Curvature in Hermitian metric

Abstract

Hermitian metric has the peculiarity of favoring negative curvature over positive curvature. We shall explain this phenomenon by pointing out that in the case of an isometric analytic imbedding the relative curvature is on the whole negative; also, by reduction to a limiting case of imbedding we shall explain why an invariant metric in the theory of Fuchsian groups is likely to be hyperbolic; see Hua []. 1 However, on the other hand, an Hermitian metric is very rigid, and the possibility of imbedding into a finitely-dimensional enveloping space is very remote. The classical conjectures about the possibility of Euclidean imbedding are rendered entirely false, but as a compensation, there are more and better theorems about equivalence and uniqueness.