The derivative of a smooth map $f : U \to V$ between open subsets $U \subset \mathbb R^m, V \subset \mathbb R^n$ at $p \in U$ is a linear map $df_p : \mathbb R^m \to \mathbb R^n$ which is charcterized by the well-known property $\lim_{h \to 0} \dfrac{\lVert (f(p +h) – (f(p) + df_p(h))\rVert}{\lVert h \rVert} = 0$. This concept relies on the linear structures of domain and range of $f$ which are given locally around $p$ and $f(p)$.
For a smooth map $f : M \to N$ from an $m$-manifold $M$ to an $n$-manifold $N$ this not work because we do not have (in general) canonical linear structures around $p$ and $f(p)$. We can of course choose charts $\phi : U \to U'$ on $M$ around $p$ and $\psi : V \to V'$ on $N$ around $f(p)$ and get a "localized derivative"
$$d_{(\phi,\psi)} f_p = d(\psi \circ f \circ \phi^{-1})_{\phi(p)} .$$
But this depends on the choice of local Euclidean coordinates around $p$ and $f(p)$ so that $d_{(\phi,\psi)} f_p$ does not seem to be a good concept of a derivative of $f$ at $p$. Indeed the derivative of $f$ at $p$ is usually understood as a linear map $df_p : T_pM \to T_{f(p)}N$ between tangent spaces.
The tangent space $T_pM$ is defined either as the set of derivations $C^\infty(M) \to \mathbb R$ at $p$ or as the set of "derivatives $u'(0)$ of curves $u : J \to M$ through $p$". Such $u$ is a smooth map defined on an open interval $J \subset \mathbb R$ with $0 \in J$ and $u(0) = p$ and the derivative (whatever its definition may be) is taken at $t =0$. The approach via derivations is very abstract; in my opinion the approach via curves is more intuitive and better for motivational purposes. However, it involves a "preliminary" concept of derivative using the following fact: Although we do not yet have an interpretation of $u'(0)$ we can at least say what it means that two curves $u,v$ have the same derivative at $0$: This can be characterized by the requirement that $(\phi \circ u)'(0) = (\phi \circ v)'(0)$ for all charts $\phi : U \to U'$ around $p$. Note that this equation holds for some chart, then it holds for all charts. Having the same derivative at $0$ is an equivalence relation for curves through $p$ and one defines $u'(0) = [u]$ = equivalence class of $u$. In my eyes this is a strange interpretation of "derivative", but formally it does make sense.
A similar phenomenon occurs in the context of the cotangent space $T^*_pM$. One approach is to define it as the set of equivalence classes of maps $f \in C^\infty(M)$ with respect to the equivalence relation of having the same derivative at $p$ which is defined via chart $\phi : U \to U'$ around $p$ by considering $d(f \circ \psi^{-1})_{\psi^{-1}(p)}$. Some authors write $df_p = [f]$ (e.g. Hitchin p. 17; see also Hitchin's definition of tangent space and tangent vectors).
My questions:
- The same thing could be done for smooth maps $f : M \to N$ by considering equivalence classes of smooth maps $M \to N$ with respect to the equivalence relation of having the same derivative at $p$, the latter being defined via charts. That is, $df_p = [f]$. Does this occur anywhere in the literature and does it have any use?
- Is there an alternative approach to define $T_pM$ (and also $T^*_pM$) not in terms of equivalence classes of curves, but in a more persuavive form? A vague idea would be that the set of derivatives of curves through a point $p$ of an open subset of $\mathbb R^m$ is nothing else than $\mathbb R^m$ itself. So why not take $T_pM = (\mathbb R^m,\phi)$, where $\phi : U \to U'$ is a fixed chart around $p$? This would involve the axiom of choice to assign charts $\phi$ to points $p$, but isn't it okay?
Best Answer
Question 1.
I have not seen it in the literature, but my knowledge is limited.
Question 2.
The idea to take $T_pM = (\mathbb R^m,\phi)$, where $\phi : U \to U'$ is a fixed chart around $p$, is nice. It means that you identify the tangent space $T_pM$ with $\mathbb R^m$ via a chart $\phi$ around $p$. This is completely legitimate, but I do not think that one should choose a specific $\phi$. Instead one should take into account all charts $\phi$ and "glue" the associated copies of $\mathbb R^m$ via the differentials of the transition functions. Let us make this precise. This will involve a lot of formalism and it is not as intuitive as you expect. The approach is purely requirements-based; its focus is to explicitly formulate our expectations regarding tangent spaces.
On an abstract level, we want to associate to each point $p$ of an $m$-dimensional manifold $M$ an $m$-dimensional tangent vector space $T_pM$ specified by the following requirements:
If $M$ is an open subset of $\mathbb R^m$, then $T_p M = \mathbb R^m$ (or more precisely, there exists a canonical isomorphism $T_pM \to \mathbb R^m$). This is a reasonable requirement: In any conceivable interpretation the set of all tangent vectors at $p$ should be nothing else than $\mathbb R^m$. It is the intuitive part of our approach.
The tangent space $T_pM$ is determined locally, i.e. $T_p M = T_pU$ for each open neigborhood $U$ of $p$. Again we can require more generally that there exists a canonical isomorphism $T_pU \to T_pM$.
Each chart $\phi : U \to U'$ around $p$ induces a specific isomorphism $d\phi_p : T_pM = T_pU \to T_{\phi(p)}U' = \mathbb R^m$.
Given two charts $\phi, \psi$ around $p$, we have $d\psi_p \circ d\phi_p^{-1} = d(\psi \circ \phi^{-1})_{\phi(p)}$. Here $\psi \circ \phi^{-1}$ is the transition map from $\phi$ to $\psi$ (which is a map between open subsets of $\mathbb R^m$) and $d(\psi \circ \phi^{-1})_{\phi(p)}$ is its standard derivative in the sense of multivariable calculus.
These requirements tell us how to construct $T_pM$. We denote by $\mathfrak C(M,p)$ the set of all charts $\phi : U \to U'$ on $M$ such that $p \in U$.
Let us begin by cumulating the vector spaces $T_{\phi(p)}U' =\mathbb R^m$ for all $\phi \in \mathfrak C(M,p)$. Formally this produces the set $$\mathbb T_pM = \mathbb R^m \times \mathfrak C(M,p) .$$ The vector spaces $\mathbb R^m \times \{\phi\} \equiv \mathbb R^m = T_pU'_\phi$ and $\mathbb R^m \times \{\psi\} \equiv \mathbb R^m = T_pU'_\psi$ for $\phi,\psi \in \mathfrak C(M,p)$ will be identified non-trivially via the isomorphism $d(\psi \circ \phi^{-1})_{\phi(p)} : \mathbb R^m \to \mathbb R^m$. That is, on $\mathbb T_pM$ we define an equivalence relation by $(v,\phi) \sim (w,\psi)$ if $w = d(\psi \circ \phi^{-1})_{\phi(p)}(v)$. Now let $$T_pM = \mathbb T_pM/\sim. $$ The equivalence classes with respect to $\sim$ are subsets of $\mathbb T_pM$ containing precisely one element of each $\mathbb R^m \times \{\phi\}$. Thus each inclusion $i_\phi : \mathbb R^m \to \mathbb T_pM, i_\phi(v) = (v,\phi)$, induces a bijection $\iota_\phi : \mathbb R^m = T_pU' \to T_pM, \iota_\phi(v) = [v,\phi]$. By construction $$\iota_\psi \circ d(\psi \circ \phi^{-1})_{\phi(p)} = \iota_\phi \tag{$*$}$$ for all $\phi,\psi \in \mathfrak C(M,p)$.
Each bijection $\iota_\phi$ induces a unique vector space structure on $T_pM$ making $\iota_\phi$ an isomorphism and because or $(*)$ all these vector space structures on $T_pM$ agree.
Let us now check the above requirements.
Each open $M \subset \mathbb R^m$ has $id_M$ as a canonical chart, thus $\iota_{id_M} : \mathbb R^m \to T_pM$ is a canonical isomorphism which allows to identify $T_pM \equiv \mathbb R^m$.
Let $U \subset M$ be an open neighborhood of $p \in M$. Then $\mathfrak C(U,p) \subset \mathfrak C(M,p)$ and thus by definition we get a canonical isomorphism $j_U : T_pU = (\mathbb T_pU/\sim) \to (\mathbb T_pM/\sim) = T_pM$.
Given $\phi \in \mathfrak C(M,p)$, we take $d\phi_p = \iota_\phi^{-1} : T_pM \to \mathbb R^m$.
This is now obvious; it is a reformulation of $(*)$.
Let us remark that our definition of $T_pM$ is a special case of a direct limit construction.
What is the relation to the "derivatives of curves" approach in your question?
To each curve $u : J \to M$ through $p$ we can associate the equivalence class $$\theta(u) = [(\phi\circ u)'(0),\phi] \in T_p M $$ where $\phi \in \mathfrak C(M,p)$ is arbitrary. Its representative in the $\mathbb R^m \times \{\psi\}$-subset of $\mathbb T_pM$ is $((\psi\circ u)'(0),\psi)$. Thus the vector $u'(0) = \theta(u)$ subsumes all "localized derivatives" of $u$.
Clearly $\theta(u) = \theta(v)$ iff $u$ and $v$ are equivalent in the sense of your question. Thus we get a bijection between the set of equivalence classes of curves through $p$ and our $T_pM$. In other words, our $T_p M$ is indeed a variant of the tangent space defined by equivalence classes of curves through $p$, but the vector $u'(0) = \theta(u)$ is certainly closer to intuition than $[u]$.
The above definition of $T_pM$ also nicely shows how to endow the tangent bundle $$T M = \bigcup_{p \in M} \{p\} \times T_p M$$ with a topology and a smooth structure. In fact, for each $\phi \in \mathfrak C(M,p)$ we get a canonical fiber-preserving bijection $$\tau_\phi : \bigcup_{p \in U} \{p\} \times T_p M \to U \times \mathbb R^m, \tau_\phi(p,v) = (p,d\phi_p(v)) .$$
If $f : M \to N$ is smooth, we have the localized derivatives $d_{(\phi,\psi)} f_p = d(\psi \circ f \circ \phi^{-1})_{\phi(p)}$. Define $$df_p = d\psi_{f(p)}^{-1} \circ d_{(\phi,\psi)}f_p \circ d\phi_p : T_pM \to T_{f(p)}N .$$
It is easy to verify that $df_p$ does not depend on the charts $\phi, \psi$. By definition the diagrams
$\require{AMScd}$ \begin{CD} T_pM @>{df_p}>> T_{f(p)}N \\ @V{d\phi_p}VV @V{d\psi_{f(p)}}VV \\ \mathbb R^m @>{d_{(\phi,\psi)} f_p}>> \mathbb R^m \end{CD} commute which nicely shows that the localized derivatives of $f$ at $p$ are not that bad as they appear at first glance. In fact, $df_p$ subsumes all these maps.
For a curve $u : J \to M$ through $p$ we get the derivative $$du_0 : \mathbb R = T_0J \to T_pM .$$ By construction we have $du_0(1) = u'(0)$.
What about the isomorphism $d\phi_p : T_pM \to \mathbb R^m = T_{\phi(p)} U'$ which was defined above in a very formal way to identify $T_pM$ with $\mathbb R^m$? We have a commutative diagram $\require{AMScd}$ \begin{CD} T_pM =T_pU @>{d\phi_p}>> T_{\phi(p)}U' \\ @V{d\phi_p}VV @V{d(id_{U'})_{\phi(p)}}VV \\ \mathbb R^m @>{d_{(\phi,id_{U'})} \phi_p}>> \mathbb R^m \end{CD} where the horizontal $d\phi_p$ is the derivative of $\phi$ at $p$ and the vertical $d\phi_p$ is our formal isomorphism. But $$d_{(\phi,id_{U'})} \phi_p= d(id_{U'} \circ \phi \circ \phi^{-1})_{\phi(p)} = id$$
which shows that both interpretations of $d\phi_p$ are identical.
Finally, let us consider smooth maps $f : (M,p) \to (N,q)$ from $M$ to $N$ mapping a fixed $p \in M$ to a fixed $q \in N$. The function $\theta$ assigning to $f$ its derivative $df_p : T_pM \to T_qN$ is easily seen to be a surjection from the set of all smooth maps $(M,p) \to (N,q)$ to the vector space of all linear maps $T_pM \to T_qN$. Clearly $\theta(f) = \theta(g)$ defines an equivalence relation $f \sim g$ for smooth maps $(M,p) \to (N,q)$. As for tangent vectors one can therefore identify $df_p$ with the equivalence class $[f]$ of $f$ which results in an alternative definition of the derivative.
This is precisely what is done for the cotangent space $T^*_pM$: The derivative $df_p : T_pM \to T_{f(p)}\mathbb R = \mathbb R$ is interpreted as the equivalence class $[f]$.
Remark 1:
Moishe Kohan comments that the above construction of $T_pM$ is closely related to an alternative description of the tangent bundle $TM$. This can be obtained by gluing all trivial bundles $B_\phi = U \times \mathbb R^m$ over the domains of charts $\phi : U \to U'$ via the transition maps of the fibers. Technically this is done via the maps $\tau_{(\phi,\psi)}: U \cap V \to GL(\mathbb R^m),\tau(p) = d(\psi \circ \phi^{-1})_{\phi(p)}$. The fiber $T_pM$ of the "glued bundle" $TM$ is thus obtained by gluing the collection of all $\{p\} \times \mathbb R^m_\phi = \{p\} \times \mathbb R^m$ with $p$ in the domain of $\phi$ via the transition maps $d(\psi \circ \phi^{-1})_{\phi(p)}$ (note that $p$ is always in the common part $U \cap V$ of all charts $\phi, \psi$ around $p$). This is in fact nothing else than the above construction.
Remark 2:
One more alternative construction of $T_pM$ goes as follows. Consider the vector space $\mathbf T_pM = (\mathbb R^m)^{\mathfrak C(M,p)} = \prod_{\phi \in \mathfrak C(M,p)} \mathbb R^m$ and define $$T_pM =\{(v_ϕ)∈\mathbf T_pM \mid d(\psi \circ \phi^{-1})_{\phi(p)}(v_ϕ)=v_ψ \text{ for all } ϕ,ψ \in \mathfrak C(M,p)\}.$$ It is easy to check that is a linear subspace of $\mathbf T_pM$. The product projections $\pi_\phi: \mathbf T_pM \to \mathbb R^m$ restrict to linear maps $d\phi_p : T_pM \to \mathbb R^m$ which are easily seen to be isomorphisms. Verification of 1. - 4. is straightforward.
Let us remark that this definition of $T_pM$ is a special case of an inverse limit construction.