Let （M,g（t）） be a compact Riemannian manifold and the metric g（t） evolve by the Ricci flow. We derive the evolution equation for the eigenvalues of geometric operator --△φ＋ cR under the Ricci flow and the normalized Ricci flow, where A, is the Witten-Laplacian operator, φ∈C∞（M）, and R is the scalar curvature with respect to the metric g（t）. As an application, we prove that the eigenvalues of the geometric operator are nondecreasing along the Ricci flow coupled to a heat equation for manifold M with some Ricci curvature 1 condition when c 〉1/4.
