The paper discusses the relations between parallel vector fields and Jacobi fields in a complete Riemannian manifold, and proves that there exists infinite closed geodesics in a local symmetric space which is orientable, compact, nonsimply connected and with nonnegative sectional curvature.

In the paper, we discuss the geometry of a complete Riemannian manifold by the concepts and theorems of Jacobi field、Ranch*s Comparison Theorem and Soul etc.

By using Jacobi field and index form, the author prives that a complete Riemannian manifold M2 of dimension 2 with nonnegative curvature must be isometric to R2, if M2 has no conjugate point.

Some Jacobians of the Transformation between Random Vectors and Matrices;

This involves computing the Jacobians of these transformations.

Then it can get a kind of model that has concentration weight and calculates the frequency with JACOBI method.

Weighted L_p~ω approximation by modified Bernstein-Durrmeyer operator with Jacobi weight;

Using the relation between the weighted modulus of smoothness and the weighted main-part modulus of smoothness,we discuss the pointwise direct and equivalent approximation theorem with Jacobi weight for Beta operator.

Using the moduli of smoothnessω_φ~Tλ(f,t)_ω,direct and inverse approximation theorems with Jacobi weight for combinations of Baskakov operators is established in the paper, and the relation between higher derivatives of the operators and the smoothness of functions to be approximated is obtained in the paper.