Let $M$ is a $2n$-dim smooth Riemannian manifold, $\eta$ is a vector field on $M$, $p\in M$ is a isolated zero of $\eta$. Then we can define a map from $\partial B_{p}(\varepsilon)$ to $S^{2n-1}(1)$ by $\eta$: $$\eta_{p}=\frac{\eta}{|\eta|}: \partial B_{p}(\varepsilon)\rightarrow S^{2n-1}(1)$$

Here $B_{p}(\varepsilon)$ is a ball about of $p$ with radius $\varepsilon$.

Let $f_1,\cdots,f_{2n-1}$ be an orthonormal basis of $T(\partial B_{p}(\varepsilon))$, let $f^{*}_1,\cdots,f^{*}_{2n-1}$ be the metric dual basis.

We can find that

$$\eta^{*}\wedge(\nabla^{TM}_{f_1}\eta)^{*}\wedge(\nabla^{TM}_{f_{2n-1}}\eta)^{*}$$

$$= (d{\rm vol}_{g^{TM}})\int^{B}\eta^{*}\wedge (\nabla^{TM}_{f_1}\eta)^{*}\wedge\cdots\wedge (\nabla^{TM}_{f_{2n-1}}\eta)^{*}$$

then

$$f^{\ast}_{1}\wedge\cdots\wedge f^{\ast}_{2n-1}\int^{B}\eta^{\ast}\wedge (\nabla^{TM}_{f_1}\eta)^{\ast}\wedge\cdots\wedge (\nabla^{TM}_{f_{2n-1}}\eta)^{\ast}=\eta^{\ast}_{p}\omega ,(1) $$

here $\omega$ be the volume form on $S^{2n-1}(1)$.

I find this formula (1) in the article “$\eta$-invariants and the Poincaré-Hopf index formula”. But I don't know how to get it. Is there some way to compute it?

isolatedzero. – Pietro Majer Mar 1 '11 at 19:09`$\eta_{p}$`

is to unitization the tangent vector on`$\partial B_{p}(\varepsilon)$`

. – Chen Mar 2 '11 at 4:19