Odd Weili Man Manifold Of Closed Curves, Morse Index Estimates

Considering geodesics in Riemannian manifold M~n, if C is the non-degenerate critical point of energy function E, we define the Morse index of C as the maximal dimension of all subspaces on which the quadratic form associated to HessianE is negative definite. In geometry, on s in pay much attention to the estimates of the Morse index.An important property of Morse index is Morse index theorem. It tells us in Riemannian manifold, the Morse index of geodesic C equals to the numbers of the conjugate points of C(0), each counted with its multiplicity.If we want to estimate Morse index, a natural idea is to find out the linearly independent vector fields that make HessianE to be negative, we study the properties of these vector fields and obtain some knowledge of Morse index. In this paper, we consider some odd dimensional Riemannian manifold satisfies this curvature condition, in these manifolds we study some special curves-closed geodesics. After study the holonomy angles of such closed geodesic, we get a family of linearly independent vector fields which makes HessianE to be negative under such curvature condition. We obtain a relation among the length of closed geodesic, holonomy angle and its Morse index . Moreover, because of Morse index theorem, we know the relation between the length of closed geodesic and holonomy angle how to reflect the numbers of the conjugate points on the closed geodesies.