Manifolds 21 | Tangent Space (Definition via tangent curves)

Tangent spaces for abstract manifolds can be defined without any ambient Euclidean space by using derivatives of curves—then identifying curves that share the same first-order behavior in local coordinates. The key move is to replace “differentiate a curve inside R^n” with “differentiate the coordinate representation of a curve,” so tangent vectors become equivalence classes of tangent curves rather than literal vectors in some surrounding space.

For an embedded submanifold M ⊂ R^n, tangent vectors come from the derivative γ′(0) of a smooth curve γ in M passing through a point p. That derivative naturally lives in R^n, so the tangent space is a subspace of R^n. The difficulty with an abstract manifold is that there may be no ambient R^n at all—only the manifold M and its smooth charts. Without an ambient space, the derivative of γ itself is not directly defined, so a substitute is needed.

The solution uses charts. Given a chart H around p (mapping into R^K, where K is the manifold’s dimension), any differentiable curve γ through p can be pushed to the coordinate level as H ∘ γ. Differentiability of γ is defined so that H ∘ γ is an ordinary differentiable map between Euclidean spaces; then the tangent vector in coordinates is simply (H ∘ γ)′(0). This produces a well-defined tangent vector in R^K for each curve.

To make the construction intrinsic, curves that produce the same coordinate derivative are treated as equivalent. If two curves γ and α through p have identical derivatives at 0 after applying a chart (and this equivalence does not depend on which chart is used, thanks to smooth transition maps), then they represent the same tangent vector. Each equivalence class—often called a “box” of curves sharing the same first-order behavior—becomes an element of the tangent space.

The tangent space T_pM is defined as the set of all such equivalence classes of differentiable curves through p, typically written using a quotient notation. For embedded submanifolds, this abstract tangent space matches the classical one: there is a natural bijection between the subspace of R^n given by γ′(0) and the equivalence class determined by γ. As a result, many calculations can treat the two tangent spaces as effectively the same.

Finally, the tangent space must carry vector space operations. Addition and scalar multiplication are made well defined by transporting equivalence classes to the coordinate level (via the induced map on tangent vectors), performing the usual operations in R^K, and then returning to the equivalence-class description. With these operations shown to be independent of chart choices, T_pM becomes a genuine vector space for abstract manifolds—setting up the machinery needed for later developments in the series.