manifold

thumb|upright=0.65|The Klein bottle immersed in three-dimensional space

AI overview

A manifold is a mathematical object that locally looks like flat, ordinary space but can have a more complex global shape—like how Earth appears flat when you're standing on it but is actually curved. Manifolds matter because they provide a framework for understanding curved spaces in physics, geometry, and other fields where objects don't fit neatly into simple flat dimensions.

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Commons category: Manifolds

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~56 min read

Article

48 sections

Contents

Motivating examples
Circle
Sphere
Other curves
Definition
Charts, atlases, and transition maps
Charts
Atlases
Transition maps
Additional structure
Manifold with boundary
Boundary and interior
Construction
Charts
Sphere with charts
Patchwork
Intrinsic and extrinsic view
''n''-Sphere as a patchwork
Identifying points of a manifold
Gluing along boundaries
{{Anchor|Cartesian products}} Cartesian products
History
Early development
Synthesis
Poincaré's definition
Topology of manifolds: highlights
Additional structure
Topological manifolds
Differentiable manifolds
Riemannian manifolds
Finsler manifolds
Lie groups
Other types of manifolds
Classification and invariants
Surfaces
Orientability
Möbius strip
Klein bottle
Real projective plane
Genus and the Euler characteristic
Maps of manifolds
Scalar-valued functions
Generalizations of manifolds
See also
By dimension
Notes
References
External links

thumb|upright=0.65|The Klein bottle immersed in three-dimensional space

thumb|right|The surface of the Earth requires (at least) two charts to include every point without plotting the same point more than once on the same chart. Here the globe is decomposed into charts around the North and [[South Poles.]]