What are Manifolds?

Differential Geometry · Axiom Academy

Discover the abstract spaces that look locally like flat Euclidean space. Step 1: The Earth Under Your Feet Imagine you're standing on Earth. The ground around you looks flat, even though Earth is actually a sphere. Explore how zooming in makes curved surfaces appear flat. Let's explore different manifolds. Click on each shape to see its properties and understand what dimension of Euclidean space it locally resembles. To do calculus on a manifold, we need coordinates. A chart is like a map that takes a patch of the manifold and assigns coordinates to it. Drag the point on the sphere to see its coordinates in different charts. Step 4: The Power of Abstraction Why do we study manifolds abstractly, without embedding them in ℝⁿ? Explore how intrinsic properties don't depend on the embedding. Intrinsic properties depend only on the manifold itself (distances, angles measured on the surface). Extrinsic properties depend on how it's embedded in a larger space (like curvature in ℝ³). Example: A flat torus (square with opposite edges identified) has zero intrinsic curvature, even though it looks curved when embedded in ℝ³! Many important manifolds can't be visualized in 3D space. The configuration space of a robot arm, the space of all possible shapes, or spacetime in general relativity are all high-dimensional manifolds. Abstract definitions let us work with these spaces mathematically.

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