Group actions on quasi-symplectic manifolds
Let M be a quasi-symplectic manifold, this means that M is a closed differentiable manifold such that the tangent bundle of M is a direct sum of quaternionic tensor products. Assume that M admits a non-trivial differentiable action of the circle group which preserves the quasi- symplectic structure. In this case it is possible to compute the rotation numbers of the normal bundle of each connected component of the fixed point set from the rotation numbers of the quaternionic bundles given by the quasi-symplectic structure of M. This is used to deduce vanishing theorems for a set of characteristic numbers of M from a result which was proved in a joint paper with \textit{R. L. E. Schwarzenberger} [Proc. Camb. Philos. Soc. 73, 439-453 (1973; Zbl 0258.58013)]. These theorems are applied to compute the Pontryagin classes of cohomology quaternionic projective spaces which are quasi-symplectic and admit a differentiable non-trivial structure preserving action of the circle group.
scientific article; zbMATH DE number 4199064