Tangent space

The tangent space $𝑇 𝑝 𝑋$ of a differentiable manifold $𝑋$ at a point $𝑝 \in 𝑋$ is a vector space corresponding to possible velocities when moving through $𝑥$ . #m/def/geo/diff A number of equivalent characterizations are useful. See also Tangent map, Tangent bundle.

Intrinsic manifold

The following characterizations of $𝑇 𝑝 𝑀$ are all useful.

As derivations at a point

Let $𝑝 \in 𝑀$ , and suppose $𝔛 (𝑀)$ is the set of vector fields viewed as derivations. We define the tangent space $𝑇 𝑝 𝑋 \subseteq ℝ 𝐶 𝛼 (𝑀)$ as the image of the map

𝔛 (𝑀) ↠ 𝑇 𝑝 𝑀 𝑣 \mapsto 𝑣 𝑝,

i.e. the set of all derivations evaluated at $𝑝$ .

Chart-free characterization as velocities

Let $𝑝 \in 𝑀$ , $𝑥 \in 𝒜$ be a chart at $𝑝$ , and

Θ = 𝖬 𝖺 𝗇 𝛼 • (((- 𝜖, 𝜖), 0), (𝑋, 𝑥))

be the set of all $𝐶 𝛼$ paths $𝜔 : (- 1, 1) \to 𝑋$ such that $𝜔 (0) = 𝑝$ . We define an equivalence relation $(\sim)$ on $Θ$ so that two paths $𝜔, 𝜗 \in Θ$ are equivalent iff

(𝑥 \circ 𝜔)' (0) = (𝑥 \circ 𝜗)' (0)

which is easily shown to be independent of choice of $𝑥$ .

Via the cotangent space

The cotangent space $𝑇 * 𝑝 𝑀$ admits an intrinsic characterization, which is applicable in other (non-differentiable) settings. The tangent space is simply the dual vector space $(𝑇 * 𝑝 𝑀) *$ .

Equivalence of characterizations

#missing/proof

Real embedded manifold

Both the following characterizations of the tangent space of a real embedded manifold is useful.

Fixed chart characterization

Let $𝑋 \subseteq ℝ 𝑁$ be a Real embedded manifold and $𝑥 \in 𝑋$ . Let

𝜑 - 1 : 𝑉 \subseteq ℝ 𝑛 \to 𝑈 \subseteq 𝑋 \subseteq ℝ 𝑁 𝑣 \mapsto 𝑥

be a local parameterization at $𝑥$ , and $𝐷 𝜑 - 1 (𝑣) : ℝ 𝑛 \to ℝ 𝑁$ be its Total derivative. Then $𝑇 𝑥 𝑋 = 𝐷 𝜑 - 1 (𝑣) (ℝ 𝑛)$ is the tangent space at $𝑥$ .

Chart-free characterization as velocities

Let $𝑋 \subseteq ℝ 𝑁$ be a Real embedded manifold and $𝑥 \in 𝑋$ . Let

Ω 𝜖 = 𝖬 𝖺 𝗇 \infty • (((- 𝜖, 𝜖), 0), (𝑋, 𝑥))

be the set of all $𝐶 \infty$ differentiable paths $𝜔 : (- 𝜖, 𝜖) \to 𝑋$ such that $𝜔 (0) = 𝑥$ . Then the set of all “velocities at $𝑥$ ”

𝑇 𝑥 𝑋 = {˙ 𝜔 (0) : 𝜔 \in Ω 𝜖, 𝜖 > 0}

is the tangent space at $𝑥$ .

The primary advantage of the fixed chart characterization is that its vector space status is clear, whereas the chart-free characterization is more intuitive and establishes chart-independence.

Equivalence of characterizations

Take a coördinate chart $𝜑 : 𝑈 \to 𝑉 \subseteq ℝ 𝑛$ with $𝜑 (𝑥) = 𝑣$ . Let $𝑥 \in 𝑋$ and let $𝑇 𝑥 𝑋$ denote the fixed-chart characterization and $˜ 𝑇 𝑥 𝑋$ denote the chart-free characterization.

Let $˙ 𝜔 (0) \in ˜ 𝑇 𝑥 𝑋$ for some $𝐶 \infty$ differentiable $𝜔 : (- 𝜖, 𝜖) \to 𝑈$ with $𝜔 (0) = 𝑥$ . Then

𝐷 𝜔 (0) = 𝐷 [𝜑 - 1 𝜑 𝜔] (0) = 𝐷 𝜑 - 1 (𝑣) 𝐷 [𝜑 𝜔] (0) \in 𝑇 𝑥 𝑋

Herefore $˜ 𝑇 𝑥 𝑋 \subseteq 𝑇 𝑥 𝑋$ .

Now let $𝐷 𝜑 - 1 (𝑣) (⃗ 𝐚) \in 𝑇 𝑥 𝑋$ for some $⃗ 𝐚 \in ℝ 𝑛$ . Define

˜ 𝜔 : (- 𝜖, 𝜖) \to ℝ 𝑛 𝑡 \mapsto 𝑣 + 𝑡 ⃗ 𝐚

and let $𝜔 = 𝜑 - 1 ˜ 𝜔 : (- 𝜖, 𝜖) \to 𝑈$ (we choose $𝜖$ so that $𝜔$ remains in $𝑈$ ). It follows $˙ 𝜔 (0) = 𝐷 [𝜑 - 1 ˜ 𝜔] (0) = 𝐷 𝜑 - 1 (𝑣) (⃗ 𝐚)$ and $˙ 𝜔 (0) \in ˜ 𝑇 𝑥 𝑋$ . Herefore $𝑇 𝑥 𝑋 \subseteq ˜ 𝑇 𝑥 𝑋$ .

Thus $𝑇 𝑥 𝑋 = ˜ 𝑇 𝑥 𝑋$ .

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