Tangent Spaces on Manifolds

Differential Geometry · Axiom Academy

LESSON Tangent Spaces on Manifolds Understanding the fundamental structure of tangent spaces through derivations and curve equivalence Given a smooth manifold M and a point p ∈ M , the tangent space T p M is the vector space of all tangent vectors at p . Intuitively, if M is a surface in ℝ³, T p M is the plane that "just touches" the manifold at p . For an n -dimensional manifold, T p M is an n -dimensional vector space. 2. Derivations as Tangent Vectors A derivation at p is a linear map v : C ∞ (M) → ℝ that satisfies the Leibniz rule (product rule): v (af + bg) = a· v (f) + b· v (g) for all a, b ∈ ℝ (linearity) v (fg) = v (f)·g( p ) + f( p )· v (g) (Leibniz rule) This abstract definition captures the idea of "directional derivative" without requiring an embedding in ℝ n . 3. Equivalence Classes of Curves An alternative approach defines tangent vectors as equivalence classes of smooth curves passing through p . The tangent vector is the equivalence class [γ] of all curves with the same velocity at p . 4. Dimension: dim(T p M) = dim(M) A fundamental theorem states that the tangent space at any point has the same dimension as the manifold itself. If M is an n -dimensional manifold, then T p M is an n -dimensional vector space for every p ∈ M . Given local coordinates (x 1 , ..., x n ) around p , we can construct a natural basis for T p M using partial derivative operators. Any tangent vector v ∈ T p M can be uniquely written as a linear combination:

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