Sharp estimates and the Dirac operator (Q1377071)

The following statement is proved: assume that \(M\) is a compact Riemannian spin manifold of dimension \(n\) and assume that there is a distance decreasing map \(f: (M,g) \rightarrow (S^n, g_0)\) of non-zero degree onto the standard sphere of normalized scalar curvature 1. Then either there is a point on \(M\) with scalar curvature \(< 1\) or \(f\) is an isometry of the standard sphere. In the case that \(f\) is the identity map of the sphere this settles a conjecture of M. Gromov. The idea of the proof is to pull back a suitable spinor bundle on the sphere to \(M\) and to use it to twist a similar spinor bundle on \(M\). Then a Bochner type argument and the Atiyah-Singer index theorem are used.