Characteristic Classes of Hermitian Manifolds

In recent years the works of Stiefel,1 Whitney,2 Pontrjagin,3 Steenrod,4 Feldbau,5 Ehresmann,6 etc. have added considerably to our knowledge of the topology of manifolds with a differentiable structure, by introducing the notion of so-called fibre bundles. The topological invariants thus introduced on a manifold, called the characteristic cohomology classes, are to a certain extent susceptible of characterization, at least in the case of Riemannian manifolds,7 by means of the local geometry. Of these characterizations the generalized Gauss-Bonnet formula of Allendoerfer-Weil8 is probably the most notable example. In the works quoted above, special emphasis has been laid on the sphere bundles, because they are the fibre bundles which arise naturally from manifolds with a differentiable structure. Of equal importance are the manifolds with a complex analytic structure which play an important role in the theory of analytic functions of several complex variables and in algebraic geometry. The present paper will be devoted to a study of the fibre bundles of the complex tangent vectors of complex manifolds and their characteristic classes in the sense of Pontrjagin. It will be shown that there are certain basic classes from which all the other characteristic classes can be obtained by operations of the cohomology ring. These basic classes are then identified with the classes obtained by generalizing Stiefel-Whitney's classes to complex vectors. In the sense of de Rham the cohomology classes can be expressed by exact exterior differential forms which are everywhere regular on the (real) manifold. It is then shown that, in case the manifold carries an Hermitian metric, these differential forms can be constructed from the metric in a simple way. This means that the characteristic classes are completely determined by the local structure of the Ilermitian metric. This result also includes the formula of Allendoerfer-Weil and can be regarded as a generalization of that formula. Concerning the relations between the characteristic classes of a complex manifold and an Hermitian metric defined on it, the problem is completely solved by the above results. It is to be remarked that corresponding questions for Rie-