# Does two different extensions of a map between manifolds have the same derivative in the manifold-domain?

It is well know that the differential calculus is developed with respect open sets but a manifold not necessarily is an open set and actually many times it not. So in Differential Geometry it is usual give the following definition of $$C^r$$-function.

Definition

A function $$f:S\rightarrow\Bbb R^n$$ where $$S\subseteq\Bbb R^k$$ is of class $$C^r$$ if it can be extended to a $$C^r$$-function defined in an open set $$U$$ containing $$S$$.

Naturally if $$\tilde {f_1}, \tilde{f_2}$$ are two different extension of $$f$$ it is possible that even their derivatives are different in $$S$$ but fortunately this do not happens if $$S$$ is a manifold as Victor Gullemin and Peter Heine show in the text Differential forms part of which I summarise to follow but if you like you can read it here.

So let be $$f:X\rightarrow Y$$ a $$C^r$$-mapping from a k-manifold $$X$$ of $$\Bbb R^n$$ and an $$l$$-manifld $$Y$$ of $$\Bbb R^m$$ and thus let be $$\tilde f$$ any $$C^r$$-extension. So we define the derivative of $$f$$ in $$X$$ to be the restriction of the derivative of $$\tilde f$$ in $$X$$, that is $$Df(x):=D\tilde f(x)$$ for any $$x\in X$$. So let's prove that this definition is consistent, that is if does not depend from the choice of a particular extension. Therefore let be $$\phi:U\rightarrow V$$ a local patch of $$X$$ and thus let be $$h:=\tilde f\circ\phi$$. Now it is possible to prove that $$D\tilde f(\phi(x))=Dh(x)$$ for any $$x\in U$$ and since $$h=\tilde f\circ\phi=f\circ\phi$$ we conclude that in $$X$$ the derivatives of two different extension of $$f$$ are equal.

So I observe that if $$\phi:U_1\rightarrow V_1$$ and $$\phi_2:U_2\rightarrow V_2$$ are two different charts then what above showed says that $$D(f\circ\phi_2)(x_1)=D(f\circ\phi_2)(x_2)$$ where $$x_2=\phi_2^{-1}(\phi(x_1))$$ but unfortunately to me it seems false this. Anyway since this was not clear to me I refer to the text Differential Topology by Victor W. Gullemin and Alan Pollack where I found this definition of derivative of a mapping between two manifold that I completely understand. However I do not understand if what showed in the link I posted implies that two different extesions of $$f$$ have the same derivative at $$X$$. So since well understand the last approach I primarily ask if with respect this it is true that two different extesions of $$f$$ have the same derivative at $$X$$ and then (only if you like) I ask to prove the equivalence of this approach with respect the first. So could anyone help me, please?

• The fact that you want to use in questions like this is the following: Given a submanifold $S^k$ in a manifold $M^n$ and $p \in S$, there exists local coordinates $(x^1, \dots, x^n)$ on $M$ such that $S$ is given by $x^{k+1} = \cdots = x^n = 0$. In that case, given a function $f$ on $M$, the differential of $f$ restricted to $S$ clearly depends only on $f$ restricted to $S$ and not on how it's extended away from $S$. – Deane Feb 20 at 21:47
• Sorry, but I do not understand what you mean: unfortunately what you claim it is not clear and obvious to me. Excuse my ignorance. – Antonio Maria Di Mauro Feb 20 at 21:55
• I'd be happy to elaborate. There are two parts to my comment. The first is about $S$ itself, and the second is about $f$. Would you like me to provide more details on both? – Deane Feb 20 at 22:13
• So, first of all I try to explain the formalism I know. If $M$ is a $k$-manifold in $\Bbb R^n$ to me a coordinate patch is a function $\phi$ from an open set of $\Bbb R^k$ or $H^k$ (the upper-half space) to an open set $V$ of $M$. So are your $x^1,...,x^n$ the coordinate functions individuate by the local chart as I above described? Then I do not know the existence of a coordinate patch such that $[\phi(x)](i)=0$ for $i=1,...,(n-k)$ and thus I can not accept this explanation if you do not prove first the result. Moreover why the existence of this local chart proves what I ask? – Antonio Maria Di Mauro Feb 20 at 22:20
• If you like to know it I say to you that I am studying by the text Analysis on Manifold by James Munkres and I refer to the texts that Munkres put in the bibliography that use substantially the same formalism. So I am sure you realise that I can accept only some particular explanation: precisely those are consistent with respect Munkres or affine formalims. – Antonio Maria Di Mauro Feb 20 at 22:29

Step 1. First prove the claim in the case when $$S\subset {\mathbb R}^k$$ is actually an open subset of $${\mathbb R}^m\subset {\mathbb R}^k$$.
Step 2. Prove that if $$S$$ is an $$m$$-dimensional submanifold in $${\mathbb R}^k$$ then for each $$x\in S$$ there exists a neighborhood $$U$$ of $$x$$ in $${\mathbb R}^k$$ and a diffeomorphism $$h: U\to {\mathbb R}^k$$ such that $$h(U\cap S)$$ is an open subset of $${\mathbb R}^m\subset {\mathbb R}^k$$.
• Well, I tried to attempt the following arguments. So let us to distinguish the case where $S$ is a $m$-manifold of $\Bbb R^k$ with empty or not empty boundary. So in the first case as you substantially stated $S$ is diffeomorphic to an open subset of $\Bbb R^m$ and thus for any $y\in S$ there exist a local patch $\alpha:U\rightarrow V$ defined in an open set $U$ of $\Bbb R^m$ such that $y=\alpha(x)$ and thus $$f(y)=\big(f\circ\alpha\big)(x)$$ for any $y\in V$ where $x:=\alpha^{-1}(y)$. – Antonio Maria Di Mauro Feb 22 at 9:05
• So if $\tilde f_1$ and $\tilde f_2$ are two different $C^r$ extension of $f$ then $$D\tilde f_1(y)=D\big(f\circ\alpha\big)(x)=D\tilde f_2(y)$$ for any $y\in V$ and thus it seems that we could claim that $D\tilde f_1(y)=D\tilde f_2(y)$ for any $y\in V$ and so for any $y\in S$. – Antonio Maria Di Mauro Feb 22 at 9:05
• However if $\alpha':U'\rightarrow V'$ is another local patch about some $y\in V$ then it would be $$D\big(f\circ\alpha'\big)(x')=D\tilde f_1(y)=D\big(f\circ\alpha\big)(x)$$ but generally $$D\big(f\circ\alpha')(x')=D\big(\tilde f\circ\alpha'\big)(x')=D\big(\tilde f\circ\alpha\big)(x)\cdot D\big(\alpha^{-1}\circ\alpha'\big)(x')\neq D\big(\tilde f\circ\alpha\big)(x)$$ and thus this (it seems) invalid my argument that is the same of Gullemin and Heine. – Antonio Maria Di Mauro Feb 22 at 9:06
• Then if the boundary of $S$ is not empty I did not able to implement your (second) hint and I did not able to prove prove the third point knowing the first and second. So could you help me, please? – Antonio Maria Di Mauro Feb 22 at 9:06