**Classical Topics in Mathematics **

Volume: 6;
2017;
214 pp;
Hardcover

MSC: Primary 53; 58;
**Print ISBN: 978-7-04-047838-9
Product Code: CTM/6**

List Price: $59.00

Individual Member Price: $47.20

# The Bochner Technique in Differential Geometry

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*Hung-Hsi Wu*

A publication of Higher Education Press (Beijing)

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.

A publication of Higher Education Press (Beijing). Distributed in North America by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in differential geometry.