Differentiable manifold

Definition

In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

Related concepts

2-form 4-manifold 5-manifold A fortiori argument Active transformation Adjoint bundle Affine bundle Affine connection Affine space Affine transformation Albert Einstein Algebra (ring theory)Algebra over a field Algebraic geometry Almost-contact manifold Almost complex manifold Almost flat manifold Almost symplectic manifold American Mathematical Society Analytic function Analytic varieties Andrew Ranicki Annals of Mathematics Antisymmetric tensor Associated bundle Atiyah–Singer index theorem Atlas (mathematics)Atlas (topology)Banach manifold Banach–Stone theorem Bernhard Riemann Biholomorphism Bump function C*-algebra Calabi–Yau manifold Calculus Carl Friedrich Gauss Cartan connection Category theory Chain rule Christoffel symbols Classical mechanics Classification of manifolds Closed and exact differential forms Closed form (calculus)Closed manifold Cofibration Collapsing manifold Commentarii Mathematici Helvetici Compact space Complete manifold Completion (algebra)Complex analytic geometry Complex differential form Complex manifold Connection (fibred manifold)Connection (mathematics)Connection (principal bundle)Connection (vector bundle)Connection form Contact (mathematics)Contact manifold Continuously differentiable Convenient vector space Coordinate transformation Cotangent bundle Cotangent space Covariance Covariance and contravariance of vectors Covariant derivative Covector Curl (mathematics)Darboux's theorem De Rham cohomology Decision problem Definite bilinear form Derivation (abstract algebra)Derivative Diffeological space Diffeology Diffeomorphism Diffeomorphisms Differentiable curve Differentiable function Differential calculus Differential calculus over commutative algebras Differential form Differential geometry Differential operator Differential structure Diffiety Dimension Direct sum of vector bundles Directional derivative Distribution (differential geometry)Divergence Divergence theorem Donaldson's theorem Dual bundle Dual numbers Dual space E8 manifold Ehresmann connection Einstein manifold Embedding Encyclopedia of Mathematics Equivalence class Equivalence principle Equivariant Euclidean space European Mathematical Society Exact differential Exotic R4 Exotic sphere Exponential map (Lie theory)Exponential map (Riemannian geometry)Exterior calculus Exterior derivative Fiber bundle Fibered manifold Fibration Fibre bundle Finsler manifold Flat manifold Foliation Frame bundle Frobenius theorem (differential topology)Fréchet manifold Frölicher space Function composition Function space Functor G-structure on a manifold Gauge theory (mathematics)General covariance General linear group General relativity Generalized Stokes theorem Geodesic Germ (mathematics)Glossary of differential geometry and topology Graded vector space Gradient Graduate Studies in Mathematics Green's theorem Gregorio Ricci-Curbastro Group (mathematics)H-cobordism theorem Habilitation Hadamard manifold Hairy ball theorem Hamiltonian mechanics Hassler Whitney Hausdorff space Hermann Weyl Hermitian manifold Hilbert manifold History of manifolds and varieties Holomorphic function Holonomic basis Homeomorphism Homotopy equivalence Hopf–Rinow theorem Hyperbolic manifold Hypercomplex manifold Hyperkähler manifold Ieke Moerdijk If and only if Immersion (mathematics)Implicit function Infinitesimal Inner product space Integral calculus Integral curve Interior product Intersection theory Introduction to the mathematics of general relativity Invariance of domain Inverse function theorem James Clerk Maxwell Jet (mathematics)Jet bundle Jet group John Milnor Journal of Differential Geometry K-theory Kenmotsu manifold Kervaire manifold Kirby–Siebenmann invariant Koszul connection Kähler manifold Lagrangian system Levi-Civita connection Lie algebra Lie algebra bundle Lie bracket of vector fields Lie derivative Lie group Lie group–Lie algebra correspondence Linear functional List of formulas in Riemannian geometry List of manifolds Local diffeomorphism Local property Local ring Locally finite collection Locally ringed space Lorentzian manifolds Lp space Manifold Manifold with boundary Maximal ideal Metric signature Michael Freedman Michel Kervaire Morse theory Mostow rigidity Moving frame Multilinear map Multilinear operator Multivariate calculus Mutatis mutandis Möbius transformation Nature (journal)Nilmanifold Noether's theorem Non-Hausdorff manifold Noncommutative geometry Nondegenerate form Normal bundle One-form Operator theory Orbifold Orientability Parallel transport Parallelizable manifold Partial derivative Partial differential equations Partition of unity Physics Poisson manifold Polynomial Presentation of a group Prime manifold Principal bundle Product rule Pseudo-Riemannian manifold Pseudogroup Pullback Pullback (differential geometry)Pushforward (differential)Quaternion Quaternionic manifold Quotient group Rank of a matrix Real line Rectifiable curve Rectifiable set Regular function Ricci-flat manifold Ricci curvature Ricci flow Riemann curvature tensor Riemann surface Riemannian geometry Riemannian manifold Riemannian metric Ringed space Rizza manifold Robin Hartshorne Sard's theorem Sasakian manifold Saunders Mac Lane Scheme (mathematics)Second countable Secondary calculus and cohomological physics Section (fiber bundle)Set (mathematics)Sheaf (mathematics)Shlomo Sternberg Shoshichi Kobayashi Simon Donaldson Simply connected Singularity theory Smooth function Smooth structure Smoothness Sobolev spaces Sophus Lie Space (mathematics)Spectrum of a C*-algebra Spinor bundle Stable normal bundle Stein manifold Stokes' theorem Stratified space Stratifold Sub-Riemannian manifold Subbundle Submanifold Submersion (mathematics)Supermanifold Support (mathematics)Surface (topology)Surgery theory Symplectic manifold Symplectomorphism Synthetic differential geometry Tame manifold Tangent bundle Tangent space Tangent vector Tautological one-form Taylor series Teichmüller space Tensor Tensor analysis Tensor bundle Tensor density Tensor field Tensor product Thurston's geometrization conjecture Topological manifold Topological space Topos Torsion tensor Total derivative Total space Transition map Tropic of Cancer Tullio Levi-Civita University of Göttingen Vector-valued differential form Vector (geometric)Vector bundle Vector field Vector flow Vector norm Vector space Velocity vector Volume form Wedge product Whitney embedding theorem Yang–Mills theory Zlil Sela

32 concepts already in your glossary