Manifolds 21 | Tangent Space (Definition via tangent curves) [dark version]

Tangent spaces for abstract manifolds can be defined without embedding the manifold into any surrounding Euclidean space by using tangent curves and an equivalence relation. The key move is to replace “take the derivative of a curve inside the ambient space” with “take the derivative after pushing the curve into a chart,” where ordinary derivatives make sense. This matters because it lets tangent vectors—and the tangent space built from them—be defined purely from the manifold’s own smooth structure, not from an external coordinate container.

For a submanifold sitting inside 2n, tangent vectors are obtained by differentiating smooth curves through a point p and then interpreting the result as a vector in 2n. The transcript highlights the obstacle for an abstract manifold M: there is no ambient 2n, so the derivative of a curve cannot be interpreted the same way. The workaround is to use charts. A chart map b7 pushes local information from M into 2K, where standard calculus applies. If a curve b3 on M passes through p, then composing with the chart gives a differentiable curve in 2K, and the derivative at the parameter value 0 produces a vector in 2K.

However, different curves through p can yield the same derivative in 2K. To capture “the tangent vector” rather than “the particular curve,” the definition groups curves into equivalence classes: two curves b3 and b1 are equivalent if their derivatives agree after pushing them through a chart (equivalently, checking one chart is enough because transition maps preserve the derivative outcome). Each equivalence class is treated as a single tangent vector. The tangent space T_pM is then the set of all such equivalence classes of differentiable curves through p.

The transcript also connects this abstract construction back to the familiar embedded case. For a submanifold, the equivalence-class tangent vector corresponds naturally to the usual derivative vector in 2n. This correspondence is described as a well-defined projection: because all curves in an equivalence class share the same derivative in the chart coordinates, the class determines the same tangent vector in the embedded picture. As a result, calculations can often be done using either viewpoint without changing the outcome.

Finally, the tangent space must be more than a set: it needs vector space operations. Addition and scalar multiplication are defined by transporting equivalence classes to the chart level (where vectors in 2K can be added and scaled), performing the operations there, and then mapping back to equivalence classes. The transcript emphasizes that these operations must be well defined—independent of which chart is used—so the smooth structure of the manifold guarantees consistency. With that, T_pM becomes a genuine vector space for abstract manifolds, setting up the machinery needed for later results in the series.