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?

Question

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.

Answer 1

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.

Answer 2

Have you looked at Mathai-Quillen's paper: "Superconnections, Thom classes, and equivariant differential forms." Topology 25: 85-100 (1986)? Where they build a form representing the Thom class (together with some very nice properties). You can also look at: "Mathai-Quillen Formalism" Siye Wu available on arXiv(hep-th) (section 2.2).

Answer 3

The following short paper is about more or less this exact question: The Gauss-Bonnet Theorem for Vector Bundles, Denis Bell, J. Geom. 85 no. 1-2 (2006) 15-21.

It is available on the author's website here: https://www.unf.edu/~dbell/GBT.pdf

Here is an arXiv link too for good measure: https://arxiv.org/abs/math/0702162

Edit: Actually, this article only constructs a specific form representing the Thom class when $V$ is a plane bundle or a direct sum of plane bundles. After that it appeals to a splitting principle which works only at the level of classes, not forms.