Computing the differential of a certain smooth map

Question

Let $M \subseteq \mathbb{R}^k$ be an embedded submanifold of $\mathbb{R}^k$, with dim$M=n$.

Let $v$ be in $\mathbb{S}^{k-1}$, and let $P_v:\mathbb{R}^k\to(\mathbb{R}v)^{\bot}$ defined by $P_v(x)=x-<x,v>v$ where $<x,v>=x^1v^1+\dots+x^kv^k$ is the Euclidean dot product and $(\mathbb{R}v)^{\bot}$ is the vector space of the vectors in $\mathbb{R}^k$ orthogonal to $v$. We know that $(\mathbb{R}v)^{\bot}$ has dimension $k-1$ as a real vector space, so it is isomorphic to $\mathbb{R}^{k-1}$.

Question 1) If I want to consider $(\mathbb{R}v)^{\bot}$ as a smooth manifold, I can choose an isomorphism between $(\mathbb{R}v)^{\bot}$ and $\mathbb{R}^{k-1}$ and declare this to be a diffeomorphism, right? Or, better, is there a canonical identification between $(\mathbb{R}v)^{\bot}$ and $\mathbb{R}^{k-1}$ ?

Now, (assuming the answer to Question 1 is affirmative), I have that $P_v:\mathbb{R}^k\to(\mathbb{R}v)^{\bot}\simeq \mathbb{R}^{k-1}$ is a map between two smooth manifolds.

Question 2) How can I show that this map is smooth? Do I have to calculate it in local coordinates? So do I have to explictly choose an isomorphism between $(\mathbb{R}v)^{\bot}$ and $\mathbb{R}^{k-1}$ and then calculate the map in local coordinates?

Let $\Phi_v:M\to (\mathbb{R}v)^{\bot}$ be $P_v\circ \iota_M$ with $\iota_M$ the inclusion map of $M$ in $\mathbb{R}^k$. So, since $P_v$ is smooth, then also $\Phi_v$ is smooth. Let $p\in M$ and $X\in T_pM$. Then $d(\iota_M)_p(X)=\sum_iw^i \partial_i|_p\in T_p\mathbb{R}^k$ for some $w\in \mathbb{R}^k$.

My notes say that $$d(\Phi_v)_p(X)=\sum_iP_v(w)^i\partial_i|_p$$

Question 3) How can I prove the above equation? How should I imagine $T_{\Phi_v(p)}((\mathbb{R}v)^{\bot})$? Who is a (canonical) basis for $T_{\Phi_v(p)}((\mathbb{R}v)^{\bot})$?

All I can see is that $$d(\Phi_v)_p(X)=d(P_v)_p(\sum_iw^i\partial_i|_p)=\sum_iw^id(P_v)_p(\partial_i|_p)$$ where the last $=$ is by the linearity of the differential.

Please use simple language since I'm a beginner in this subject, do full calculation if needed, and also, if you think I lack some knowledge of some topic of smooth manifold theory (useful to better understand your answer to my question), please let me know.

Answer 1

1. There is no canonical choice. Probably the best way to deal with it is to treat it as a regular submanifold. You can then treat its tangent space as a subspace of the ambient manifold, and only choose coordinates once.

2. The definition of smooth mapping involves local coordinates so... yes. However having chosen slice coordinates the map is just cutting off the last few coordinates, and since the coordinate functions are smoot you are in.

3. Once again this is just a statement in local coordinates, depending on the first question.

Loring Tu’s Book on manifolds is the one most beginners find easiest to read. The exercises are straightforward plug and chug based on the material being studied.