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- Manifold abstract "In mathematics, a manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded in three dimensional real space, but also the Klein bottle and real projective plane which cannot.Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane (in the context of manifolds they are called charts). When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map.The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds.This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.".
- Manifold thumbnail BoysSurfaceTopView.PNG?width=300.
- Manifold wikiPageExternalLink Dim_E.htm.
- Manifold wikiPageExternalLink www.map.mpim-bonn.mpg.de.
- Manifold wikiPageExternalLink Geom.
- Manifold wikiPageExternalLink Grund.
- Manifold wikiPageExternalLink www.mpim-bonn.mpg.de.
- Manifold wikiPageID "2073470".
- Manifold wikiPageLength "67535".
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- Manifold wikiPageRevisionID "669627064".
- Manifold wikiPageWikiLink 1-manifold.
- Manifold wikiPageWikiLink 2-manifold.
- Manifold wikiPageWikiLink 3-manifold.
- Manifold wikiPageWikiLink 4-manifold.
- Manifold wikiPageWikiLink 5-manifold.
- Manifold wikiPageWikiLink Abelian_variety.
- Manifold wikiPageWikiLink Absolute_value.
- Manifold wikiPageWikiLink Affine_connection.
- Manifold wikiPageWikiLink Algebraic_geometry.
- Manifold wikiPageWikiLink Algebraic_topology.
- Manifold wikiPageWikiLink Algebraic_variety.
- Manifold wikiPageWikiLink Ambient_space.
- Manifold wikiPageWikiLink Analysis_Situs_(paper).
- Manifold wikiPageWikiLink Analytic_continuation.
- Manifold wikiPageWikiLink Analytic_function.
- Manifold wikiPageWikiLink Analytic_manifold.
- Manifold wikiPageWikiLink Analytical_mechanics.
- Manifold wikiPageWikiLink Andrey_Markov,_Jr..
- Manifold wikiPageWikiLink Andrey_Markov_(Soviet_mathematician).
- Manifold wikiPageWikiLink Angle.
- Manifold wikiPageWikiLink Arc_length.
- Manifold wikiPageWikiLink Area.
- Manifold wikiPageWikiLink Atiyah–Singer_index_theorem.
- Manifold wikiPageWikiLink Atlas.
- Manifold wikiPageWikiLink Atlas_(topology).
- Manifold wikiPageWikiLink Ball_(mathematics).
- Manifold wikiPageWikiLink Banach_manifold.
- Manifold wikiPageWikiLink Banach_space.
- Manifold wikiPageWikiLink Basis_(linear_algebra).
- Manifold wikiPageWikiLink Bernhard_Riemann.
- Manifold wikiPageWikiLink Betti_number.
- Manifold wikiPageWikiLink Betti_numbers.
- Manifold wikiPageWikiLink Bijection.
- Manifold wikiPageWikiLink Boundary_(topology).
- Manifold wikiPageWikiLink CR_manifold.
- Manifold wikiPageWikiLink CW_complex.
- Manifold wikiPageWikiLink Calculus.
- Manifold wikiPageWikiLink Carl_Friedrich_Gauss.
- Manifold wikiPageWikiLink Carl_Gustav_Jacob_Jacobi.
- Manifold wikiPageWikiLink Carl_Gustav_Jacobi.
- Manifold wikiPageWikiLink Cartesian_coordinate_system.
- Manifold wikiPageWikiLink Cartesian_product.
- Manifold wikiPageWikiLink Category:Manifolds.
- Manifold wikiPageWikiLink Category_theory.
- Manifold wikiPageWikiLink Characteristic_class.
- Manifold wikiPageWikiLink Characteristic_classes.
- Manifold wikiPageWikiLink Chart_(mathematics).
- Manifold wikiPageWikiLink Chart_(topology).
- Manifold wikiPageWikiLink Circle.
- Manifold wikiPageWikiLink Classical_mechanics.
- Manifold wikiPageWikiLink Classification_of_manifolds.
- Manifold wikiPageWikiLink Closed_manifold.
- Manifold wikiPageWikiLink Cohomology.
- Manifold wikiPageWikiLink Combinatorial_manifold.
- Manifold wikiPageWikiLink Compact_space.
- Manifold wikiPageWikiLink Complement_(set_theory).
- Manifold wikiPageWikiLink Complete_set_of_invariants.
- Manifold wikiPageWikiLink Complex_manifold.
- Manifold wikiPageWikiLink Complex_number.
- Manifold wikiPageWikiLink Connected_space.
- Manifold wikiPageWikiLink Conservation_law.
- Manifold wikiPageWikiLink Conservation_laws.
- Manifold wikiPageWikiLink Contact_geometry.
- Manifold wikiPageWikiLink Continuous_function.
- Manifold wikiPageWikiLink Continuous_function_(topology).
- Manifold wikiPageWikiLink Continuously_differentiable.
- Manifold wikiPageWikiLink Convenient_vector_space.
- Manifold wikiPageWikiLink Countable_set.
- Manifold wikiPageWikiLink Covering_space.
- Manifold wikiPageWikiLink Cross-cap.
- Manifold wikiPageWikiLink Cubic_curve.
- Manifold wikiPageWikiLink Cubic_plane_curve.
- Manifold wikiPageWikiLink Curvature.
- Manifold wikiPageWikiLink Curvature_of_Riemannian_manifolds.
- Manifold wikiPageWikiLink Curve.
- Manifold wikiPageWikiLink Cylinder_(geometry).
- Manifold wikiPageWikiLink Derivative.
- Manifold wikiPageWikiLink Diffeomorphic.
- Manifold wikiPageWikiLink Diffeomorphism.
- Manifold wikiPageWikiLink Differentiable_function.
- Manifold wikiPageWikiLink Differentiable_manifold.
- Manifold wikiPageWikiLink Differentiable_structure.
- Manifold wikiPageWikiLink Differential_form.
- Manifold wikiPageWikiLink Differential_geometry.
- Manifold wikiPageWikiLink Differential_structure.
- Manifold wikiPageWikiLink Differential_topology.
- Manifold wikiPageWikiLink Digital_manifold.
- Manifold wikiPageWikiLink Digital_topology.
- Manifold wikiPageWikiLink Dimension.