The tangent space and differentials

 Preliminary exam prep

Let $M,N$ be smooth $n$-manifolds. Here we discuss different definitions of the tangent space and differentials, or pushforwards, of smooth maps $f:M\to N$.

Derivations (Lee)

Definition: A derivation of $M$ at $p\in M$ is a linear map $v:C^\infty(M)\to \R$ such that for all $f,g\in C^\infty(M)$,
$$v(fg) = f(p)v(g) + g(p)v(f).$$
The tangent space $T_pM$ to $M$ at $p$ is the set of all derivations of $M$ at $p$.

Given a smooth map $F:M\to N$ and $p\in M$, define the differential $dF_p:T_pM\to T_{f(p)}N$, which, for $v\in T_pM$ and $f\in C^\infty(N)$ acts as
$$dF_p(v)(f) = v(f\circ F)\in \R.$$

Dual of cotangent (Hitchin)

Definition: Let $Z_p\subset C^\infty(M)$ be the functions whose derivative vanishes at $p\in M$. The cotangent space $T_p^*M$ to $M$ at $P$ is the quotient space $C^\infty(M)/Z_p$. The tangent space to $M$ at $P$ is the dual of the cotangent space $T_pM = (T_p^*M)^* = \Hom(T_p^*M,\R)$.

Given a smooth map $F:M\to N$ and $p\in M$, define the differential
$$\begin{array}{r c l} dF_p\ :\ T_pM & \to & T_{F(p)}N, \\ \left(f:C^\infty(M)/Z_p \to \R\right) & \mapsto & \left(\begin{array}{r c l} g\ :\ C^\infty(N)/Z_{F(p)} & \to & \R, \\ h & \mapsto & f(h\circ F). \end{array}\right) \end{array}$$
This definition makes clear the relation to the first approach. Since $h\not\in Z_{F(p)}$, the derivative of $h$ does not vanish at $F(p)$. Hence the derivative of $h\circ F$ at $p$, which is the derivative of $h$ at $F(p)$ multiplied by the derivative of $F$ at $p$, does not a priori vanish at $p$.

Derivative of chart map (Guillemin and Pollack)

Definition: Let $f:\R^n\to \R^m$ be a smooth map. Then the derivative of $f$ at $x\in \R^n$ in the direction $y\in \R^n$ is defined as
$$df_x(y) = \lim_{h\to 0}\left[\frac{f(x+yh)-f(x)}h\right].$$
Given $x\in M$ and charts $\varphi:\R^n\to M\subset \R^m$, the tangent space to $M$ at $p$ is the image $T_pM = d\varphi_0(\R^n)$, where we assume $\varphi(0)=p$.

Given a smooth map $F:M\to N$ and charts $\varphi:\R^n\to M$, $\psi:\R^n\to N$, with $\varphi(0)=p$ and $\psi(0)=F(p)$, define the differential $dF_p:T_pM\to T_{F(p)}N$ via the diagrams below.

Here $h = \psi^{-1}\circ F\circ \varphi$, so $dh_0$ is well-defined. Hence $dF_p = d\psi_0\circ dh_0\circ d\varphi_0^{-1}$ is also well-defined.

Sometimes the differential is referred to as the pushforward, in which case it is denoted by $(F_*)_p$.

References: Lee (Introduction to Smooth Manifolds, Chapter 3), Hitchin (Differentiable manifolds, Chapter 3.2), Guillemin and Pollack (Differential topology, Chapter 1.2)