Some Differential Operators in Real and Complex Geometry

Jerzy Kalina and Antoni Pierzchalski
Additional contact information
Jerzy Kalina: Polish Academy of Sciences, Institute of Mathematics
Antoni Pierzchalski: University of Łódź, Institute of Mathematics

A chapter in Deformations of Mathematical Structures, 1989, pp 3-28 from Springer

Abstract: Abstract The famous Laplace-Beltrami operator △ acting on differential forms on a Riemannian manifold M determines in some sense the geometry of M. For example the Hodge decomposition theorem implies that in the compact case $$ X(M) = Trace {e^{{ - t{\Delta_{{even}}}}}} - Trace {e^{{ - t{\Delta_{{odd}}}}}} $$ where X(M) is the Euler characteristic of M and $$ {\Delta_{\text{even}}} = \Delta {\left| {_{p{even}}^{ \oplus }\Lambda P(M),{ }{\Delta_{\text{odd}}} = \Delta } \right|_p}_{\text{odd}}^{ \oplus }\Lambda P $$ Of course the theory of the operator in the complex case is much richer. We are going to give a short review of the theory of the Laplace-Beltrami operator on compact complex manifolds. In particular, the Hodge decomposition and its applications will be given. The case of a compact Kähler manifold will also be mentioned. Some other elliptic operators essentially connected with the geometry of M will be introduced. One of them is the so-called Ahlfors-Laplacian S*S acting on 1-forms. S is the Ahlfors’ operator which arises naturally in the theory of quasiconformal deformations of M. S*S is strongly influenced by the geometry of M. It behaves nicely both in the real and in the complex cases. Before passing to the operators some necessary information from the theory of real and complex geometry will be given.

Keywords: Riemannian Manifold; Complex Manifold; Cohomology Class; Real Vector Space; Complex Vector Space (search for similar items in EconPapers)
Date: 1989
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-2643-1_1

Ordering information: This item can be ordered from
http://www.springer.com/9789400926431

DOI: 10.1007/978-94-009-2643-1_1

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().