Group actions on quasi-symplectic manifolds

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The purpose of this paper is to prove a vanishing theorem for characteristic numbers of quasi-symplectic manifolds, which admit a differentiable structure preserving action of the circle group S ~. Here a dosed differentiable manifold is called a quasi-symplectic manifold when the tangent bundle is a direct sum of quaternionic tensor products. Examples of this class of manifolds are the quaternionic flag manifolds, and particularly the quaternionic projective spaces. For an S 1_action which preserves a quasi-symplectic structure it is easy to compute the rotation numbers of the normal bundle over each connected component of the fixed point set from the rotation numbers of the quaternionic bundles naturally given by the quasi-symplectic structure. This is used to deduce a vanishing theorem for characteristic numbers from a result which was proved in a common paper with R. Schwarzenberger [11], and which is a consequence of the AtiyahSinger fixed point theorem. In Sect. 2 we recall a result from [11]. In Sect. 3 we prove the announced theorem. Sect. 4 contains applications to the Pontryagin classes of cohomology quaternionic projective spaces.