The Bochner technique and modification of the Ricci tensor

Limoncu M.

ANNALS OF GLOBAL ANALYSIS AND GEOMETRY, vol.36, no.3, pp.285-291, 2009 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 36 Issue: 3
  • Publication Date: 2009
  • Doi Number: 10.1007/s10455-009-9163-y
  • Journal Name: ANNALS OF GLOBAL ANALYSIS AND GEOMETRY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.285-291
  • Keywords: Bochner technique, Modification of the Ricci tensor, Vanishing theorems, Eigenvalue estimate
  • Anadolu University Affiliated: Yes

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.