Some Implications of the Generalized

1. Introduction. Perhaps the most significant aspect of differential geometry is that which deals with the relationship between the curvature properties of a Riemannian manifold M and its topological structure. One of the beautiful results in this connection is the (generalized) Gauss-Bonnet theorem which relates the curvature of compact and oriented even-dimensional manifolds with an important topological invariant, viz., the Euler-Poincar6 characteristic x(M) of M. In the 2-dimensional case, the sign of the Gaussian curvature determines the sign of x(M). Moreover, if the Gaussian curvature vanishes identically, so does #(M). In higher dimensions, the Gauss-Bonnet formula (cf. §3) is not so simple, and one is led to the following important Question. Does a compact and oriented Riemannian manifold of even dimension n = 2m whose sectional curvatures are all non-negative have non-negative Euler-Poincare characteristic, and ij the sectional curvatures are nonpositive is (-l)m*(M) = 0?