The Bochner technique and modification of the Ricci tensor
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In the well-known vanishing theorems of Bochner, the assumptions Ric ≥ 0 and Ric ≤ 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 ≥ 0 and Ric ≤ 0. We prove that the assertions and results of Bochner’s vanishing theorems still hold under these modified assumptions. Additionally, the assumption Ric ≥ 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.
KeywordsBochner technique Modification of the Ricci tensor Vanishing theorems Eigenvalue estimate
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