  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 nonzero at any point, then this linear combination
must be a multiple of the $\hat{A}$-genus, which improves on a result of Gromov
and Lawson. Our proof combines an idea of Atiyah and Hirzebruch and the
celebrated Calabi-Yau theorem.
