Two days ago, I gave a seminar talk on Chern‘s proof of the generalized Gauss-Bonnet theorem. Here I record the answer to a question asked by one of my colleague during the talk. Although not directly related to the proof of the generalized Gauss-Bonnet theorem, I think it’s quite interesting itself.
Let me first give some background before stating the question.
Roughly speaking, the idea of proof goes like this: Let’s assume for simplicity. Usually the curvature form (or more appropriately, the Pfaffian of the curvature form) is not exact, for otherwise the integral of the curvature form on a closed surface is zero. However, Chern observed that the pullback of the curvature form onto the unit sphere bundle is exact, and a smooth non-degenerate vector field on naturally induces a diffeomorphism from onto a submanifold in . Here is the open subset of where . By pulling back the curvature form onto and applying the Stokes theorem, we can localize the curvature integral into a sum of line integrals on small loops around the singularities of , which turn out to give the sum of the index of the vector field. Finally by the Poincare-Hopf theorem, this would give the Euler characteristic of .
While introducing the concept of the unit sphere bundle, I was asked by one of my colleague whether the unit sphere bundle of is the Hopf fibration. I didn’t know the answer at that time. But then I thought about it again and found that the answer is quite obviously no. However, I found it quite interesting that the Hopf fibration is actually the double cover of . I think this is a good exercise in geometry and I am recording it here.
For a Riemannian manifold , the unit sphere bundle is defined to be
with projection . This is an -bundle over . In particular, if is two-dimensional, then is a circle bundle.
Proposition 1 The circle bundle is diffeomorphic to .
Proof: We regard is the standard unit sphere, and regard as a column vector. We can define by
where is the cross product of and in . Then
Clearly this is a diffeomorphism.
Recall that the Hopf fibration is given by the quotient of the action on , where the action is given by . It is clear that the quotient space is , which is diffeomorphic to . The Hopf fibration is defined to be this quotient: .
From the above discussion, it is clear that is not the Hopf fibration. In fact, as is simply connected, it is not diffeomorphic to . However, the Hopf fibration can be regarded as the double cover of , in the sense that this diagram commutes
Here is the double covering map from to . To see this, first identify with the compact symplectic group where is the set of quaternions, where . We identify with the space of purely imaginary quaternions . Then we define by
It is clear that . Under this identification, then as we identify with . Let , where . Suppose , . Then by a direct calculation (done by Mathematica here)
On the other hand, for , can be defined to be
Expanding the above, we have
Comparing with (1), we have proved the commutativity.