Let $V \rightarrow M$ be an oriented vector bundle over a compact 
oriented manifold $M$ equipped with a metric $h$ (the metric $h$
is a metric on the Vector bundle $V$, not on the manifold $M$). 
Is there some ``natural'' differential $\omega\_T$ form 
representing the Thom Class of $V$? In particular I want the 
following properties: 


1) If $(M,g)$ is a compact two dimensional Riemannian manifold,  

and $V = TM$, the tangent 
bundle of $M$ and $X\_0 : M \rightarrow TM$ the zero vector field, then 
$$ X\_0^{\*} (\omega\_T) = \frac{K}{2 \pi} dA $$
equality holding on the level of forms, where $K$ is the 
Gaussian curvature and $dA$ is the area form. 


2) If $V\rightarrow M$ is a complex vector bundle with a hermitian metric $h$ and 
$s\_o : M \rightarrow V$ the zero section then 
$s\_0^\*(\omega\_T)$ is the differential form for the 
top Chern class obtained by Chern Weil theory 
(again equality holds on the level of forms). 


Notice that on the level of cohomology, the pull back via the 
zero section of the 
Thom class gives us the Euler class of $V$. My basic question 
is that what should one take the Thom class to be, to obtain 
equality on ``the level of forms'' when there is a natural form 
representing the Euler class. 



Is there a natural form representing the Thom class of a vector bundle, which when pulled back via the zero section represents the Euler class on the level of forms?
Here is another construction which goes back to Chern's proof of the Gauss-Bonnet theorem.


Suppose that $\pi: E\to M$ is an oriented rank $2k$ real vector bundle over the manifold $M$. Assume additionally that $E$ is equipped with a metric $g$, and a connection $\nabla$ compatible with the metric. 


In your case $2k=\dim M =2$, $E= TM$, $g$ is a Riemann metric and the connection $\nabla$ is the Levi-Civita connection.


Denote by $S(E)$ the unit sphere bundle of $E$. Using the connection $\nabla$ one can **explicitly** construct a form


$$\Psi(\nabla) \in \Omega^{2k-1}(S(E)) $$


with the property that the integral over each fiber of $S(E)$ is equal to $-1$. The form $\Psi(\nabla)$ is known as the *global angular form* determined by $g$ and $\nabla$.


The connection $\nabla$ also **explicitly** defines via the Chern-Weil construction a closed $2k$-form $\newcommand{\be}{\boldsymbol{e}}$


$$\be(\nabla)\in \Omega^{2k}(M). $$


This the the so-called *Euler form* which in your case is $\frac{1}{2\pi} K dA$. 


$\newcommand{\bR}{\mathbb{R}}$ Now choose a smooth function $\rho :[0,\infty)\to [0,\infty)$ which ir equal to $-1$ near $0$ and equal to $0$ on $[1, \infty)$. Let $r: E\to \bR$ denote the radial distance along the fibers of the vector bundle. Define


$$\omega(\nabla)=-\rho'(r) dr\wedge \Psi(\nabla) + \rho(r)\pi^\*\be(\nabla). $$


One can then show the following.


**1.** $\omega(\nabla)$ is a closed form representing the Thom class of $E$.


**2.** If $\zeta\_0: M\to E$ denotes the zero section, then


$$\zeta\_0^\* \omega(\nabla)=\be(\nabla). $$


**3.** Suppose that $\zeta:M\to E$ is a section with transversal zero set. For $t>0$ we set $\zeta\_t:=t\zeta$. Note that $\zeta$ and $\zeta\_t$ have the same zero sets. Let $Z=\zeta^{-1}(0)$ so that $Z$ is a codimension $2k$-submanifold of $M$ whose normal bundle carries a natural orientation. Then the following hold.


**a.** For any neighborhood $\newcommand{\eN}{\mathscr{N}}$ $\eN$ of $Z$ in $M$ there exists $T=T(\eN)$ such that the support of $\zeta\_t^\* \omega(\nabla)$ is contained in $\eN$ for any $t>T$.


**b.** The form $\zeta\_t^\*\omega(\nabla)$ is Poincare dual to the cohomology class determined by $Z$.


**c.** As $t\to \infty$ the forms $\zeta\_t^\*\omega(\nabla)$ converge in the sense of currents to the current of integration determined by $SZ$.


In your case, the zero set consists of finitely many points, the $2$-form $\zeta\_t^\*\omega(\nabla)$ has the description


$$\zeta\_t^\*\omega(\nabla)=\rho\_t dA $$


where the function $\rho\_t$ is concentrated near the zeros of the section, and forms high (positive or negative) peaks around these points. 


The statement **3.c.** above contains as a special case the Poincare-Hopf theorem.


For more details see [Section 8.3.2 of these notes.](http://www3.nd.edu/~lnicolae/Lectures.pdf)


