微分几何中的Bochner技术 英文版
作者：伍鸿熙 著
出版时间： 2017年版
内容简介
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 1940's for proving on compact Riemannian manifolds that certain objects of geometric interest (e.g. harmonic forms, harmonic spinor fields, etc.) must satisfy additional differential equations when appropriate curvatureconditions are imposed. In 1953, K. Kodaira applied this method to prove the vanishing theorem that now bears his name for harmonic forms with values in a holomorphic vector bundle; this 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 andharmonic 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. Thebrief introductions to the elementary portions of spinor geometry and harmonic maps may be especially useful to beginners.
Bochner 技术是数学中经典和有效的技术，可以用来证明数学中非常重要的消失定理和刚性性质。伍鸿熙教授著书众多，写作经验丰富，本书是di一次系统介绍Bochner技术及其应用的著作。
目录
1 Coordinates and Frames Normal at a Point
2 The Weitzenbock Formulas
3 Some Results in the Compact Case
4 Some Results in the Noncompact Case
5 Harmonic Spinor Fields
5.1 Algebra
5.2 Topology
5.3 Geometry
6 Harmonic Mapping
6.1 Riemannian vector bundles
6.2 The definition of a harmonic map and the first consequences .
6.3 Existence results
6.4 First applications of the Bochner technique
6.5 Strong rigidity theorems
6.6 Miscellaneous remarks
References
Appendix: Vector Fields and Pdcci Curvature by S. Bochner
A.1 Real spaces
A.2 Hermitian metric
A.3 Complex spaces
Index
