本项目将黎曼几何、李群和李代数与数值分析、数值优化问题有机的统一起来，充分利用黎曼几何的内在性质及线性空间中已有的关于Newton法收敛性的研究结果及黎曼流形上依赖于测地线和平行移动的Newton法收敛性分析的已有结果，对黎曼流形上的基于回拉和向量移动的Newton法的局部和半局部的收敛性进行分析和研究。本项目将研究黎曼流形上基于回拉和向量移动的Newton法及奇异情形下的Newton法的收敛半径的估计，收敛判据和Smale点估计理论；研究李群上基于回拉和向量移动的Newton法的收敛性；最后将运用我们的研究结果解决一些具体的实际问题如特征值问题、脊椎问题、模式识别问题等。本项目是属于黎曼几何、李群和李代数、数值分析、数值计算、优化理论等多个分支的交叉学科，无论在理论上还是在应用前景上都有重要的研究价值和学术意义。

In this project, we will combine Riemannian geometry, Lie group and Lie algebra, numerical analysis and optimization theory, and then study the problems about the Newton method on Riemannian manifolds, which depends on retraction and vector transport. Making use of known convergence analysis results about the Newton method in linear spaces and manifolds which is relied on geodesics and parallel transport, we will give the estimations of convergence radius of the Newton method depended on retraction and vector transport,provide the uniform convergence criteria of the method, and the Samle's point estimation theory and so on. We will also study the Newton method for singular vector field on manifolds. We try to give the local and semi-local convergence analysis of various Newton methods on Lie groups which depends on retarction and vector transport, and then establish the well-known Smale point estimate theory and so on. At the end, we will use our results to solve some concrete problems such as eigenvalue problems, humane spine problems, pose-estimation problems. Our project is a combination of Riemannian geometry, Lie group and Lie algebra, numerical analysis, optimization theory. Hence, in view of application and theoretical development, our project is very meaningful and valuable.