The Bochner Technique in Differential Geometry

Generally speaking, the Bochner-Technique is a method to relate the Laplace operator of a Riemannian manifold to its curvature tensor. It is often used to derive topological consequences from curvature conditions through analysis.

 

This book appeared originally in 1988, and the new edition, under review here, is slightly expanded from the first. Despite its specialized title, this book should appeal not only to researchers in the subject but also to graduate students who want to learn the basic computational techniques and main results in geometric analysis or complex differential geometry. 

 

The first chapter contains a lucid introduction to coordinate free computations using normal frames, i.e. locally defined vector fields that are parallel at a chosen point. Wu treats the case of real manifolds and that of Kähler manifolds in parallel, making the significance of the Kähler condition very clear. This method is used throughout the book.

 

The second chapter proves the basic Bochner formulas, again both in the real and complex case in parallel, thus giving a perfect illustration of the usefulness of normal frames in computations. 

 

The following two chapters give examples of applications for compact and non-compact manifolds, beginning with the most classical results and ending with more complicated applications. Each section explains a result, its background and significance, and gives detailed proofs. One highlight is a very transparent proof of Meyer’s homology sphere theorem. Whatever cannot be proven in the scope of the book is extensively explained and referenced (for instance, results from Hodge theory are used throughout).

 

The first two chapters together with a selection of topics from chapters 3 and 4 are an ideal source for parts of a second semester course in Riemannian geometry or a graduate student seminar, providing an analytic contrast to the more geometric Jacobi field approach to classical theorems involving curvature and topology.

 

Chapter 5 explains the proof of Lichnerowicz’s theorem that the  Â-genus of a compact spin manifold with positive scalar curvature vanishes. Again, everything surrounding the use of the Bochner formula is proven in detail, including the background on Clifford algebras, and the required topology is explained. This material is now also available elsewhere (e.g. Roe’s book on Elliptic Operators), but the exposition here is still worth reading.

 

Chapter 6 turns to harmonic maps between Riemannian manifolds. This longer chapter builds up slowly, beginning with the foundations of harmonic maps, stating the basic existence results (like the theorem of Eells and Sampson), first applications of the Bochner formula for harmonic maps, and then culminates in a proof of Siu’s Strong Rigidity Theorem for strong negatively curved compact Kähler manifolds. 

 

Since the first edition, the Bochner technique has seen further applications, some of which are surveyed in this new edition.  Most notably, this includes the role of the Bochner technique in Margulis’ Strong Rigidity Theorem.

 

The clarity of the book is exceptional, both in the computational parts that provide detailed proofs of everything Bochner, and in the expositional parts that explain the background.

---

 

Matthias Weber is a Professor of Mathematics at Indiana University. He received his PhD at the University of Bonn in 1993. His diplom thesis from 1989 about Siu’s Strong Rigidity Theorem builds upon chapter 6 of Wu’s book.