TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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Curve contact classes are canonically isomorphic to derivation tangent vectors

Statement

At each point $p$ of a smooth manifold, the set of contact classes of smooth curves through $p$ is canonically isomorphic to the tangent space $T_{p} M$ of derivations.

Facts & Assumptions

Given: A smooth manifold point $p$ .

[L1]

A contact class has a well-defined velocity derivation (Curve velocity depends only on the contact class).

[L2]

Coordinate derivations form a basis of $T_{p} M$ (Coordinate derivations form a basis of the tangent space).

Proof

technique · direct

1.1

By [L1], sending a contact class $[γ]$ to its velocity derivation $\dot{γ} (0)$ defines a map from curve classes to $T_{p} M$ .

L1given

2.1

Choose a chart $(U, x)$ around $p$ . If $v = \sum_{i} a_{i} \partial_{x^{i}} ∣_{p} \in T_{p} M$ , define a curve in the chart by $t \mapsto x (p) + t (a_{1}, \dots, a_{n})$ and transport it back by $x^{- 1}$ . Its contact class maps to $v$ , so the map of step 1.1 is surjective by [L2].

L2step 1.1construct

2.2

If two curve classes have the same velocity derivation, then they have the same values on each coordinate germ $[x^{i}]$ , so their coordinate velocity vectors agree; hence the classes are equal. Thus the map is injective.

L2step 1.1

3.1

The map of step 1.1 is therefore a canonical bijection between curve contact classes and derivation tangent vectors.

step 2.1step 2.2∎

Depends on

Used by

Every tangent vector is the velocity of a smooth curve Corollary
The tangent bundle of the circle is a cylinder Example
The tangent space of Euclidean space Example
The tangent space of the sphere from curve velocities Example

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)