On some geometrical properties of Seifert bundles

In this paper we use the integration along the leaves introduced by Haefliger in 1980 to obtain a differentiable version of the Gysin sequence and Euler class for compact Hausdorff orientable foliations with generic leaf the sphereS ^p. From this we give a geometrical significance to the vanishing of the Euler class on Seifert bundles. We also obtain an integral formula on Seifert bundles similar to the Gauss-Bonnet one.