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The tensor products of the exterior powers of X have Chi = 0 for a given i) =} (the tensor products of all 'l/Jk(X) have Chi = 0) =} (all homogeneous polynomials in chj of degree i equal zero) =} (all homogeneous polynomials in Cj of degree i equal zero).

Admitting no metrics with Sc > O. In fact, closed spin manifolds of dimensions 4k with A -=I- 0 do exist as was indicated earlier. ) The spinor power of Lichnerowicz' theorem cannot be matched by traditional devices of Riemannian geometry even if one strengthens the condition Sc > 0 to K > o. The simplest example where the theorem applies is the famous K 3 -surface, which is, topologically speaking, a 4-manifold V presented by a non-singular complex surface in CP3 of degree 4, say given by the following equation in the homogeneous coordinates in Cp3, This 4-manifold V is simply connected (by the Lefschetz theorem) and admits a (Kahler) metric with Ricci = 0 by Yau's solution to the Calabi conjecture.

Uniform contractibility A metric space V is called uniformly contractible if every ball Bv(R) in V is contractible inside some concentric ball Bv(p), p 2:: R, where p = peR) does not depend on x (but may depend on V). For instance, the Euclidean space is uniformly contractible with peR) = R. g. discrete) action of an isometry group with compact quotient is (obviously) uniformly contractible. On the other hand the (contractible) surface in Fig. 3 below is not uniformly contractible. ::.... ------,...

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