【作者】 罗志坤；

【导师】 吴发恩；

【作者基本信息】 北京交通大学 ， 计算数学， 2010， 硕士

【摘要】 变分法是17世纪末发展起来的一门数学分支。其理论完整,在力学、物理学、光学、摩擦学、经济学、宇航理论、信息论和自动控制论等诸多方面有广泛的应用。我们可以看到变分法在经典微分几何中的重要作用。有些文献通过活动标架得到关于度量的黎曼曲率张量、里奇曲率张量以及数量曲率张量的第一、第二变分公式,从而得到度量的体积变分与子流形的体积变分之间的关系。有了这些公式,我们把这些公式应用于作用在1-形式上热不变量的变分。从而得出一个结论：对于4维紧致黎曼流形M,如果度量g是共形平坦并且是热核不变量的临界点,那么M是数量平坦或空间形式。更多还原

【Abstract】 The variational method is a branch of mathematics, and was developed in the 17th century. Its theory is complete and has extensive application in mechanics, physics, optics, tribological, economics, aerospace theory and automatic control theory, etc. We can also see the variational method plays an important role in classical differential geometry. Some literatures give the first and the second variation formulas of the Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using the moving frame method, so there is a relation between the variation of the volume of a metric and that of a submanifold. We give an application of these formulas to the variations of heat invariants which function on one-form. We prove that a conformally flat metric g is a critical point of the third heat invariant functional for a compact 4-dimensional manifold M, then (M, g) is either scalar flat or a space form.更多还原