Sario L., Nakai M., Wang C., Chung L.O. Classification Theory of Riemannian Manifolds

Springer, 1977. — 517 p.The purpose of the present research monograph is to systematically develop a classification theory of Riemannian manifolds toased on the existence or nonexistence of harmonic, quasiharmonic, and toiharmonic functions with various tooundedness properties. By definition, a function u is harmonic, quasiharmonic, or toiharmonic, according as Au = 0, Au = 1, or AAu = 0, where A signifies the Laplace-Beltrami operator -div grad = dô + Ôd. With two exceptions, all results presented herein are new in that they have not appeared in any other took. The exceptions are the classical definitions of A in Chapter 0 and the toasic harmonic inclusion relations 1,1.11 and 1,2.1.
This monograph, the result of an eight year research project, is offered to the mathematical public with some serious claims to Toeing a new theory, the object of which we have found exceptionally fascinating,
Six phases can toe distinguished in the historical development of classification theory:
Riemann" s mapping theorem.
The classical type problem.
Harmonic and analytic classification of Riemann surfaces.
Harmonic classification of Riemannian manifolds.
Biharmonic classification of Riemannian manifolds.
The 'biharmonic type problem.
We shall briefly discuss each of these six phases, the last three of which are the topic of the present took.