# Hitchin's definition of tangent space and tangent vectors

In page 18 of N. Hitchin the tangent space $$T_pM$$ of a manifold $$M$$ in a point $$p$$ is defined as the dual of $$T^\star_p M$$ where $$T^\star_pM$$ is the quotient space: $$C^\infty(M)/Z_p(M) \quad\text{and}\, Z_p(M)=\big\{f\in C^\infty(M): d(f\circ \varphi^{-1 })_p=0 \text{ for all }(\mathcal U,\varphi) \textrm{ local chart in }p\big\}$$ A tangent vector at $$p$$ is not defined as an element of $$T_pM$$. Instead it is defined as a linear map $$X_p:C^\infty(M)\to \mathbb R$$ that satisfies the Leibniz rule: $$X_p(fg)=X_p(f)g(p)+f(p)X_p(g)$$ Then Hitchin proves that $$T_pM$$ is isomorphic to the annihilator $$Z_p(M)^\circ$$ of $$Z_p(M)$$ in $$C^\infty(M)$$ and each tangent vector belongs to $$Z_p(M)^\circ$$.

1.- Why a tangent vector cannot be defined simply as an element of $$T_pM$$?

2.- Are there elements of $$T_pM$$ that do not satisfy the Leibniz Rule?

• (0) it is not on page 12. (1) it is just one of the four or so equivalent ways to do this. (2) No. Aug 22 '21 at 9:11
• Here's the approach you want to take. (1) Let $(x_1,\ldots,x_d)=\varphi$. Prove $\mathrm{d}x_i$ for $i=1,\ldots,d$ spans $T_p^\ast M$. (2) Argue the dual $T_pM$ is therefore $d$ dimensional and spanned by $\partial/\partial x_i$ for $i=1,\ldots,d$. (3) Argue that all elements of $T_pM$ satisfy the Leibniz rule by showing the basis vectors do.
– zzz
Aug 22 '21 at 10:23

I think Hitchin's exposition may be a little bit confusing. You are absolutely right, Hitchin begins by defining the tangent space $$T_pM$$ in a certain way (as the dual $$(T^*_pM)^*$$of $$T^\star_p M$$) and therefore a tangent vector should be defined to be an element of $$T_pM$$.

However, he admits that the above definition of $$T_pM$$ is somewhat counterintuitive. Quotation:

This is admittedly a roundabout way of defining $$T_pM$$ [...] This definition at first sight seems far away from our intuition about the tangent space to a surface in $$\mathbb R^3$$ [...] The problem arises because our manifold $$M$$ does not necessarily sit in Euclidean space and we have to define a tangent space intrinsically. There are two ways around this: one would be to consider functions $$f : \mathbb R \to M$$ and equivalence classes of these, instead of functions the other way $$f : M \to \mathbb R$$. Another, perhaps more useful, one is provided by the notion of directional derivative.

This "more useful" approach is made precise in Definition 10. Note that a linear map $$X_p : C^\infty(M)\to \mathbb R$$ satisfying the Leibniz rule is usually called a derivation at $$p$$.

But what is the relation between the elements of $$T_pM$$ and tangent vectors understood as derivations?

Formally the elements of $$T_pM$$ are linear maps $$\xi_p : T^\star_p M = C^\infty(M)/Z_p(M) \to \mathbb R$$. Composing with the (linear) quotient map $$\pi_p : C^\infty(M) \to C^\infty(M)/Z_p(M)$$ gives linear maps $$\xi'_p = \xi_p\pi_p: C^\infty(M) \to \mathbb R$$. In other words, $$\pi_p$$ induces a function $$\pi^*_p : (T^\star_p M)^* \to C^\infty(M)^* .$$ Here $$C^\infty(M)^*$$ denotes the dual space of $$C^\infty(M)$$. It is easily verified that $$\pi^*_p$$ is linear. It is an injection because $$\pi_p$$ is a surjection. Therefore $$\pi^*_p$$ identifies $$T_p M$$ with its image in $$C^\infty(M)^*$$ which is a linear subspace of $$C^\infty(M)^*$$.

Hitchin proves that the image of $$\pi^*_p$$ is nothing else than the set of derivations at $$p$$. And that is the whole "secret": The elements of $$T_pM$$ can canonically be identified with the derivations at $$p$$. Yes, formally the elements of $$T_pM$$ are no derivations, but it is common practise not to distinguish between these formally distinct maps. In that sense the elements of $$T_pM$$ are derivations at $$p$$.

Update:

I would have preferred to define first the tangent space $$T_pM$$ via derivations at $$p$$ and then the cotangent space $$T^*_pM = (T_pM)^*$$ as the dual of the tangent space. It would then have been a theorem that $$T^*_pM$$ can be identified with $$C^\infty(M)/Z_p(M)$$ which is the set of equivalence classes of smooth functions $$M \to \mathbb R$$, where functions are equivalent if the have the same derivative at $$p$$.