Manifold

A manifold is a space that locally resembles Euclidean space, though its global shape may differ. This local structure makes calculus possible even when no single coordinate system describes the whole space.

Topological space: A set equipped with a collection of open subsets that specifies which points are nearby. Manifolds add local Euclidean structure to the more general notion of topological space.
Differentiable manifold: A manifold whose chart transitions are differentiable to a specified degree. Differentiability of transitions allows calculus to be defined independently of the chosen chart.
Circle: The set of points in a plane at a fixed positive distance from a center. The circle is the simplest compact one-dimensional manifold without boundary.
Riemannian manifold: A smooth manifold equipped with an inner product on each tangent space, varying smoothly. A Riemannian metric adds lengths, angles, and distances to a smooth manifold.
Differential geometry: The study of smooth spaces using calculus, geometric structures, and differential equations. Manifolds are the central spaces on which differential geometry is developed.
Homeomorphism: A continuous bijection between topological spaces whose inverse is also continuous. Charts identify manifold neighborhoods with Euclidean open sets through homeomorphisms.
Smooth manifold: A manifold with infinitely differentiable coordinate transitions. Smooth manifolds provide the standard setting for differential geometry.
Sphere: The set of points in three-dimensional space at a fixed distance from a center. The sphere is a compact surface that cannot be covered by one coordinate chart.
Symplectic manifold: A smooth manifold equipped with a closed, nondegenerate differential two-form. Symplectic structure captures the geometry underlying classical Hamiltonian mechanics.
General relativity: Einstein's theory of gravity, in which spacetime geometry is shaped by matter and energy. Its spacetime is modeled as a four-dimensional Lorentzian manifold.

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