Seifert invariants of left-quotients of 3-dimensional simple Lie groups

This paper is concerned with a result of Raymond and Vasquez that gives a classification of compact left-quotients of 3-dimensional Lie groups in terms of their Seifert invariants. By a simple argument involving rotation numbers of a vector field transverse to the Seifert fibration, we prove their main result, the classification of left-quotients of the simple Lie groups S̃L2 and SU(2). As an application of this proof, we give a classification of Seifert fibred 3-manifolds which admit a contact form such that the fibration is a Legendre fibration.