Matches in DBpedia 2016-04 for { <http://dbpedia.org/resource/Affine_connection> ?p ?o }
- Affine_connection abstract "In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space Rn by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.On any manifold of positive dimension there are infinitely many affine connections. If the manifold is further endowed with a Riemannian metric then there is a natural choice of affine connection, called the Levi-Civita connection. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (linearity and the Leibniz rule). This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle. A choice of affine connection is also equivalent to a notion of parallel transport, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the frame bundle. Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group or as a principal connection on the frame bundle.The main invariants of an affine connection are its torsion and its curvature. The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection. Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry; the main differences are encapsulated in the curvature of the connection.".
- Affine_connection thumbnail Parallel_transport_sphere.svg?width=300.
- Affine_connection wikiPageExternalLink item?id=ASENS_1924_3_41__1_0.
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- Affine_connection wikiPageRevisionID "696569769".
- Affine_connection wikiPageWikiLink Affine_group.
- Affine_connection wikiPageWikiLink Affine_space.
- Affine_connection wikiPageWikiLink Affine_transformation.
- Affine_connection wikiPageWikiLink Albert_Einstein.
- Affine_connection wikiPageWikiLink Associated_bundle.
- Affine_connection wikiPageWikiLink Atlas_(topology).
- Affine_connection wikiPageWikiLink Basis_(linear_algebra).
- Affine_connection wikiPageWikiLink Bernhard_Riemann.
- Affine_connection wikiPageWikiLink Bilinear_map.
- Affine_connection wikiPageWikiLink Bundle_map.
- Affine_connection wikiPageWikiLink Cartan_connection.
- Affine_connection wikiPageWikiLink Cartesian_product.
- Affine_connection wikiPageWikiLink Category:Connection_(mathematics).
- Affine_connection wikiPageWikiLink Category:Differential_geometry.
- Affine_connection wikiPageWikiLink Christoffel_symbols.
- Affine_connection wikiPageWikiLink Complex_manifold.
- Affine_connection wikiPageWikiLink Connection_(affine_bundle).
- Affine_connection wikiPageWikiLink Connection_(fibred_manifold).
- Affine_connection wikiPageWikiLink Connection_(mathematics).
- Affine_connection wikiPageWikiLink Connection_(principal_bundle).
- Affine_connection wikiPageWikiLink Connection_(vector_bundle).
- Affine_connection wikiPageWikiLink Connection_form.
- Affine_connection wikiPageWikiLink Convex_set.
- Affine_connection wikiPageWikiLink Covariant_derivative.
- Affine_connection wikiPageWikiLink Covariant_transformation.
- Affine_connection wikiPageWikiLink Curvature.
- Affine_connection wikiPageWikiLink Curve.
- Affine_connection wikiPageWikiLink Derivative.
- Affine_connection wikiPageWikiLink Development_(differential_geometry).
- Affine_connection wikiPageWikiLink Differentiable_manifold.
- Affine_connection wikiPageWikiLink Differential_(infinitesimal).
- Affine_connection wikiPageWikiLink Differential_form.
- Affine_connection wikiPageWikiLink Differential_geometry.
- Affine_connection wikiPageWikiLink Differential_geometry_of_surfaces.
- Affine_connection wikiPageWikiLink Dot_product.
- Affine_connection wikiPageWikiLink Ehresmann_connection.
- Affine_connection wikiPageWikiLink Einstein_notation.
- Affine_connection wikiPageWikiLink Elwin_Bruno_Christoffel.
- Affine_connection wikiPageWikiLink Encyclopedia_of_Mathematics.
- Affine_connection wikiPageWikiLink Equivariant_map.
- Affine_connection wikiPageWikiLink Erlangen_program.
- Affine_connection wikiPageWikiLink Euclidean_geometry.
- Affine_connection wikiPageWikiLink Euclidean_space.
- Affine_connection wikiPageWikiLink Euclidean_vector.
- Affine_connection wikiPageWikiLink Exponential_map_(Riemannian_geometry).
- Affine_connection wikiPageWikiLink Exterior_algebra.
- Affine_connection wikiPageWikiLink Exterior_derivative.
- Affine_connection wikiPageWikiLink Felix_Klein.
- Affine_connection wikiPageWikiLink Fiber_bundle.
- Affine_connection wikiPageWikiLink Foundations_of_Differential_Geometry.
- Affine_connection wikiPageWikiLink Frame_bundle.
- Affine_connection wikiPageWikiLink Frame_of_reference.
- Affine_connection wikiPageWikiLink Gauge_covariant_derivative.
- Affine_connection wikiPageWikiLink Gauge_theory.
- Affine_connection wikiPageWikiLink General_linear_group.
- Affine_connection wikiPageWikiLink General_relativity.
- Affine_connection wikiPageWikiLink Geodesic.
- Affine_connection wikiPageWikiLink Gregorio_Ricci-Curbastro.
- Affine_connection wikiPageWikiLink Group_action.
- Affine_connection wikiPageWikiLink Group_homomorphism.
- Affine_connection wikiPageWikiLink Group_representation.
- Affine_connection wikiPageWikiLink Hermann_Weyl.
- Affine_connection wikiPageWikiLink Homogeneous_space.
- Affine_connection wikiPageWikiLink Infinity.
- Affine_connection wikiPageWikiLink Integrability_conditions_for_differential_systems.
- Affine_connection wikiPageWikiLink Introduction_to_the_mathematics_of_general_relativity.
- Affine_connection wikiPageWikiLink Jean-Louis_Koszul.
- Affine_connection wikiPageWikiLink Kernel_(algebra).
- Affine_connection wikiPageWikiLink Klein_geometry.
- Affine_connection wikiPageWikiLink Levi-Civita_connection.
- Affine_connection wikiPageWikiLink Lie_algebra.
- Affine_connection wikiPageWikiLink Lie_bracket_of_vector_fields.
- Affine_connection wikiPageWikiLink Lie_derivative.
- Affine_connection wikiPageWikiLink Lie_group.
- Affine_connection wikiPageWikiLink Linear_differential_equation.
- Affine_connection wikiPageWikiLink Linear_independence.
- Affine_connection wikiPageWikiLink Linear_map.
- Affine_connection wikiPageWikiLink Linearity.
- Affine_connection wikiPageWikiLink List_of_formulas_in_Riemannian_geometry.
- Affine_connection wikiPageWikiLink Manifold.
- Affine_connection wikiPageWikiLink Mathematics.
- Affine_connection wikiPageWikiLink Maurer–Cartan_form.
- Affine_connection wikiPageWikiLink Metric_connection.
- Affine_connection wikiPageWikiLink Moving_frame.
- Affine_connection wikiPageWikiLink Ordinary_differential_equation.
- Affine_connection wikiPageWikiLink Origin_(mathematics).
- Affine_connection wikiPageWikiLink Overdetermined_system.
- Affine_connection wikiPageWikiLink Parallel_transport.
- Affine_connection wikiPageWikiLink Parallelizable_manifold.
- Affine_connection wikiPageWikiLink Partial_differential_equation.
- Affine_connection wikiPageWikiLink Picard–Lindelöf_theorem.
- Affine_connection wikiPageWikiLink Point_(geometry).
- Affine_connection wikiPageWikiLink Pre-Lie_algebra.