The Tangent Bundle

Differential Geometry · Axiom Academy

Understanding how tangent spaces assemble into a smooth manifold structure 1. Definition: The Tangent Bundle The tangent bundle TM of a manifold M is the disjoint union of all tangent spaces at every point: Each element of TM is a pair (p, v) where p ∈ M is a point on the manifold and v ∈ T p M is a tangent vector at that point. 2. TM as a Manifold (Dimension 2n) The tangent bundle TM is itself a smooth manifold! If M has dimension n, then TM has dimension 2n - we need n coordinates for the base point p and n coordinates for the tangent vector v. 3. The Projection Map π: TM → M There's a natural projection map π that sends each tangent vector back to its base point: This map is smooth and surjective. The preimage π⁻¹(p) of any point p is exactly the tangent space T p M - a vector space of dimension n. Over any chart (U, φ) on M, the tangent bundle looks like a product space: This means locally, TM looks like U × ℝⁿ. The tangent bundle is a vector bundle - it's locally trivial but may be globally twisted. 5. Example: Tangent Bundle of S² Consider the 2-sphere S² embedded in ℝ³. At each point p on the sphere, the tangent space T p S² is a 2-dimensional plane perpendicular to the radius vector. The tangent bundle TS² is a 4-dimensional manifold (dim S² = 2, so dim TS² = 4). It's diffeomorphic to the unit tangent vectors in ℝ³ that are perpendicular to radial directions.

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