flow case, it is shown that any solution to tile hyperbolic geometric flow that develops a singularity in finite time has unbounded Ricci curvature.展开更多
文摘The notion of finite type submanifolds was introduced by B. Y. Chen. In this paper the conjectures on scalar curvature of Veronese generating submanifolds in E~σ and the minimal conjecture on Veronese space-like submanifold Σ and Veronese pseudo-Riemannian submanifold in E_1~σ are proved. We have Σ is minimal in H^5. is minimal in S_1~5, Σ and are of 1-type in E_1~σ.
基金Project supported by the National Natural Science Foundation of China (No. 10431040, No.10501025) and the Liu Hui Center for Applied Mathematics.
文摘A Riemannian manifold （M, g） is called Einstein manifold if its Ricci tensor satisfies r=c.g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.
文摘The author considers the hyperbolic geometric flow introduced by Kong and Liu. Using the techniques and ideas to deal with the evolution equations along the Ricci flow by Brendle, the author derives the global forms of evolution equations for Levi-Civita connection and curvature tensors under the hyperbolic geometric flow. In addition, similarly to the Ricci flow case, it is shown that any solution to tile hyperbolic geometric flow that develops a singularity in finite time has unbounded Ricci curvature.
