Manifold

Definition

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.

Related concepts

1-manifold 2-manifold 3-manifold 4-manifold 5-manifold Abelian variety Absolute value Abstract index notation Adjoint bundle Affine bundle Affine connection Albert Einstein Algebraic curve Algebraic geometry Algebraic topology Algebraic variety Almost-contact manifold Almost complex manifold Almost flat manifold Almost symplectic manifold Ambient space An Introduction to Manifolds Analysis Situs (paper)Analytic continuation Analytic function Analytic manifold Analytical mechanics Andrey Markov (Soviet mathematician)Angle Angular momentum Antisymmetric tensor Area Associated bundle Atiyah–Singer index theorem Atlas Atlas (topology)Augustin-Louis Cauchy Ball (mathematics)Banach manifold Banach space Basis (linear algebra)Bernhard Riemann Betti number Bijection Boy's surface CR manifold CT scan CW complex Calabi–Yau manifold Calculus Calculus on Manifolds (book)Carl Friedrich Gauss Carl Gustav Jacobi Cartan connection Cartan formalism (physics)Cartesian coordinate system Cartesian product Categories of manifolds Category theory Cauchy stress tensor Cell decomposition Characteristic classes Chart (mathematics)Circle Classical mechanics Classification of manifolds Closed and exact differential forms Closed manifold Cofibration Cohomology Collapsing manifold Combinatorial manifold Comment. Math. Helv.Compact space Complement (set theory)Complete manifold Complete set of invariants Complex differential form Complex geometry Complex manifold Complex number Computer vision Connected space Connection (fibred manifold)Connection (mathematics)Connection (principal bundle)Connection (vector bundle)Connection form Conservation laws Contact geometry Contact manifold Continuous function (topology)Continuously differentiable Continuum mechanics Convenient vector space Coordinate chart Coordinate system Cotangent bundle Cotangent space Countable set Covariance and contravariance of vectors Covariant derivative Covering space Cross-cap Cubic curve Curvature Curvature of Riemannian manifolds Curve Cylinder (geometry)Darboux's theorem De Rham cohomology Diffeology Diffeomorphic Diffeomorphism Differentiable curve Differentiable manifold Differentiable structure Differential calculus over commutative algebras Differential form Differential geometry Differential structure Differential topology Diffiety Digital manifold Digital topology Dimension Directional statistics Disjoint union Disk (mathematics)Distance Distribution (differential geometry)Divergence Dot product Dual bundle Dyadics Earth Ehresmann connection Einstein manifold Einstein notation Einstein tensor Electromagnetic tensor Electromagnetism Elliptic geometry Elliptic integral Elwin Bruno Christoffel Embedding Encyclopedia of Mathematics Engineering Equivalence class Euclid Euclidean geometry Euclidean space Euler characteristic European Mathematical Society Exponential map (Lie theory)Exponential map (Riemannian geometry)Exterior calculus Exterior covariant derivative Exterior derivative Exterior form Exterior product Fiber bundle Fibered manifold Fibration Filtration (mathematics)Finite group Finsler manifold Flat manifold Foliation Foundations of mathematics Frobenius theorem (differential topology)Fréchet manifold Fréchet space Functional (mathematics)Functional analysis G-structure on a manifold Gauge theory (mathematics)Gaussian curvature Gauss–Bonnet theorem General linear group General relativity General topology Generalized Poincaré conjecture Generalized Stokes theorem Generalized coordinate Genus (mathematics)Geodesic Geometric topology Geometrization conjecture Geometry German language Giovanni Gerolamo Saccheri Globe Glossary of differential geometry and topology Glossary of tensor theory Gluon field strength tensor Graph of a function Gregorio Ricci-Curbastro Grigori Perelman Group (mathematics)Group action (mathematics)Hadamard manifold Hamiltonian mechanics Handlebody Harmonic analysis Harmonic function Hassler Whitney Hausdorff space Hearing the shape of a drum Heat kernel Henri Poincaré Hermann Grassmann Hermann Weyl Hermitian manifold Hilbert manifold Hilbert spaces History of manifolds and varieties Hodge star operator Holomorphic function Homeomorphic Homeomorphism Homeomorphisms Homology (mathematics)Homology manifold Homotopy Homotopy theory Hopf–Rinow theorem Hyperbola Hyperbolic geometry Hyperbolic manifold Hypercomplex manifold Hyperkähler manifold Hypersphere Immersion (mathematics)Implicit equation Implicit function theorem Index notation Inner product space Integral curve Interior product Intrinsic and extrinsic properties Invariant (mathematics)Inverse function Inversion in a sphere Isolated point Isometric embedding Isometric immersion James Munkres Jan Arnoldus Schouten Jet bundle John Milnor Joseph-Louis Lagrange János Bolyai K-theory Katsumi Nomizu Kenmotsu manifold Klein bottle Knot theory Koszul connection Kronecker delta Kähler manifold Lagrangian mechanics Laplace operator Lemniscate Leonhard Euler Level set Levi-Civita connection Levi-Civita symbol Lie algebra bundle Lie bracket of vector fields Lie derivative Lie group Lie group–Lie algebra correspondence Line (geometry)Line with two origins Linear algebra Linear map Liouville tori List of manifolds Local diffeomorphism Local homeomorphism Locally connected space Locally constant Locally ringed space Locus (mathematics)Long line (topology)Lorentzian manifold Manifold (disambiguation)Manifold with boundary Map Map (mathematics)Mapping (mathematics)Maps of manifolds Mathematical analysis Mathematical induction Mathematical physics Mathematical singularity Mathematician Mathematics Mathematics of general relativity Matrix (mathematics)Max Dehn Metric (mathematics)Metric tensor Metric tensor (general relativity)Michael Freedman Michael Spivak Mixed tensor Moment of inertia Morin surface Morris Hirsch Morse function Morse theory Moving frame Multi-index notation Multilinear algebra Multivector Möbius strip Nash embedding theorem Nautical chart Neighbourhood (mathematics)Niels Henrik Abel Nikolai Ivanovich Lobachevsky Nilmanifold Noether's theorem Non-Euclidean geometry Non-Hausdorff manifold Nonmetricity tensor Normal bundle Normal vector Normed space North Pole One-form Open (topology)Open ball Open interval Open subset Operation (mathematics)Orbifold Orientability Orientable manifold Orthogonal group Parabola Parallel postulate Parallel transport Parallelizable manifold Partial differential equations Penrose graphical notation Phase space Physics Piecewise linear manifold Plane (geometry)Poincaré conjecture Point-set topology Poisson bracket Poisson manifold Polar coordinates Polytope Poul Heegaard Power series Prime manifold Principal bundle Product topology Projection (mathematics)Projective plane Pseudo-Riemannian manifold Pseudogroup Pseudotensor Pullback (differential geometry)Pushforward (differential)Quasitoric manifold Quaternionic manifold Quotient space (topology)Raising and lowering indices Ramified covering space Real number Real projective plane Real projective space Rectifiable curve Rectifiable set Regular function René Thom Ricci-flat manifold Ricci calculus Ricci curvature Ricci flow Riemann curvature tensor Riemann surface Riemannian manifold Riemannian metric Riemannian submersion Rizza manifold Robion Kirby Sard's theorem Sasakian manifold Scheme (mathematics)Second-countable space Second countable Secondary calculus and cohomological physics Section (fiber bundle)Semialgebraic set Sergei Novikov (mathematician)Sheaf (mathematics)Simon Donaldson Simplicial complexes Simply connected Siméon Poisson Singular homology Singularity theory Smooth function Smooth structure Smoothness Solution of the Poincaré conjecture Sophus Lie South Pole Spacetime Sphere Sphere eversion Spherical harmonics Spin tensor Spinor Spinor bundle Square Stable normal bundle Stein manifold Stephen Smale Stratified space Stratifold Stress–energy tensor Sub-Riemannian manifold Subanalytic set Subbundle Submanifold Submersion (mathematics)Supermanifold Surface (topology)Surgery theory Symmetric tensor Symmetrization Symmetry group Symplectic manifold Symplectomorphism Synonymous Systems of equations Table of Lie groups Tame manifold Tangent bundle Tangent space Tangent vectors Tangential Tensor Tensor (intrinsic definition)Tensor algebra Tensor bundle Tensor calculus Tensor contraction Tensor density Tensor field Tensor operator Tensor product Tensors in curvilinear coordinates Tetrad (index notation)Theorema egregium Timeline of manifolds Topological invariant Topological manifold Topological map Topological property Topological space Topological vector space Topologically stratified space Topology Torsion (differential geometry)Torsion tensor Torus Transport phenomena Transpose Tullio Levi-Civita Two-point tensor Uniformization theorem Unit circle Unit disc Unit sphere Van der Waerden notation Vector-valued differential form Vector (mathematics and physics)Vector bundle Vector field Vector flow Vector space Vertex (geometry)Victor Guillemin Vladimir Arnold Voigt notation Volume Volume form Wedge product Well-behaved Weyl tensor Whitney conditions Whitney embedding theorem Whitney immersion theorem William Kingdon Clifford William Rowan Hamilton William S. Massey William Thurston Woldemar Voigt Yang–Mills theory Élie Cartan

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