Smooth manifold

A space locally modeled on Euclidean space whose coordinate charts have smoothly compatible transition maps. This structure makes derivatives and other tools of calculus well-defined across the space.

Coordinate chart: A homeomorphism from an open subset of a space to an open subset of Euclidean space. Charts provide the local coordinates whose compatibility defines smoothness.
Smooth map: A map between smooth manifolds whose coordinate expressions have derivatives of every order. These maps preserve the differentiable structure between manifolds.
Exotic sphere: A smooth manifold homeomorphic but not diffeomorphic to a standard sphere. It shows that one topological space can carry inequivalent smooth structures.
General relativity: A theory of gravity in which spacetime geometry is dynamical and matter determines its curvature. Its spacetime is modeled as a four-dimensional smooth manifold with a Lorentzian metric.
Atlas (mathematics): A collection of coordinate charts covering a manifold. An atlas assembles local descriptions into a global smooth structure.
Differential of a map: The linear map between tangent spaces induced by a differentiable map at a point. It extends the ordinary derivative to maps between manifolds.
Topological manifold: A space locally homeomorphic to Euclidean space and satisfying specified separation and countability conditions. Unlike a smooth manifold, it need not have smoothly compatible coordinates.
Configuration space: The space of all possible configurations of a physical system. Constraints often make a system's configuration space a smooth manifold.
Transition map: A coordinate change obtained by composing one chart with the inverse of another on their overlap. Smoothness requires these coordinate changes to be differentiable.
Vector field: A choice of tangent vector at each point of a manifold, varying smoothly with the point. Vector fields describe smooth directions of motion over the space.

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