Tangent Space Examples

Differential Geometry · Axiom Academy

EXAMPLE Tangent Space Examples Working through concrete examples of tangent spaces and differentials in differential geometry Excellent work! You've explored several fundamental examples of tangent spaces. Here's what we learned: Euclidean Tangent Spaces: For , the tangent space at any point is canonically isomorphic to itself. Tangent vectors can be identified with directional derivatives. Tangent to Submanifolds: For the 2-sphere , the tangent space at the north pole is the xy-plane. We can characterize it using the gradient condition: vectors perpendicular to . Curves Perspective: Tangent vectors are equivalence classes of curves through a point, where two curves are equivalent if they have the same velocity at that point. This provides a coordinate-free definition. The Differential: For a smooth map , the differential is a linear map between tangent spaces. It captures how the map deforms infinitesimal directions. These perspectives on tangent spaces—as derivations, velocity vectors of curves, and domains of differentials—are all equivalent but each provides unique insights. Mastering these viewpoints is essential for understanding differential geometry!

This is the written version of the interactive lesson above. See the full Differential Geometry course.