I'm reading a paper and it says
The differential form $\psi$ is zero to infinite order at all points of $F$.That is,all coefficients of $\psi$ are zero to infinite order at points of $F$.(Here $\psi$ is a differential form defined on a complex manifold $N$,and $F\subset N$ is a submanifold.)
Also,let $J$ be the complex structure on $N$.Suppose $N\cong \Delta\times F$,where $\Delta$ is the unit disc in the complex plane.Let $J_0$ be the "product" complex structure on $N\cong \Delta\times F$.Then $J-J_0$ dies to infinite order along $F$.
What does it mean by saying all coefficients of $\psi$ are zero to infinite order at points of $F$ and $J-J_0$ dies to infinite order along $F$?
If you know,please tell me.Thank you!
A smooth function $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ is usually said to vanish up to order $k$ at a point $p$ if $(\partial^{\alpha} f)(p) = 0$ for all $|\alpha| \leq k$. That is, the function and all possible partial derivatives of order $\leq k$ vanish at $p$. This can be extended to functions defined on a manifold by saying that $f \colon M \rightarrow \mathbb{R}$ vanishes up to order $k$ at a point $p$ if the local representation of $f$ in a coordinate system around $p$ (a function $U \rightarrow \mathbb{R}$ where $U \subseteq \mathbb{R}^{\dim M}$) vanishes up to order $k$. You can check directly that this notion is independent of the coordinate system used around $p$.
Finally, we can extend this notion for various tensors (such as differential forms and complex structures) by requiring that all the local coefficients of the tensor with respect to some coordinate system around $p$ vanish up to order $k$.