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Construct allowing differentiation of tangent vector fields of manifolds
In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent
Affine_connection
differential geometry, a connection on an affine bundle is a specialisation to affine bundles of the more general notion of a connection on a principal bundle
Connection_(affine_bundle)
Mathematical formulation of gravity theory with a general affine connection
physics, a metric-affine gravitation theory is a mathematical formulation of the theory of gravity involving a general affine connection on the world manifold
Metric-affine gravitation theory
Metric-affine_gravitation_theory
Straight path on a curved surface or a Riemannian manifold
vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain
Geodesic
Canonical connection on a pseudo-Riemannian manifold
Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the
Levi-Civita_connection
Theoretical attempts to unify the forces of nature
the affine connection as the fundamental structure field rather than the metric tensor which was the original focus of general relativity. Affine connection
Classical unified field theories
Classical_unified_field_theories
Connection on a spinor bundle
mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded
Spin_connection
Object in differential geometry
geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors X , Y {\displaystyle
Torsion_tensor
System of moving vectors in differential geometry
manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors
Parallel_transport
Euclidean geometry without distance and angles
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance
Affine_geometry
Topics referred to by the same term
Look up affine in Wiktionary, the free dictionary. Affine may describe any of various topics concerned with connections or affinities. It is used in mathematics
Affine
Generalization of affine connections
of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization
Cartan_connection
Array of numbers describing a metric connection
array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed
Christoffel_symbols
Function in mathematics
various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary
Connection_(mathematics)
In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold
Affine_manifold
Gauge theory with affine connections
In mathematical physics, an affine gauge theory is a classical gauge theory in which the gauge fields are affine connections on the tangent bundle over
Affine_gauge_theory
Classical theory of gravitation
the affine connection and then separately posing a constraint that forces both the torsion and contorsion to be zero, which thus forces the affine connection
Einstein–Cartan_theory
Tensor in differential geometry
unparameterized geodesics). If ∇ {\displaystyle \nabla } denotes an affine connection, then the curvature tensor R {\displaystyle R} is the (1,3)-tensor
Ricci_curvature
depends on the metric through the affine connection. Whereas the covariant derivative requires an affine connection to allow comparison between vectors
Mathematics of general relativity
Mathematics_of_general_relativity
differential geometry uses the affine or Blaschke normal, the induced affine connection, the affine fundamental form, the affine shape operator, and related
Affine_differential_geometry
Special coordinate system in differential geometry
a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying
Normal_coordinates
Map from tangent space to the manifold
canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection. Let M {\displaystyle
Exponential map (Riemannian geometry)
Exponential_map_(Riemannian_geometry)
Quantity in general relativity
}\left({\sqrt {-g}}g^{\nu \beta }\right)_{,\rho }\right]\end{aligned}}} Affine connection version: t LL μ ν + Λ g μ ν κ = 1 2 κ [ ( 2 Γ α β σ Γ σ ρ ρ − Γ α
Stress–energy–momentum pseudotensor
Stress–energy–momentum_pseudotensor
Topics referred to by the same term
vector field on a manifold Connection (affine bundle) Connection (composite bundle) Connection (fibred manifold) Connection (principal bundle), gives the
Connection
Geometric transformation that preserves lines but not angles nor the origin
affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space
Affine_transformation
Algebraic operation on coordinate vectors
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Dot_product
Concept in differential geometry
applied to real-world data. Affine holonomy groups are the groups arising as holonomies of torsion-free affine connections; those which are not Riemannian
Holonomy
Topological space that locally resembles Euclidean space
Riemannian manifold and the torsion of a manifold equipped with an affine connection. This distinction between local invariants and no local invariants
Manifold
Tensor index notation for tensor-based calculations
A Koszul connection on the tangent bundle of a differentiable manifold is called an affine connection. A connection is a metric connection when the covariant
Ricci_calculus
Branch of mathematics
connection serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which
Differential_geometry
Type of transport in differential geometry
equivalence class of torsion-free affine connections having the same unparametrized geodesics. Projective connections are modeled on the geometry of projective
Projective_connection
Tensor field in Riemannian geometry
pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity
Riemann_curvature_tensor
Type of physical quantity
according to the first definition. The Christoffel symbols of an affine connection on a manifold can be thought of as the correction terms to the partial
Pseudotensor
Defines a notion of parallel transport on a bundle
and define a notion of torsion for affine connections which are not the Levi-Civita connection. Given two connections ∇ 1 , ∇ 2 {\displaystyle \nabla _{1}
Connection_(vector_bundle)
Topics referred to by the same term
Torsion-free affine connection, an affine connection whose torsion tensor vanishes Torsion-free metric connection or Levi-Civita connection, a unique symmetric
Torsion-free
Specification of a derivative along a tangent vector of a manifold
approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional
Covariant_derivative
Hypothetical object of spacetime
general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamical
White_hole
Object in differential geometry
metric-compatible affine connection with Christoffel symbol Γ k i j {\displaystyle {\Gamma ^{k}}_{ij}} and the unique torsion-free Levi-Civita connection for the
Contorsion_tensor
Dirac equation for self-interacting fermions
momentum (spin). This theory removes a constraint of the symmetry of the affine connection and treats its antisymmetric part, the torsion tensor, as a variable
Nonlinear_Dirac_equation
Chern's conjecture for affinely flat manifolds was proposed by Shiing-Shen Chern in 1955 in the field of affine geometry. As of 2025, it remains an unsolved
Chern's conjecture (affine geometry)
Chern's_conjecture_(affine_geometry)
Manifold upon which it is possible to perform calculus
satisfy some of the usual features of directional differentiation. An affine connection, which is not uniquely defined, but generalizes in a more complete
Differentiable_manifold
Concept relating to general relativity
{\displaystyle {g_{\mu \nu }}} but also the forty components of the affine connection Γ β μ α {\displaystyle {\Gamma _{\,\beta \mu }^{\alpha }}} , assuming
Palatini_variation
such a connection is equivalently given by a Cartan connection for the affine group of affine space, and is often called an affine connection. The two
Linear_connection
Principal fiber bundle
1-form A, the electromagnetic four-potential, (equivalently, the affine connection) such that π ∗ F = d A . {\displaystyle \pi ^{*}F=dA.} Given a circle
Circle_bundle
Property of a mathematical space
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Dimension
Math/physics concept
vector on M, and d denotes the pushforward. Ehresmann connection Cartan connection Affine connection Curvature form Griffiths & Harris (1978), Wells (1980)
Connection_form
Generalization of the Levi-Civita connection
{\displaystyle \gamma } (see Weyl transformation). A Weyl connection is a torsion free affine connection on M {\displaystyle M} such that, for any g ∈ [ g ]
Weyl_connection
Topics referred to by the same term
Lie algebra to a Lie group, More generally, in a manifold with an affine connection, X ↦ γ X ( 1 ) {\displaystyle X\mapsto \gamma _{X}(1)} , where γ X
Exponential_map
Covariant derivative of the metric tensor
metric g {\displaystyle g} , and let ∇ {\displaystyle \nabla } be an affine connection on the tangent bundle T M {\displaystyle TM} . The nonmetricity tensor
Nonmetricity_tensor
Euclidean space without distance and angles
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent
Affine_space
Mathematical operation on vector spaces
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Tensor_product
Topics referred to by the same term
of an affine connection or covariant derivative (on tensors); the curvature form of an Ehresmann connection: see Ehresmann connection, connection (principal
Curvature_tensor
Concept in mathematics
(more generally) if it has a solder form, then the connection is an example of an affine connection, and the curvature is not the only invariant, since
Connection_(principal_bundle)
Hypothetical topological feature of spacetime
general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamic
Wormhole
Map from a Lie algebra to its Lie group
right-invariant affine connection on G {\displaystyle G} . This is usually different from the canonical left-invariant connection, but both connections have the
Exponential_map_(Lie_theory)
Matrix operation which flips a matrix over its diagonal
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Transpose
Cosmological model in which the observable universe is the interior of a black hole
gravitational field as general relativity but without the condition that the affine connection be symmetric. Fermions, described by Dirac spinor fields, are the
Black_hole_cosmology
Smooth manifold with an inner product on each tangent space
matters that the metric space came from a Riemannian manifold. An (affine) connection is an additional structure on a Riemannian manifold that defines differentiation
Riemannian_manifold
Branch of mathematics
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Multilinear_algebra
Quadratic form related to curvatures of surfaces
manifold and n a field of normal vectors on the hypersurface. (If the affine connection is torsion-free, then the second fundamental form is symmetric.) The
Second_fundamental_form
Differential geometry construct on fiber bundles
linear when applied to connections, is sometimes used (like the word affine – see Affine connection) to refer to connections defined on the tangent bundle
Ehresmann_connection
Generalization of straight line to a curved space time
are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) symmetric in the two lower indices
Geodesics in general relativity
Geodesics_in_general_relativity
Method for specifying point positions
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Coordinate_system
Theory of gravitation as curved spacetime
are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower
General_relativity
Japanese mathematician
University and published joint papers with his friend Shigeru Ishihara on affine connections and conformal transformations from 1953 to 1955. In 1956, he established
Morio_Obata
Construct in differenital geometry
In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of
Metric_connection
Set of vectors used to define coordinates
notions of an affine space, projective space, convex set, and cone have related notions of basis. An affine basis for an n-dimensional affine space is n
Basis_(linear_algebra)
Application of Lagrangian mechanics to field theories
because F is an exact form. The A field can be understood to be the affine connection on a U(1)-fiber bundle. That is, classical electrodynamics, all of
Lagrangian_(field_theory)
Mathematical function, in linear algebra
numbers, the map x ↦ x + 1 {\textstyle x\mapsto x+1} is not linear (but is an affine transformation). If A {\displaystyle A} is a m × n {\displaystyle m\times
Linear_map
Attempt to extend Yang–Mills theory to gravity
identified with the translation part of an affine connection on a world manifold X {\displaystyle X} . Any such connection is a sum K = Γ + Θ {\displaystyle K=\Gamma
Gauge_gravitation_theory
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Kronecker_delta
Isomorphism between the tangent and cotangent bundles of a manifold
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Musical_isomorphism
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Moment_of_inertia
Cosmological fine-tuning problem
general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamical
Flatness_problem
Local rescaling of a metric tensor
Levi-Civita connection and associated spin connections are not invariant under Weyl transformations. Weyl connections are a class of affine connections that
Weyl_transformation
Algebraic object with geometric applications
of a tensor density is the current density of electromagnetism. Under an affine transformation of the coordinates, a tensor transforms by the linear part
Tensor
Unique existence of the Levi-Civita connection
a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-)Riemannian connection of the given
Fundamental theorem of Riemannian geometry
Fundamental_theorem_of_Riemannian_geometry
Continuous surjection satisfying a local triviality condition
input." This property is formally captured in the notion of dependent type. Affine bundle Algebra bundle Characteristic class Covering map Equivariant bundle
Fiber_bundle
Technique in statistics
function). In this case, the manifold naturally inherits two flat affine connections, as well as a canonical Bregman divergence. Historically, much of
Information_geometry
Shorthand notation for tensor operations
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Einstein_notation
Mathematical statistics distance measure
similar to the Hellinger metric (in the sense that it induces the same affine connection on a statistical manifold). Furthermore, the Jensen–Shannon divergence
Kullback–Leibler_divergence
Mathematical structure
in two quite different ways in connection with the affine building X for SLn(Qp): The link of each vertex L in the affine building corresponds to submodules
Building_(mathematics)
Levi-Civita connection parallel transport Development (differential geometry) connection form Cartan connection affine connection conformal connection projective
List of differential geometry topics
List_of_differential_geometry_topics
metric [ g ] {\displaystyle [g]} , then a Weyl connection is by definition a torsion-free affine connection ∇ {\displaystyle \nabla } such that ∇ g = α ⊗
Einstein–Weyl_geometry
X {\displaystyle {\mathcal {T}}/{\mathcal {X}}} . One recovers an affine connection in the projective class from a section X {\displaystyle X} of X {\displaystyle
Tractor_bundle
(pseudo-)Riemannian manifold whose geodesics are reversible
regarded as affine symmetric spaces. If M = G / H is a symmetric space, then Nomizu showed that there is a G-invariant torsion-free affine connection (i.e.
Symmetric_space
Universal construction in multilinear algebra
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Tensor_algebra
Tensor that describes the 4D geometry of spacetime
there is a unique connection ∇ on any semi-Riemannian manifold that is compatible with the metric and torsion-free. This connection is called the Levi-Civita
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Differential form of degree one or section of a cotangent bundle
manifold. The prototypical example of this application is a connection one-form. A connection one-form takes values in a Lie algebra corresponding to the
One-form
Mathematical notation
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Multi-index_notation
Riemannian manifold or more generally any manifold equipped with an affine connection, the notion of "straight line" generalizes to that of a geodesic.
Levi-Civita_parallelogramoid
Type of derivative in differential geometry
derivation, which can be applied to scalars, vectors, tensors and affine connections and which proved to be a powerful instrument in the study of groups
Lie_derivative
Set of a ring's prime ideals
The adjective "affine" in the phrase "affine scheme" comes from the fact that an affine algebraic variety can be identified with the affine scheme built
Spectrum_of_a_ring
Array of numbers
matrices to represent objects; to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a three-dimensional
Matrix_(mathematics)
Decomposition in multilinear algebra
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Tensor_rank_decomposition
(Ricci tensor, Weyl tensor). SageManifolds can also deal with generic affine connections, not necessarily Levi-Civita ones. More documentation is on doc.sagemath
Sage_Manifolds
Concepts in mathematics
defined. Let M be a pseudo-Riemannian manifold (or any manifold with an affine connection) and let p be a point in M. Then for every V in TpM there exists a
Vector_flow
Antisymmetric permutation object acting on tensors
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Levi-Civita_symbol
Assignment of a tensor continuously varying across a region of space
space R n {\displaystyle \mathbb {R} ^{n}} may be subjected to arbitrary affine transformations: x k ↦ A j k x j + a k {\displaystyle x^{k}\mapsto A_{j}^{k}x^{j}+a^{k}}
Tensor_field
Exterior algebraic map taking tensors from p forms to n-p forms
Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors
Hodge_star_operator
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