# Equivalent notions of tangent spaces and the differential of a smooth map using different definitions of manifold

This is a follow up to this question, in which the context is provided. I would like to see the equivalence of two notions of tangent spaces and differentials of smooth maps.

Guillemin and Pollack define the tangent space as follows. Let $$\phi: U \subseteq \mathbb{R}^n \to M\subseteq \mathbb{R}^N$$ be a coordinate map for the embedded submanifold $$M$$ s.t.\ $$0 \in U$$ and $$\phi(0) = p$$. Then $$D\phi|_0$$ is the derivative of $$\phi$$ at $$0$$. Then let $$T_pM := D\phi|_0( \mathbb{R}^n)$$ i.e. a certain $$n$$ dimensional vector subspace of $$\mathbb{R}^N$$. It is then proved that this definition is independent of $$\phi$$.

Tu defines the tangent space of the vector space of derivations of germs of $$C^\infty$$ functions at a point $$p$$.

How do these definitions correspond?

Under this correspondence, we should get a correspondence between notions of the differential of a smooth map. Let $$M, N \subseteq \mathbb{R}^N$$ be $$m, n$$ resp. dimensional embedded submanifolds. Let $$f: M \to N$$ be a smooth map between the two manifolds. From this we want to construct a vector space homomorphism from $$T_pM \to T_{f(p)}N$$ for any $$p \in M$$. Tu and G&P do this in two different ways.

G&P define $$df_p$$ as follows. Let $$\phi: U \to M$$ and $$\psi: V \to N$$, where $$U,V$$ are open subsets of $$\mathbb{R}^m, \mathbb{R}^n$$ resp., be coordinate maps centered at $$p$$ and $$f(p)$$ resp. Then let $$h = \psi^{-1} \circ f \circ \phi: U \cap f^{-1}(\psi(V)) \to \mathbb{R}^n$$. Then $$h$$ is smooth and we define $$df_p = d\psi_0 \circ dh_0 \circ d\phi^{-1}_0$$, where $$d\psi_0$$ is just the total derivative of $$\psi$$ at $$0$$. It is then verified this definition is in fact coordinate independent.

Tu defines the differential of $$f$$ by defining its action on $$C^\infty$$ real-valued functions (as this will uniquely determine a derivation): $$f_*(v)(g) =v(g\circ f)$$.

How do these definitions correspond?

• The natural candidate for associating a derivation to a vector in G&P's definition of a tangent space is taking the directional derivative along that vector. This only makes sense though if smooth functions from the manifold to $\mathbb{R}$ can be extended to smooth functions on an neighborhood of any point. This is intrinsically possible in G&P's definition of smooth though it is not clearly possible in Tu's definition (this is part of the linked question). Jan 17 '20 at 0:46
• Though for this to work, first we have to show we get a well defined derivation that is independent of choice of extension. Then we would have to verify it actually gives a vector space isomorphism. Jan 17 '20 at 0:49