The following are a few paragraphs from Section 1.1 of Carlo Mantegazza's Lecture Notes on Mean Curvature Flow (Birkhäuser):
In this section, we introduce some basic notation and facts about Riemannian manifolds and their submanifolds. A good reference is Riemannian geometry (Gallot, Hulin, and Lafontaine).
... (some paragraphs are omitted)
The main objects we will consider are $n$-dimensional, complete hypersurfaces immersed $\mathbb{R}^{n+1}$, that is, pairs $(M,\phi)$ where $M$ is an $n$-dimensional, smooth manifold with empty boundary and $\phi:M\to\mathbb{R}^{n+1}$ is a smooth immersion (the rank of the differential $\mathrm{d}\phi$ is equal to $n$ everywhere on $M$).
...
Since $\phi$ is locally an embedding in $\mathbb{R}^{n+1}$, at every point $p\in M$ we can define up to a sign a unit normal vector $\nu(p)$. Locally, we can always choose $\nu$ to be smooth.
If the hypersurface $M$ is compact and embedded, that is, the map $\phi$ is one-to-one, the inside of $M$ is easily defined and we will consider $\nu$ to be the inner pointing unit normal vector at every point of $M$. In this case the vector field $\nu:M\to\mathbb{R}^{n+1}$ is globally smooth.
For the record, I do not use the recommended textbook (GHL) much. My knowledge of Riemannian geometry comes mainly from Introduction to Riemannian Manifolds written by John M. Lee, but I think they both work well for us in terms of what is going to be discussed. And I want to remind you that $M$ may not be a subset of $\mathbb{R}^{n+1}$ although Mantegazza says that immersed hypersurfaces in $\mathbb{R}^{n+1}$ are to be considered. I guess this is probably an expression of viewing $M$ as a hypersurface in $\mathbb{R}^{n+1}$ via $\phi$.
My question is, why does Mantegazza point out that $\phi$ is locally an embedding in $\mathbb{R}^{n+1}$ when he defines unit normal vectors $\nu(p)$ at some point $p\in M$? Based on what I have learned, I don't really think we have to have an embedding around $p$ in order to define unit normal vectors. The fact that $\phi$ is an immersion assures us of an injective linear transformation $\mathrm{d}\phi_p$, so the linear subspace $\mathrm{d}\phi_p(T_p M)$ is $n$-dimensional, thereby making the orthogonal complement $(\mathrm{d}\phi_p(T_p M))^\perp$ become $1$-dimensional. And that brings us exactly two unit normal vectors after we normalize the vector in the basis for the orthogonal complement, and take also the negative of the result, right? Why do we need to notice that $\phi$ is locally an embedding? What is the argument for? Thank you.
Edit. I know why $\phi$ is locally an embedding, which is merely a property every immersion has. My question is really about why the author mentions it in defining unit normal vectors.