This monograph is a detailed survey of an area of differential geometry surrounding the Bochner technique. This is a technique that falls under the general heading of “curvature and topology” and refers to a method initiated by Salomon Bochner in the 1940s for proving that on compact Riemannian manifolds, certain objects of geometric interest (e.g., harmonic forms, harmonic spinor fields, etc.) must satisfy additional differential equations when appropriate curvature conditions are imposed.

In 1953 K. Kodaira applied this method to prove the vanishing theorem for harmonic forms with values in a holomorphic vector bundle. This theorem, which bears his name, was the crucial step that allowed him to prove his famous imbedding theorem. Subsequently, the Bochner technique has been extended, on the one hand, to spinor fields and harmonic maps and, on the other, to harmonic functions and harmonic maps on noncompact manifolds. The last has led to the proof of rigidity properties of certain Kähler manifolds and locally symmetric spaces.

This monograph gives a self-contained and coherent account of some of these developments, assuming the basic facts about Riemannian and Kähler geometry as well as the statement of the Hodge theorem. The brief introductions to the elementary portions of spinor geometry and harmonic maps may be especially useful to beginners.