Tangent map

The tangent map is a generalization of the total derivative to an arbitrary differentiable manifold. #m/def/geo/diff See also Differential pushforward.

Tangent map on tangent spaces

Real embedded manifold

All three of the following characterizations of tangent space maps on real embedded manifolds are useful. Compare with the different definitions of the tangent space.

Fixed chart characterization

Let $𝑋 \subseteq ℝ 𝑁$ and $𝑌 \subseteq ℝ 𝑀$ be real embedded manifolds and $𝑓 : 𝑋 \to 𝑌$ be a $𝐶 \infty$ differentiable function with $𝑓 : 𝑥 \mapsto 𝑦$ and take local parameterizations $𝜓, ˜ 𝜓$ about $𝑥$ and $𝑦$ respectively so that the following diagrams commute in $𝖬 𝖺 𝗇 \infty$ , $𝖬 𝖺 𝗇 \infty$ , and $𝖵 𝖾 𝖼 𝗍 ℝ$ respectively.

Then $𝑇 𝑥 𝑓 : 𝑇 𝑥 𝑋 \to 𝑇 𝑦 𝑌$ is the tangent space map of $𝑓$ at $𝑥$ .

Chart-free characterization

Let $𝑋 \subseteq ℝ 𝑁$ and $𝑌 \subseteq ℝ 𝑀$ be real embedded manifolds and $𝑓 : 𝑋 \to 𝑌$ be a $𝐶 \infty$ differentiable function with $𝑓 : 𝑥 \mapsto 𝑦$ . Then the tangent space map $𝑇 𝑥 𝑓 : 𝑇 𝑥 𝑋 \to 𝑇 𝑦 𝑌$ of $𝑓$ at $𝑥$ is defined such that

𝑇 𝑥 𝑓 (˙ 𝜔 (0)) = 𝐷 [𝑓 𝜔] (0)

for any $𝐶 \infty$ path $𝜔 : (- 𝜖, 𝜖) \to 𝑋$ with $𝜔 (0) = 𝑥$ .

Fixed extension characterization

Let $𝑋 \subseteq ℝ 𝑁$ and $𝑌 \subseteq ℝ 𝑀$ be real embedded manifolds and let $𝑓 : 𝑋 \to 𝑌$ be a $𝐶 \infty$ differentiable function with $𝑓 : 𝑥 \mapsto 𝑦$ . with $𝐶 \infty$ extension $𝐹 : 𝑈 \to ℝ 𝑚$ for some open neighbourhood $𝑈$ of $𝑥$ in $ℝ 𝑁$ .

Let $𝐷 𝐹 (𝑥) : ℝ 𝑁 \to ℝ 𝑀$ be the total derivative of $𝐹$ at $𝑥$ . Then $𝑇 𝑥 𝑓 = 𝐷 𝐹 (𝑥) ↾ 𝑇 𝑥 𝑋 : 𝑇 𝑥 𝑋 \to 𝑇 𝑦 𝑌$ is the tangent space map of $𝑓$ at $𝑥$ .

Together these definitions firmly establish that the differential tangent space map exists, is independent from any choice of chart or extension, and is a linear map between tangent spaces.

Equivalence of characterizations

Let $⃗ 𝐚 \in 𝑇 𝑥 𝑋$ . Then $⃗ 𝐚 = 𝐷 𝜔 (0)$ for some $𝐶 \infty$ path $𝜔 : (- 𝜖, 𝜖) \to 𝑋$ with $𝜔 (0) = 𝑥$ . The fixed-extension characterization gives

𝑇 𝑥 𝑓 ⃗ 𝐚 = 𝐷 𝐹 (𝑥) ⃗ 𝐚 = 𝐷 𝐹 (𝑥) 𝐷 𝜔 (0) = 𝐷 [𝐹 𝜔] (0) = 𝐷 [𝑓 𝜔] (0)

which matches the chart-free characterization where we have used the chain rule for the total derivative and the fact $𝐹 𝜔 = 𝑓 𝜔$ . Now consider both fixed charts with a compatible fixed extension, so that the following diagram commutes

Note that since $Φ 𝜓 = i d 𝑉$ , it follows $𝐷 Φ (𝑥) 𝐷 𝜓 (𝑣) = 𝐷 [Φ 𝜓] (𝑣) = 𝟙$ , so $𝐷 Φ (𝑥) ↾ 𝑇 𝑥 𝑋 = 𝐷 𝜓 (𝑣) - 1$ . Thus the following diagram commutes

and the fixed chart characterization concurs with the fixed extension characterization.

Tangent map on tangent bundles

#to/complete

Properties

Let $𝑓 : 𝑋 \to 𝑌 : 𝑥 \mapsto 𝑦$ and $𝑔 : 𝑌 \to 𝑍 : 𝑦 \mapsto 𝑧$ be $𝐶 \infty$ differentiable maps between $𝐶 \infty$ differentiable manifolds of dimensions $𝑛, 𝑚, 𝑘$ respectively.

Chain rule: $𝑇 𝑥 [𝑔 𝑓] = 𝑇 𝑦 𝑔 𝑇 𝑥 𝑓$

Proof for real embedded manifolds

Take the fixed chart characterization so that

It follows from the chain rule for the total derivative that the following diagram commutes.

as required.

Even more straightforwardly, taking the chart-free characterization

𝑇 𝑥 [𝑓 𝑔] 𝐷 𝜔 (0) = 𝐷 [𝑓 𝑔 𝜔] (0) = 𝑇 𝑦 𝐷 [𝑔 𝜔] (0) = 𝑇 𝑦 𝑓 𝑇 𝑥 𝑔 𝐷 𝜔 (0)