manifold
Sign in to savethumb|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.
AI-generated from the Wikipedia summary — may contain errors.
Article
48 sectionsContents
- 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.]]