On the Variation of a Metric and Its Application

Some of the variation formulas of a metric were derived in the literatures by using thelocal coordinates system. In this paper, We give the first and the second variation formulas of theRiemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using themoving frame method. We establish a relation between the variation of the volume of a metric andthat of a submanifold. We find that the latter is a consequence of the former. Finally we give anapplication of these formulas to the variations of heat invariants. We prove that a conformally flatmetric g is a critical point of the third heat invariant functional for a compact 4-dimensional manifoldM, then (M, g) is either scalar flat or a space form.