The Bochner Technique
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Aside from the variational techniques we’ve used in prior sections one of the oldest and most important techniques in modern Riemannian geometry is that of the Bochner technique. In this chapter we prove the classical theorem of Bochner about obstructions to the existence of harmonic 1-forms. We also explain in detail how the Bochner technique extends to forms and other tensors by using Lichnerowicz Laplacians. This leads to a classification of compact manifolds with nonnegative curvature operator in chapter 10 To establish the relevant Bochner formula for forms, we have used a somewhat forgotten approach by Poor. It appears to be quite simple and intuitive. It can, as we shall see, also be generalized to work on other tensors including the curvature tensor.
KeywordsBochner Technique Lichnerowicz Laplacian Nonnegative Curvature Operator Bochner Formula Forgotten Approach
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