A characterization of the Â-genus as a linear combination of Pontrjagin numbers
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 Â-genus, which improves a result of Gromov and Lawson. Our proof combines an idea of Atiyah and Hirzebruch and the celebrated Calabi–Yau theorem.
KeywordsÂ-genus Spin manifold Ricci curvature Positive scalar curvature Pontrjagin number
Mathematics Subject Classification53C21 57R20 53C23 57R75
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