In this paper, we study the question of which compact manifolds admit a metric with positive scalar curvature. Scalar curvature is perhaps the weakest invariant among all the well-known invariants constructed from the curvature tensor. It measures the deviation of the Riemannian volume of the geodesic ball from the euclidean volume of the geodesic ball. As a result, it does not tell us much of the behavior of the geodesics in the manifold. Therefore it was remarkable that in 1963, Lichnerowicz [i] was able to prove the theorem that on a compact spin manifold with positive scalar curvature, there is no harmonic spinor. Applying

We give a complete classification of locally strongly convex affine hypersurfaces of Rn+1 with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric. It turns out that all such affine hypersurfaces are quadrics or can be obtained by applying repeatedly the Calabi product construction of hyperbolic affine hyperspheres, using as building blocks either the hyperboloid, or the standard immersion of one of the symmetric spaces SL(m,R)/SO(m), SL(m,C)/SU(m), SU 2m /Sp(m), or E6(−26)/F4.

We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive MV-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

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