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I am reading Differential Forms in Algebraic Topology by Bott & Tu. The authors constructed the global angular form $\psi$ for an oriented $k$-sphere bundle $E$ over a smooth manifold $M$. It has the properties:

     1. $\psi$, when restricted to each fiber, is the generator for the top 
cohomology of the fiber. To be more precise, $\psi|_{E_p}$ is a closed $k$-
form and $\int_{E_p}\psi|_{Ep}=1$. The integral makes sense since the sphere 
bundle is oriented.
     2. $d\psi=-\pi^*(e)$, where $\pi:E\to M$ is the natural projection and 
$e=e(E)$ is the Euler class of the sphere bundle.

The global angular form is clearly not uniquely characterized by the above two properties. For any closed form $\eta$ on $M$, $\psi+\pi^*(\eta)$ will satisfy 1 and 2 if $\psi$ does. Note that $\psi$ is in general not closed, hence does not represent a cohomology class. Now my question is

    To What extent is the global angular form of an oriented sphere bundle 
unique, satisfying 1 and 2 above? 

Since the global angular form is in some sense choosing a polar coordinate system for each fiber and the Euler class indicates the obstruction against this. (I am sorry for my imprecision at this point. I may need some help here.) So I have conjectured the following proposition:

With notations as above. Let $\omega$ be a $k$-form on $E$ such that 
 1. $\omega$ restricts to the generator of the top cohomology of each fibre and 
 2. $d\omega=\pi^*(\eta)$ for some *closed* form $\eta$ on $M$. 
Then $\eta$ is the Euler class of the sphere bundle. 

Can someone prove of disprove this proposition? Any comment on the intuitive understanding of the global angular forms is appreciated. Thanks!

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Why doesn't LaTeX work in the grey area? Could somebody help me fix it? – Yang Zhou Jan 9 '13 at 12:01

1 Answer 1

A global angular form determines an orientation of a sphere bundle. The orientations then determines the Euler class and the global angular form back again.So every global angular form determines the Euler class.

For an n-1 dimensional sphere bundle that comes from a vector bundle, one can see this directly by integrating the exterior derivative of the global angular form over a smooth section that has only finitely many singularities. By removing small open balls around each singularity one can replace the integral by the integral of the global angular form on the boundaries of these balls (Stokes theorem). Letting the radii of the balls go to zero shows that these integrals are computing the sum of the the indexes of each singularity i.e. the Euler number of the sphere bundle by computing the number of times that the vector field winds around the fiber sphere above each singularity.

An important example of this is the case of the tangent circle bundle of an oriented surface. Here any connection 1-form is a global angular form. Its exterior derivative is the curvature 2-form. So the integral of the curvature 2-form over a section gives the Euler characteristic. This is the Gauss-Bonnet theorem.

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