The Bochner technique and modification of the Ricci tensor

Abstract

In the well-known vanishing theorems of Bochner, the assumptions Ric a parts per thousand yen 0 and Ric a parts per thousand currency sign 0 are modified by using Hessian and Laplacian of a smooth positive function such that, when this function is constant, these assumptions return to Ric a parts per thousand yen 0 and Ric a parts per thousand currency sign 0. We prove that the assertions and results of Bochner's vanishing theorems still hold under these modified assumptions. Additionally, the assumption Ric a parts per thousand yen H > 0 given in the eigenvalue estimate theorem of Lichnerowicz is also modified in the same way, and we obtain estimates for the first positive eigenvalue of the Laplace operator.