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I can't understand a passage from A Course in Differential Geometry by T. Aubin. First, there is Definition 2.6., which I posted in this question. And now there's this:

$(\Phi_*)_P$ is nothing else than $(d\Phi)_P.$

($\Phi:M_n\to W_p$, where $M_n$ and $W_p$ are manifolds of dimensions $n$ and $p$ respectively. $P\in M_n.$)

I don't understand what this sentence means. I can't find a definition of $(d\Phi)_P$ earlier in the book. There is a definition for the case of $\Phi$ being a map between open subsets of Euclidean spaces, but not for general manifolds. So how can we establish that the two objects are the same if we haven't defined one of them? The author goes on to explain thus:

Indeed, consider a local chart at $P$ with coordinates $\{x^i\}$ and a local chart at $Q$ with coordinates $\{y^a\}$. $\Phi$ is defined in a neighbourhood of $P$ by $p$ real-valued functions $\Phi^a(x^1,x^2,\cdots,x^n),\;a=1,2,\cdots,p.$

I don't get it. What are the $\Phi^a?$ If they're real-valued, then $(\Phi^1,\cdots,\Phi^p)$ cannot be equal to $\Phi$ because $\Phi$ goes from $M_n$ to $W_p$, not to $\Bbb R^p$. To get to $\Bbb R^p$, I need to compose $\Phi$ with a chart. And aren't the coordinates $x^i$ real too? I understand that when I have a chart $(\Omega,\varphi)$ for an open set $\Omega\subseteq M_n$ and $\varphi:\Omega\to\Bbb R^n,$ then the coordinates $x^i$ of a point $m\in M_n$ are the coordinates of $\varphi(m)$ in $\Bbb R^n$. So that would mean that $\Phi^a$ are actually functions from $\Bbb R^n\supseteq\Omega$ to $\Bbb R$. And probably $(\Phi^1,\cdots,\Phi^p)=\psi\circ\Phi\circ\varphi^{-1}$ for some charts $\phi$ and $\psi$. Is that correct? If so, isn't the notation very counterintuitive?

The author now says the following. I don't understand it at all. He uses the symbol $(d\Phi)_P$ again, and I think it still hasn't been defined.

Using intrinsic notations to simplify, we get $$X(f\circ\Phi)=d(f\circ\Phi)_P\circ X=(df)\circ(d\Phi)_P\circ X=(df)\circ(\Phi_*)_PX.$$ Indeed, $\{X^i\}$ being the components of $X$ in the basis $\{(\partial/\partial x^i)_P\},$ the components of $Y=(\Phi_*)_PX$ are $$Y^a=\sum_{i=1}^n \frac{\partial\Phi^a}{\partial x^i}$$ in the basis $\{\partial/\partial y^a)_Q\}.$ When we use intrinsic notation, we do not specify the local charts. In the coordinate systems $\{x^i\}$ and $\{y^a\}$, the equality above shows that $(d\Phi)_P=((\partial\Phi^a/\partial x^i))_P=(\Phi_*)_P.$

I don't understand what the intrinsic notations are. I can't see how the notation is intrinsic if we use the coordinates $x^i$ and $y^a$ which are real.

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I will try and answer what appear to be the questions in some sort of systematic order. I don't have a copy of Aubin's text (and will try to use the notation presented in the original post), but I think that the questions are relatively clear and can be addressed despite this.

  • $ \left(\Phi_{*}\right)_{P}$ is nothing other than $\left(d\Phi\right)_{P}$:

Without seeing the textbook, I imagine that Aubin is trying to say that the push forward of $\Phi$ at $P \in M$ (i.e. $\left(\Phi_{*}\right)_{P})$) is the same as the differential of $\Phi$ at $P$ (i.e. $\left(d\Phi\right)_{P}$), or even that $\left(\Phi_{*}\right)_{P}$ is the same in spirit as $\left(d\Phi\right)_{P}$ in the case that $\Phi$ is a differentiable map between Euclidean spaces. The matter of choosing coordinates about $P \in M$ and $\Phi(P) = Q \in W$ is intended to illuminate this matter.

Choosing coordinate charts $(U, \varphi = (x^{1}, \ldots x^{n})$ and $(V, \psi = (y^{1}, \ldots , y^{p})$ about $P$ and $Q = \Phi(P)$ respectively allows one to do exactly what you are suggesting: form the coordinate representative of $\Phi$. In the given coordinate systems, the coordinate representative of $\Phi$ is precisely $$\psi \circ \Phi \circ \phi^{-1} : \phi(U) \subseteq \mathbb{R}^{n} \to \psi(V) \subseteq \mathbb{R}^{p}.$$

Chasing around the appropriate maps shows that the $\Phi^{a} = y^{a} \circ \Phi\circ \varphi^{-1} : x(U) \to \mathbb{R}$, $a = 1..p$, are real-valued functions that give the coordinate functions of the map $\Phi$ w.r.t. the chosen coordinate systems.

So, ``And probably $(\Phi^{1},⋯,\Phi^{p})$=ψ∘Φ∘φ−1 for some charts ϕ and ψ. Is that correct?'' Yes.

  • ``I don't understand what the intrinsic notations are.'':

You have some understanding of this, as the intrinsic notations are precisely the notations that do not depend on coordinate charts. Thus the line

$$ X(f\circ\Phi)=d(f\circ\Phi)_P\circ X=(df)\circ(d\Phi)_P\circ X=(df)\circ(\Phi_*)_PX, $$

is given entirely in intrinsic notation assuming that you will accept the notation $(\Phi_{*})_{P} = (d\Phi)_{P}.$ Note that the line is trying to tell you how a vector field $X$ on $M$ pushes forward to a vector field on $W$ be defining how the push forward of $X$ acts on real valued functions defined on $W$.

Now, when one passes to the chosen coordinate systems and one analyzes $(\Phi_{*})_{P} X$, which is now a vector field on $W$, then the identification of $(\Phi_{*})_{P} = (d\Phi)_{P}$ becomes clear: Expanding $(\Phi_{*})_{P}X$ as a vector field in the coordinate bases $\left\{ \frac{\partial}{\partial y^{a}}\right\}$ shows that the components of $(\Phi_{*})_{P}X$ are $Y^{a} = X^{i}\frac{\partial \Phi^a}{\partial x^{i}}$ (There seems to be an error in what you have posted above). Taking $X$ to be the coordinate bases vector fields themselves, then it should be clear that the linear map $(\Phi_{*})_{P} : T_{P}M \to T_{Q}W$ is described in the chosen coordinates by the matrix $\left( \frac{\partial\Phi^{a}}{\partial x^{j}}\right)$, $a = 1..p$, $j = 1 . . n$.

The point that I believe Aubin is trying to make is that this matrix for $(\Phi_{*})_{P}$ in the chosen coordinates is the same as the matrix $(d\Phi)_{P}$ for the map $\psi \circ \Phi \circ \varphi^{-1} : \varphi(U) \subset \mathbb{R}^{n} \to \psi(V) \subseteq \mathbb{R}^{p}$, the coordinate representative of $\Phi$ as a map between Euclidean spaces.

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