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.