奇数维黎曼流形中闭曲线的Morse指标估计

【作者】 张弘；

【导师】 李庆忠；

【作者基本信息】 首都师范大学 ， 基础数学， 2006， 硕士

【摘要】 考虑黎曼流形M~n中的测地线C，若C是能量函数的E的临界点，对于E的非退化临界点C，我们把HessianE的最大的负定子空间的维数称为测地线C的Morse指标。在几何学中，我们十分关注Morse指标的估计。 Morse指标的一个重要的性质就是Morse指标定理。它告诉我们黎曼流形中的测地C的Morse指标等于C上的共轭点的个数(个数按重数计算)。 如果我们要去估计Morse指标，最自然的方法就是找出使Hessian为负的线性无关的向量场，然后通过研究向量场的性质去刻画Morse指标。本文中，我们考虑满足某些曲率条件的奇数维黎曼流形中特殊的曲线——闭测地线，通过研究闭测地线的完整角，用构造的方法给出了一族在这种曲率条件下使HessianE为负的特殊的线性无关的向量场，从而给出了闭测地线长度，完整角和它的Morse指标之间的关系，给出了通过闭测地线长度去估计Morse指标的方法，然后通过Morse指标定理，说明了在奇数维黎曼流形中闭测地线长度与完整角的关系对闭测地线上共轭点个数的影响。更多还原

【Abstract】 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.更多还原