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A topological manifold equipped with a compatible smooth atlas.
wp-smooth-manifoldnlab-smooth-manifoldlee-smooth-manifoldsEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Holomorphic transition maps are smooth, so a complex atlas canonically determines an underlying smooth manifold of twice the complex dimension.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
wp-complex-manifoldnlab-smooth-manifoldlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Choose a maximal atlas whose transition functions are smooth.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
nlab-smooth-manifoldlee-smooth-manifoldsEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Add a maximally nonintegrable hyperplane field.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-contact-manifoldnlab-contact-manifoldlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Differential forms are smooth sections of exterior powers of the cotangent bundle.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-differential_formnlab-differential_formThis is an authored directed relation from the source endpoint to the target endpoint.
Smooth coordinate changes make the exterior derivative of differential forms globally well defined, independently of the local chart used to calculate it.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
lee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Add an integrable tangent distribution or foliation atlas.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-foliationnlab-foliationlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Use the smooth-manifold structure on the common carrier.
Feed several structures into a construction junction and impose compatibility between them.
nlab-lie-grouplee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the Smooth manifold structure as one jointly required input.
Feed several structures into a construction junction and impose compatibility between them.
wp-lie-groupoidnlab-lie-groupoidlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Supply the Smooth manifold structure as one jointly required input.
Feed several structures into a construction junction and impose compatibility between them.
wp-vector-bundlenlab-vector-bundlelee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Add a covariant derivative on vector fields.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-affine-connectionlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Choose a consistent orientation.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-orientable-manifoldnlab-oriented-manifoldlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Add a Lie bracket on smooth functions that is a derivation in each argument.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-poissonlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Add a smoothly varying nondegenerate symmetric bilinear form.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-pseudo-riemannianlee-smooth-manifoldslee-riemannian-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Choose a real-analytic atlas compatible with the underlying smooth structure.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-analytic-manifoldlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
A smooth curve is a smooth map from a one-dimensional parameter domain into a smooth manifold.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
wp-smooth_curvenlab-smooth_curvelee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Relativistic spacetime is modeled as a smooth manifold so that fields, curves, and differential equations are defined locally.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
einstein-general-relativitywp-physics-spacetimenlab-physics-spacetimenlab-smooth-manifoldlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Add a differential 2-form that is closed and nondegenerate.
Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.
nlab-symplecticlee-smooth-manifoldsThis is an authored directed relation from the source endpoint to the target endpoint.
Each smooth manifold canonically determines its tangent bundle.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-tangent_bundlenlab-tangent_bundle