On The Variation Of A Metric And Its Application In Riemannian Geometry

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.