The Bochner Technique

Abstract

One of the oldest and most important techniques in modern Riemannian geometry is that of the Bochner technique. In this chapter we shall prove some of the classical theorems Bochner proved about obstructions to the existence of Killing fields and harmonic 1-forms. We also explain how the Bochner technique extends to forms. This will in the next chapter lead us to a classification of compact manifolds with nonnegative curvature operator. To establish the relevant Bochner formula for forms, we have used the language of Clifford multiplication. It is, in our opinion, much easier to work consistently with Clifford multiplication, rather than trying to keep track of wedge products, interior products, Hodge star operators, exterior derivatives, and their dual counterparts. In addition, it has the effect of preparing one for the world of spinors, although we won’t go into this here. In the last section we give a totally different application of the Bochner technique. In effect, we try to apply it to the curvature tensor itself. The outcome will be used in the next chapter, where manifolds with nonnegative curvature operator will be classified. The Bochner technique on spinors is only briefly mentioned in this chapter, but Appendix C is devoted to this subject.