Cotangent space using various definition of tangent space $T^{\text{glue}}_pM$, $T^{\text{paths}}_pM$ and $T^{\text{der}}_pM$

Question

There are several ways we can define tangent space at point $p$ on a manifold $M$.

Definition-1: $$T^{\text{glue}}M=\sqcup_i(U_i\times\mathbb R^n)/\sim$$ where for $(x,v)\in U_i\times\mathbb R^n$, $(y,w)\in U_j\times\mathbb R^n$ we have $(x,v)\sim(y,w)$ if and only if $y=\phi_j\phi_i^{-1}(x)$ and $w=d(\phi_j\phi_i^{-1})_x(v)$. We define the tangent space at $p\in M$ as $T^{\text{glue}}_pM=\{[p,v]:v\in\mathbb R^n\}$ and $\pi:T^{\text{glue}}M\rightarrow M,\pi([p,v])=p$.

Definition-2: $$T^{\text{path}}_pM=\{\text{paths }\gamma:(-\epsilon,\epsilon)\rightarrow M:\gamma(0)=p\}/\sim$$ where $\alpha\sim\beta$ if $(\phi_i\circ\alpha)'(0)=(\phi_i\circ\beta)'(0)$ for every $i$ such that $p\in U_i$. $T^{\text{path}}M=\sqcup_{p\in M} T^{\text{path}}_pM$ and $\pi:T^{\text{path}}M\rightarrow M,\pi(\gamma)=\gamma(0)$.

Definition-3: A derivation of germs is an $\mathbb R$-linear map $X:\operatorname{Germs}_p\rightarrow \mathbb R$ which satisfies, $$X(fg)=f(p)X(g)+X(f)g(p)$$ We define $T^{\text{der}}_p M$ to be the set of derivations of germs at $p$.

Maybe I can show the equivalence of Definition 1 and 2 using,

$$ \begin{align*} \Phi:T^{\text{path}}_p M &\rightarrow T^{\text{glue}}_p M\\ [\gamma]&\mapsto [\phi_i\circ\gamma(0),(\phi_i\circ\gamma)'(0)] \end{align*} $$

Question 1: How to show the equivalence for derivation definition with other ones?

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So far, I understand all the definitions of the tangent space. But I was wondering how to define the cotangent space in the context of each definition. Most the resource which I follow define the cotangent space as the dual vector space of the tangent space, like in the Differential Geometry by Loring W.Tu, differential 1-from was defined as,

A $1$-form on $M$ is the assignment of a linear function $\omega_p:T_p M\rightarrow \mathbb R$ to each $p$ in $M$.

I got another weird definition of $1$-form (Though it was for holomorphic $(1,0)$-form, source: 4.3.2, Riemann surfaces by Dror Varolin, I just replace holomorphic term with differentiable).

If $f_i:U_i\rightarrow\mathbb R$ are differentiable functions on domain $U_i$ containing $p$, we say that $(f_1,U_1)$ and $(f_2,U_2)$ define the same germ if there is an open subset $U\subset U_1\cap U_2$ such that $f_1\mid_U=f_2\mid_U$. The equivalence class of all $(f,U)$ that define the same germs is denoted by $[f]_p$ and is called a germ of a differentiable function. The set of all germs of differential functions is denoted by $\mathcal O_{M,p}$. I (guess I) knew that it is the sheaf of differentiable function at point $p$. Now they define the $1$-form as,

Two germs $[f]_p=(f,U)$ and $[g]_p=(g,V)$ are said to be cotangent, written $[f]_p\sim_1 [g]_p$, if there is a coordinate chart $\varphi:W\rightarrow\mathbb R$ such that $p\in W\subset U\cap V$ and $$(f\circ\varphi^{-1})'(\varphi(p))=(g\circ\varphi^{-1})'(\varphi(p))$$
The set of equivalence classes of mutually tangent germs is denoted by $K_{M,p}$, and the elements of $K_{M,p}$ are called $1$-form. The equivalence class of germs tangent to $[f]_p$ is denoted by $df(p)$.

Question 2: I guess I couldn't relate to this definition or failed to understand its motivation. How is this definition related to the covector?

I was searching for the definition of cotangent space in the context of Definition-1,2 and 3 given for tangent space above. Because I want to know how the dualization idea arises for each case, another reason is, that we can show the equivalency of each definition for tangent space, I want to see the same thing for cotangent space $(T^{\text{glue}}M)^*$, $(T^{\text{paths}}M)^*$ and $(T^{\text{der}}M)^*$.

Question 3: How do we define the duality for each definition mentioned for tangent space above?

Any resource/help will be appreciated as I was struggling with these too much. Though I restricted everything to smooth manifold settings, if anyone enlightens me about complex manifolds, you are welcome.

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I understand my question 2, hence remove it. Hence, question-3 is the only one which I want to resolve.

For question-3, I guess couldn't come up with natural ideas in the following places:

• $T^{\text{glue}*}_pM=\{\omega_p:T^{\text{glue}}_pM\rightarrow\mathbb R\}$, then for any $[p,v]\in T^{\text{glue}}_pM$, $\omega_p([p,v])=?$
• $T^{\text{path}*}_pM=\{\omega_p:T^{\text{path}}_pM\rightarrow\mathbb R\}$, then for any $[\gamma]\in T^{\text{path}}_pM$, $\omega_p([\gamma])=?$
• $T^{\text{der}*}_pM=\{\omega_p:T^{\text{der}}_pM\rightarrow\mathbb R\}$, then for any $X\in T^{\text{der}*}_pM$, $\omega_p(X)=?$

Okay, after getting the dual vector space how to glue them together to construct the cotangent bundle? As the definition 1,2,3 we have some sort of equivalence relation which helps us for gluing. But now, I don't see something naturally arising from these information.

Answer 1

Here is a terse take on this:

The definition of $T^{\operatorname{glue}}_p$ is based on the fact that local coordinates $(x^1, \dots, x^n)$ induce a basis $(\partial_1, \dots, \partial_n)$ of $T^{\operatorname{glue}}_p$. This in turn induces a dual coordinate basis of the cotangent space, defined as the dual space.

Using the path definition, I would start with a smooth function $f$ and observe that it defines a linear function $df$ on $T^{\operatorname{path}}$ given by $$ df([\gamma]) = (f\circ \gamma)'(0) $$ and define the cotangent space to be the set of equivalence classes. This is essentially the same as the definition $\mathfrak{m}/\mathfrak{m}^2$. This approach also works with $T^{\operatorname{der}}_pM$.

I would suggest that if you are introducing these concepts from scratch to use the definition of a tangent vector as a velocity vector of a curve, observe that the directional derivative of a function is a linear function of the velocity vector, use this to define the differential of a function at a point, and finally define the cotangent space is the space of all possible differentials of functions. I suggest avoding the other two approaches. The derivation approach is an elegant but somewhat opaque way to define the tangent space rigorously. The local trivialization via coordinates approach is messy and confusing.