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Manifold

★★★★★Graduate+

📖Definition

A manifold is a topological space locally resembling Euclidean space. Smooth manifolds additionally have differentiable structure.

∀ p ∈ M, ∃ U ∋ p : U ≅ ℝⁿ

Locally Euclidean property

φ_β ∘ φ_α⁻¹ : ℝⁿ → ℝⁿ

Transition functions (diffeomorphism)

예제 1

Explain why sphere S² is a 2-manifold.

예제 2

What is the dimension of a torus?

Discovered by: Bernhard Riemann (1854)

Riemann laid foundations of manifold theory by introducing general n-dimensional spaces.

Geometry of spacetime

Manifold learning, dimensionality reduction

Configuration spaces

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