A characterization of the (A)over-cap-genus as a linear combination of Pontrjagin numbers

摘要

We show in this short note that if a rational linear combination of Pontrjagin numbers vanishes on all simply-connected 4k-dimensional closed connected and oriented spin manifolds admitting a Riemannian metric whose Ricci curvature is nonnegative and not identically zero, then this linear combination must be a multiple of the (A) over cap -genus, which improves a result of Gromov and Lawson. Our proof combines an idea of Atiyah and Hirzebruch and the celebrated Calabi-Yau theorem.