Manifolds Summary

Differential Geometry · Axiom Academy

Let's review the key concepts from Unit 6: the foundation of differential geometry. Local Euclidean Property: A manifold looks like ℝ n in a neighborhood around every point Global vs Local: Globally complex (like a sphere or torus), but locally flat and simple Topological Space: Hausdorff and second-countable, ensuring nice separation and structure Dimension: An n -manifold is locally homeomorphic to ℝ n Charts, Atlases & Smooth Structure Chart: A homeomorphism φ: U → V ⊂ ℝ n mapping an open set to Euclidean space Atlas: A collection of charts whose domains cover the entire manifold Transition Maps: Functions φ β ∘ φ α -1 relating overlapping charts Smooth Structure: An atlas where all transition maps are smooth (infinitely differentiable) Definition: The tangent space T p M at point p consists of all directional derivatives at p Vector Space: T p M is an n -dimensional vector space Geometric Intuition: Think of all velocity vectors of curves passing through p Coordinate Basis: In local coordinates, the basis is ∂/∂x 1 , ..., ∂/∂x n Construction: The tangent bundle TM is the union of all tangent spaces Formula: TM = ⊔ p∈M T p M (disjoint union) Dimension: If M is n -dimensional, then TM is 2n -dimensional Natural Structure: TM itself is a smooth manifold with projection π: TM → M Definition: A smooth assignment of a tangent vector to each point on the manifold Notation: A vector field X assigns X(p) ∈ T p M for all p ∈ M

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