单位球丛上L(g)泛函变分问题

The Variational Problem of L(g) on Unit Tangent Sphere Bundles

摘要: 本文研究了紧切触度量流形(M,η,g)上的L(g)泛函，该泛函是对Reeb向量场方向的里奇曲率在切触度量流形上的积分。特别地，我们考虑了紧黎曼流形的单位球丛这一特殊的切触度量流形。首先计算了L(g)泛函在球丛的底流形上的泛函形式。然后通过计算L泛函在底流形上的变分，我们发现当底流形是二维时，具有平坦性的黎曼度量是L泛函在底流形上的临界点。

Abstract: This paper investigates theL(g)functional on contact metric manifold(M,η,g), which is the integral of the Ricci curvature along the Reeb vector field direction on the contact metric manifold. In particular, we consider the special case of the unit Tangent Sphere Bundles of a Riemannian manifold as a contact metric manifold. Firstly, we compute the functional form ofL(g)functional on the base manifold of the Tangent Sphere Bundles. Then, by computing the variation of the L functional on the base manifold, we find that Riemannian metrics with flatness are critical points of the L functional on the base manifold when the base manifold is two-dimensional.