On Curvature and Characteristic Classes of a Riemann Manifold

Abstract

The development in recent years of the theory of fiber bundles in differential geometry has led to relations between the curvature of a compact RIwMANN manifold and certain global invariants, the so-called characteristic classes. The simplest among these relations is the GAussBO~N~,T formula1), which expresses the EULER-PoI~c~a~ characteristic of a compact orientable I~I~M~N manifold as an integral involving the components of the I~IEMA_,NTN-CHttISTOFFEL curvature tensor. There are other relations, to be given below, between the PON~AGIN characteristic classes and differential forms constructed from the curvature tensor. The purpose of our paper is twofold: To express these relations in a convenient form and to draw some conclusions from them. The differential forms which express the EULE~-PoI~cA~ characteristic and the PO~TRZAGI~ classes bear a close relationship with skew-symmetric determinants and P F ~ T I ~ functions. A theorem concerning them, which we will find useful, is given in w 2. In w 3 we shall introduce notions by which our relations can be best formulated. In w 4 we derive some implications of the curvature properties on the characteristic classes. In particular, we give a proof of a theorem of J. MIT,NOR ~) tO the effect tha t an orientable compact tCI~,M~_~N manifold of four dimensions has positive EULE~-PoINCAR~ characteristic, if its sectional curvature is always positive or always negative. Among the implications of these relations is the fact that they provide a way for effective computation of the characteristic classes. For instance, it will be easy to compute the PO~TVZAGI~ classes of the complex projective space, using its elliptic t termit ian metric. Although we will not carry this out, we shall make an application of the knowledge of these classes by deriving the following theorem: The complex projective space P~ of (complex) dimension m cannot be differentiably imbedded in a real Euclidean space of dimension 3m ~ 1 or 3m ~ 2, according as m is even or odd.