Betti numbers of 3-Sasakian manifolds
A 3-Sasakian manifold is a \((4n+3)\)-dimensional Riemannian manifold \((S,g)\) that admits three distinct Sasakian structures whose vector fields \(\xi^1\), \(\xi^2\), \(\xi^3\) are mutually orthogonal and satisfy a curvature condition. One says that \(S\) is regular if the vector fields \(\xi^i\) are complete and the corresponding 3-dimensional foliation is regular. (Then the space of leaves is a smooth \(4n\)-dimensional manifold \(M\) and the induced metric on \(M\) is quaternion-Kähler with positive scalar curvature.) In the paper under review, the authors prove a vanishing theorem and give constraints for the Betti numbers of compact 3-Sasakian manifolds. In particular, they prove that if \(S\) is a compact regular 3-Sasakian manifold of dimension \(4n+3\), then: (i) the Betti numbers of \(S\) satisfy \(\sum^n_{k=1} k(n+1-k)(n+1-2k)b_{2k}=0\), (ii) if \(b_4=0\) and \(n=3\) or 4, then \(S\) is the sphere or real projective space.
scientific article; zbMATH DE number 942739