Differentiable manifold

A topological manifold equipped with compatible differentiable coordinate charts. These charts make smooth functions, tangent spaces, and calculus well-defined independently of coordinates.

Coordinate chart: A homeomorphism from an open subset of a topological space to an open subset of Euclidean space. Each chart assigns local coordinates in which smoothness can be tested.
Transition map: A coordinate change obtained by composing one chart with the inverse of another on overlapping domains. Its differentiability determines whether charts belong to a compatible differentiable atlas.
Riemannian manifold: A differentiable manifold equipped with a smoothly varying positive-definite inner product on its tangent spaces. Adding a metric enables lengths, angles, curvature, and geodesics.
Topological equivalence: Equivalence of spaces under homeomorphisms, which preserve topology but need not preserve differentiability. Manifolds can be topologically equivalent while carrying inequivalent smooth structures.
Smooth four-dimensional Poincaré conjecture: The question of whether every smooth four-dimensional manifold homeomorphic to the four-sphere is diffeomorphic to it. It asks whether the four-sphere has an exotic smooth structure.
Smooth map: A map between differentiable manifolds whose coordinate representations have derivatives of every order. Chart compatibility makes this coordinate-based definition independent of chart choices.
Tangent bundle: The bundle whose fiber at each point of a manifold is that point's tangent space. It assembles the pointwise tangent spaces enabled by smooth coordinates.
Lie group: A group that is also a differentiable manifold, with smooth multiplication and inversion operations. Its smooth structure makes continuous symmetries accessible to differential calculus.
Piecewise-linear manifold: A manifold described by compatible triangulations and piecewise-linear coordinate changes. It encodes a different kind of local regularity from differentiable charts.
Smooth structures on R4: The collection of differentiable structures on four-dimensional Euclidean space, including uncountably many exotic examples. Their abundance contrasts with the uniqueness of smooth structures on Euclidean spaces of other dimensions.

Linked from 3 pages