I came across submanifolds of $\Bbb R^n$ of finite type (whatever that means) in Stein's Harmonic Analysis (PMS-43): Real-Variable Methods, Orthogonality, and Oscillatory Integrals. I'm trying to understand what the "type" of a submanifold really is. I shall quote the book:
Here $S$ will be a smooth $m$-dimensional submanifold of $\Bbb R^n$, with $1\le m\le n-1$, and our assumptions regarding curvature will be replaced by the more general assumption that, at each point, $S$ has at most a finite order of contact with any affine hyperplane. We say that such submanifolds are of finite type.
- What is meant by "order of contact"? Are there easy-to-see examples of submanifolds of finite type, where it's possible to explicitly calculate the order of contact with a given affine hyperplane?
The precise definitions we require are as follows. We consider $S$ in a sufficiently small neighborhood of a given point and write $S$ as the image of a smooth mapping $\phi: U \to \Bbb R^n$, where $U$ is a neighborhood of the origin in $\Bbb R^m$. To ensure that $S$ is smoothly embedded, we suppose that the vectors $\partial\phi/\partial x_1, \ldots, \partial\phi/\partial x_m$ are linearly independent for each $x\in U$.
- Why do we assume $\partial\phi/\partial x_1, \ldots, \partial\phi/\partial x_m$ to be linearly independent? By a "smooth embedding", I understand that the inclusion map $i: S \hookrightarrow \Bbb R^n$ is smooth, and the corresponding differential $di_p: T_pS \to T_p(\Bbb R^n)$ is injective for all $p\in S$.
Now fix a point $x_0 \in U$, and a unit vector $\eta\in \Bbb R^n$. We assume that the function $$[\phi(x) - \phi(x_0)] \cdot \eta$$ does not vanish to infinite order as $x\to x_0$; that is, for each $x_0 \in U$ and each unit vector $\eta\in \Bbb R^n$, there is a multi-index $\alpha$, with $|\alpha| \ge 1$, so that $$\partial^\alpha_x[\phi(x)\cdot \eta] \vert_{x = x_0} \ne 0.$$
- What does it mean for a function to vanish to infinite order as $x\to x_0$? Trying to see where $\partial^\alpha_x[\phi(x)\cdot \eta] \vert_{x = x_0} \ne 0$ comes from.
Notice that if $(x', \eta')$ is sufficiently close to $(x_0,\eta)$, then also $$\partial^\alpha_x[\phi(x)\cdot \eta'] \vert_{x = x'} \ne 0.$$ The smallest $k$ so that, for each unit vector $\eta$, there exists an $\alpha$ with $|\alpha| \le k$ for which $$\partial^\alpha_x[\phi(x)\cdot \eta] \vert_{x = x_0} \ne 0,$$ is called the type of $\phi$ (and the type of $S$) at $x_0$. Also, if $U_1 \subset U$ is a compact set, the type of $\phi$ in $U_1$ is defined to be the maximum of the types of the $x_0 \in U_1$.
- The author has defined the type of $S$ at $x_0\in S$, but I don't think I really understand the motivation and the concept. Its relation with the "order of contact" business isn't clear either. Additionally, the choice of $\phi: U \to \Bbb R^n$ depends on the $x_0 \in S$ under consideration, and is not unique - so explicitly computing the type of $S$ at $x_0$ seems to be like shooting arrows in the dark.
I'd appreciate any help with the above questions and understanding what the type of a submanifold really means. Thank you!