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Type of fiber bundle
mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine. Let π ¯ : Y ¯
Affine_bundle
Construct allowing differentiation of tangent vector fields of manifolds
methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry and tensor
Affine_connection
Let Y → X be an affine bundle modelled over a vector bundle Y → X. A connection Γ on Y → X is called the affine connection if it as a section Γ : Y →
Connection_(affine_bundle)
Gauge theory with affine connections
Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold X {\displaystyle
Affine_gauge_theory
Continuous surjection satisfying a local triviality condition
type. Affine bundle Algebra bundle Characteristic class Covering map Equivariant bundle Fibered manifold Fibration Gauge theory Hopf bundle I-bundle Natural
Fiber_bundle
Topics referred to by the same term
constrained linear combination Affine connection, a connection on the tangent bundle of a differentiable manifold Affine coordinate system, a coordinate
Affine
Euclidean space without distance and angles
associated adjoint bundle. For any non-empty subset X of an affine space A, there is a smallest affine subspace that contains it, called the affine span of X.
Affine_space
Straight path on a curved surface or a Riemannian manifold
More precisely, an affine connection gives rise to a splitting of the double tangent bundle TTM into horizontal and vertical bundles: T T M = H ⊕ V . {\displaystyle
Geodesic
Function in mathematics
manifold) Connection (principal bundle) Connection (vector bundle) Connection (affine bundle) Connection (composite bundle) Connection (algebraic framework)
Connection_(mathematics)
Topics referred to by the same term
a manifold Connection (affine bundle) Connection (composite bundle) Connection (fibred manifold) Connection (principal bundle), gives the derivative of
Connection
Mathematical object studied in the field of algebraic geometry
An irreducible affine algebraic set is also called an affine variety. (Some authors use the phrase affine variety to refer to any affine algebraic set
Algebraic_variety
In comparison with General Relativity, dynamic variables of metric-affine gravitation theory are both a pseudo-Riemannian metric and a general linear connection
Metric-affine gravitation theory
Metric-affine_gravitation_theory
aforementioned isomorphism. Explicitly, the tractor bundle can be represented in a given affine chart by pairs ( μ i ρ ) {\displaystyle (\mu ^{i}\
Tractor_bundle
Concept in algebraic geometry
X_{s}} is an affine scheme. For example, the trivial line bundle O X {\displaystyle {\mathcal {O}}_{X}} is ample if and only if X is quasi-affine. In general
Ample_line_bundle
associated bundle — such a connection is equivalently given by a Cartan connection for the affine group of affine space, and is often called an affine connection
Linear_connection
Mathematical construct of fiber bundles
soldering on an affine bundle E is a choice of isomorphism of E with the tangent bundle of M. Often one speaks of a solder form on a vector bundle, where it
Solder_form
Generalization of vector bundles
forms on X {\displaystyle X} . For example, a section of the canonical bundle of affine space A n {\displaystyle \mathbb {A} ^{n}} over k {\displaystyle k}
Coherent_sheaf
Concept in mathematics
principal G-connections is an affine space for this space of 1-forms. For the trivial principal G {\displaystyle G} -bundle π : E → X {\displaystyle \pi
Connection_(principal_bundle)
Assignment of a tensor continuously varying across a region of space
language of vector bundles, the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles w times. While locally
Tensor_field
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)
Principal fiber bundle
four-potential, (equivalently, the affine connection) such that π ∗ F = d A . {\displaystyle \pi ^{*}F=dA.} Given a circle bundle P over M and its projection
Circle_bundle
Concept in differential geometry
in a principal bundle with the curvature form of the connection. To make this theorem plausible, consider the familiar case of an affine connection (or
Holonomy
System of moving vectors in differential geometry
If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to
Parallel_transport
Grassmann bundle Grassmann dimensions Grassmann graph Grassmann integral Grassmann number Grassmann variables Grassmannian Affine Grassmannian Affine Grassmannian
List of things named after Hermann Grassmann
List_of_things_named_after_Hermann_Grassmann
Defines a notion of parallel transport on a bundle
covariant derivative or affine connection defines a connection on the tangent bundle of M, or more generally on any tensor bundle formed by taking tensor
Connection_(vector_bundle)
Algebraic variety defined within an affine space
geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More
Affine_variety
Generalization of algebraic variety
finitely generated module on each affine open subset of X. Coherent sheaves include the important class of vector bundles, which are the sheaves that locally
Scheme_(mathematics)
Concept in mathematics
is simpler for a vector bundle (i.e., a variety corresponding to a locally free sheaf of constant rank). We say a vector bundle E on an algebraic variety
Equivariant_sheaf
Generalization of a vector bundle
{Spec} _{X}R} of a quasi-coherent graded OX-algebra R is called the cone or affine cone of R. Similarly, the relative Proj P ( C ) = Proj X R {\displaystyle
Cone_(algebraic_geometry)
{K}})/G({\mathcal {O}})} . Alexander Schmitt (11 August 2010). Affine Flag Manifolds and Principal Bundles. Springer. pp. 3–6. ISBN 978-3-0346-0287-7. Retrieved
Affine_Grassmannian
Specification of a derivative along a tangent vector of a manifold
contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded
Covariant_derivative
Type of Kac–Moody algebras
the integers. Central extensions of the affine Lie group by a single generator are topologically circle bundles over this free loop group, which are classified
Affine_Lie_algebra
Application of Lagrangian mechanics to field theories
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 its effects
Lagrangian_(field_theory)
isomorphic affine schemes. It can be noted that the ring of global sections of this scheme is a field, which implies that the scheme is not affine. Any two
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
with the spectrum of a ring. Important affine morphisms are vector bundles, and finite morphisms. 5. The affine cone over a closed subvariety X of a projective
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Operation on fibered manifolds
=\Gamma _{\lambda }^{i}\,,} of the jet bundle J1Y → Y, and vice versa. It is an affine bundle modelled on a vector bundle There are the following corollaries
Connection_(fibred_manifold)
Array of numbers describing a metric connection
describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing
Christoffel_symbols
Object in differential geometry
of Einstein–Cartan theory. Let M be a manifold with an affine connection on the tangent bundle (aka covariant derivative) ∇. The torsion tensor (sometimes
Torsion_tensor
Concept in algebraic geometry
a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion
Smooth_scheme
geometry, affine differential geometry is the study of differential invariants of curves, surfaces, and higher-dimensional submanifolds under affine transformations
Affine_differential_geometry
Mathematical concept
intersection matrix of the components. This is either a 1×1 zero matrix, or an affine Cartan matrix, whose Dynkin diagram is given. The multiplicities of each
Elliptic_surface
Generalization of affine connections
geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept
Cartan_connection
Theorem in algebraic geometry that builds a homotopy equivalent affine variety
X is a scheme quasi-projective over an affine scheme, then there exists a vector bundle E over X and an affine E-torsor W. By the definition of a torsor
Jouanolou's_trick
Type of ringed space
affine if Y {\displaystyle Y} has an open affine cover U i {\displaystyle U_{i}} 's such that f − 1 ( U i ) {\displaystyle f^{-1}(U_{i})} are affine.
Sheaf_of_algebras
Concept in economics
the quantity of goods. An example is affine pricing. A nonlinear price schedule is a menu of different-sized bundles at different prices, from which the
Nonlinear_pricing
Partial differential equations whose solutions are instantons
of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the
Yang–Mills_equations
Manifold upon which it is possible to perform calculus
bundle. One can also define the tangent bundle as the bundle of 1-jets from R (the real line) to M. One may construct an atlas for the tangent bundle
Differentiable_manifold
Universal construction in multilinear algebra
contraction Tensor product Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and
Tensor_algebra
{\displaystyle \mathbf {F} _{q}} and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by Bun G ( X ) {\displaystyle
Moduli stack of principal bundles
Moduli_stack_of_principal_bundles
Branch of mathematics
vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be spacetime and the bundles and
Differential_geometry
Affine connection on the tangent bundle of a manifold
relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric
Levi-Civita_connection
Algebraic structure in linear algebra
counterpart to vector bundles. Roughly, affine spaces are vector spaces whose origins are not specified. More precisely, an affine space is a set with a
Vector_space
Scheme in algebraic geometry
normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry. The normal cone CXY or
Normal cone (algebraic geometry)
Normal_cone_(algebraic_geometry)
Attempt to extend Yang–Mills theory to gravity
general relativity and metric-affine gravitation theory as the gauge ones. In terms of gauge theory on natural bundles, gauge fields are linear connections
Gauge_gravitation_theory
Tensor in differential geometry
Kähler manifold. The Ricci tensor can also be generalized to arbitrary affine connections, where it is an invariant that plays an especially important
Ricci_curvature
Vector bundle existing over a Grassmannian
line bundle as defined before if k is the field of real or complex numbers. In more concise terms, L is the blow-up of the origin of the affine space
Tautological_bundle
Generalisation of a sheaf; a fibered category that admits effective descent
2009), "Lectures on the Moduli Stack of Vector Bundles on a Curve", Affine Flag Manifolds and Principal Bundles, Basel: Springer Basel (published 2010), pp
Stack_(mathematics)
Differential geometry construct on fiber bundles
word affine – see Affine connection) to refer to connections defined on the tangent bundle or frame bundle. An Ehresmann connection on a fiber bundle (endowed
Ehresmann_connection
are affine. Then X is a quotient stack if and only if it has the resolution property; i.e., every coherent sheaf is a quotient of a vector bundle. Earlier
Quotient_stack
Mathematical operation on vector spaces
contraction Tensor product Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and
Tensor_product
Isomorphism between the tangent and cotangent bundles of a manifold
isomorphism) is an isomorphism between the tangent bundle T M {\displaystyle \mathrm {T} M} and the cotangent bundle T ∗ M {\displaystyle \mathrm {T} ^{*}M} of
Musical_isomorphism
Number used in algebraic geometry
In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the variety with n hyperplanes in
Degree of an algebraic variety
Degree_of_an_algebraic_variety
Projective analogue of the spectrum of a ring
Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective
Proj_construction
Connection on a spinor bundle
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 as the gauge
Spin_connection
Method of defining surface detail on a computer-generated graphic or 3D model
triangles for rendering and affine mapping is used on them. The reason this technique works is that the distortion of affine mapping becomes much less noticeable
Texture_mapping
Object in differential geometry
contorsion tensor expresses the difference between a metric-compatible affine connection with Christoffel symbol Γ k i j {\displaystyle {\Gamma ^{k}}_{ij}}
Contorsion_tensor
Concept in algebraic geometry
canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle
Canonical_bundle
Study of complex manifolds and several complex variables
cohomology of line bundles on compact Kähler manifolds, and Cartan's theorems A and B for the cohomology of coherent sheaves on affine complex varieties
Complex_geometry
Concept in algebraic geometry
consequence of the vanishing of cohomology for affine schemes: for a separated scheme X {\displaystyle X} , an affine open covering { U i } {\displaystyle \{U_{i}\}}
Coherent_sheaf_cohomology
Generalizations of codimension-1 subvarieties of algebraic varieties
by analysing its codimension-1 subvarieties and the corresponding line bundles. On singular varieties, this property can also fail, and so one has to
Divisor_(algebraic_geometry)
Algebra associated to any vector space
(oriented) k {\displaystyle k} -dimensional volume and exterior algebra is affine space. This is also the intimate connection between exterior algebra and
Exterior_algebra
Relates the geometric vector bundles to algebraic projective modules
(1955, §50) applies to vector bundles in the category of affine varieties over an algebraically closed field. Let X be an affine variety with structure sheaf
Serre–Swan_theorem
Structure group sub-bundle on a tangent frame bundle
1-form on M with values in the adjoint bundle AdQ. That is to say, the space AQ of adapted connections is an affine space for Ω1(AdQ). The torsion of an
G-structure_on_a_manifold
Shorthand notation for tensor operations
contraction Tensor product Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and
Einstein_notation
Algebraic variety in a projective space
by open affine subvarieties and satisfies the separation axiom. Thus, the local study of X (e.g., singularity) reduces to that of an affine variety.
Projective_variety
Study of vector bundles, principal bundles, and fibre bundles
theory is the space of connections on a vector bundle or principal bundle. This is an infinite-dimensional affine space A {\displaystyle {\mathcal {A}}} modelled
Gauge_theory_(mathematics)
Type of physical quantity
density 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
Pseudotensor
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
contraction Tensor product Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and
Kronecker_delta
Exterior algebraic map taking tensors from p forms to n-p forms
can play a role in differential geometry when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms. This
Hodge_star_operator
Ruled surface over the projective line
this P 1 {\displaystyle \mathbb {P} ^{1}} -bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts
Hirzebruch_surface
on a jet bundle J 1 Q → R {\displaystyle J^{1}Q\to \mathbb {R} } . This equation also is represented by a connection on an affine jet bundle J 1 Q → Q
Non-autonomous system (mathematics)
Non-autonomous_system_(mathematics)
Physical theory with fields invariant under the action of local "gauge" Lie groups
theory shows that affine representations (i.e., affine modules) of the gauge transformations can be classified as sections of a jet bundle satisfying certain
Gauge_theory
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)
Most general completion of a commutative square given two morphisms with same codomain
product over R, and Spec is a contravariant functor, the pullback of two affine schemes Spec(A) and Spec(B) over Spec(R), usually called fiber product,
Pullback_(category_theory)
Commutative algebra theorem
geometric setting it is a statement about the triviality of vector bundles on affine space. The theorem states that every finitely generated projective
Quillen–Suslin_theorem
Set on which a group acts freely and transitively
left or right multiplication. Another example is the affine space concept: the idea of the affine space A underlying a vector space V can be said succinctly
Principal_homogeneous_space
Covariant derivative of the metric tensor
{\displaystyle g} , and let ∇ {\displaystyle \nabla } be an affine connection on the tangent bundle T M {\displaystyle TM} . The nonmetricity tensor is defined
Nonmetricity_tensor
Algebraic operation on coordinate vectors
contraction Tensor product Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and
Dot_product
Mathematical space
Grassmannian plays a similar role for algebraic K-theory. Affine Grassmannian Grassmann bundle Grassmann graph Lee 2012, p. 22, Example 1.36. Shafarevich
Grassmannian
Operation that pairs a left and a right R-module into an abelian group
bundles T M , T ∗ M {\displaystyle TM,T^{*}M} are viewed as locally free sheaves on M. The exterior bundle on M is the subbundle of the tensor bundle
Tensor_product_of_modules
Generalization of an ordered basis of a vector space
geometry, p. 18 Griffiths 1974 "Affine frame" Proofwiki.org See Cartan (1983) 9.I; Appendix 2 (by Hermann) for the bundle of tangent frames. Fels and Olver
Moving_frame
Concept in algebraic geometry
there are open affine subsets U = Spec A of X containing x and V = Spec B of Y such that ƒ(U) ⊆ V. Then ƒ: U → V is a morphism of affine schemes and thus
Morphism_of_schemes
Type of functor
the trivial bundle A k n {\displaystyle \mathbb {A} _{k}^{n}} over Spec ( k ) {\displaystyle \operatorname {Spec} (k)} . A quasi-affine morphism between
Morphism_of_algebraic_stacks
Method for specifying point positions
contraction Tensor product Transpose (2nd-order tensors) Related abstractions Affine connection Basis Cartan formalism (physics) Connection form Covariance and
Coordinate_system
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
Topics referred to by the same term
manifold — see also Curvature of Riemannian manifolds; the curvature of an affine connection or covariant derivative (on tensors); the curvature form of an
Curvature_tensor
Topological space that locally resembles Euclidean space
from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields. Schemes are likewise glued together from affine schemes
Manifold
Type of derivative in differential geometry
of Lie derivation, which can be applied to scalars, vectors, tensors and affine connections and which proved to be a powerful instrument in the study of
Lie_derivative
Theory of gravity
π : M → M be the Minkowski fiber bundle over the spacetime manifold M. For each point p ∈ M, the fiber Mp is an affine space. In a fiber chart (V, ψ),
Teleparallelism
Concept in mathematics
on the affine line A1 (that is, on the complex numbers C). This equation corresponds to a flat connection on the trivial algebraic line bundle over A1
Riemann–Hilbert correspondence
Riemann–Hilbert_correspondence
AFFINE BUNDLE
AFFINE BUNDLE
Female
Scandinavian
Scandinavian form of Hebrew Adiyna, ADINE means "slender."
Female
English
English pet form of Latin Euphemia, EFFIE means "Well I speak."
Girl/Female
French
Blond.
Girl/Female
Armenian
Valuable.
Girl/Female
French
May Jehovah add. Addition (to the family). A feminine form of Joseph.
Girl/Female
English Latin
Warm.
Female
English
Variant spelling of English Aline, ALLINE means "little Eve."Â
Girl/Female
German
Soldier. Army Man. from the Old German Hariman.
Girl/Female
Latin
Red haired.
Female
English
 Variant spelling of English Aileen, ALINE means "little Eve." Compare with another form of Aline.
Male
English
Middle English form of Anglo-Saxon Ealdwine, ALDINE means "old friend."
Girl/Female
Irish American Celtic English French
Oath.
Girl/Female
Irish
In charge.
Female
English
Pet form of English Saffron, SAFFIE means "saffron (the spice)."
Male
English
Pet form of English Alfred, ALFIE means "elf counsel."
Female
French
 Contracted form of French Adeline, ALINE means "little noble." Compare with another form of Aline.
Male
English
English name, probably derived from the vocabulary word alpine, ALPINE means "of the Swiss Alps."
Female
Hebrew
Variant spelling of Hebrew Amina, AMINE means "faithful, trusted."
Girl/Female
Italian
Famous bearer: Alcine is mistress of alluring enchantments and sensual pleasures in the Orlando...
Girl/Female
Irish French
Beautiful.
AFFINE BUNDLE
AFFINE BUNDLE
Girl/Female
Tamil
Kokilapriya | கோகீலாபà¯à®°à®¿à®¯à®¾
Name of a Raga
Girl/Female
Arabic, British, English, Farsi, Gujarati, Hindu, Indian, Kannada, Sindhi, Tamil, Telugu, Urdu
Evening; Night; Reward; Smile; Happy; Peaceful; Love Affection; Stars of Sky; Beauty
Boy/Male
Bengali, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Telugu, Traditional
Calm and Composed
Boy/Male
Norse
Eternal king's son.
Boy/Male
Arabic
Darling; Dear
Boy/Male
Gujarati, Hindu, Indian
Fearless Lord Shiva
Girl/Female
Tamil
Position
Girl/Female
Hindu
Moonlit night
Boy/Male
American, British, English, French
Riverbank; Derived from Place-name Deverel
Girl/Female
Indian, Traditional
Singer
AFFINE BUNDLE
AFFINE BUNDLE
AFFINE BUNDLE
AFFINE BUNDLE
AFFINE BUNDLE
n.
The company or corporation, or persons collectively, whose place of business is in an office; as, I have notified the office.
v. t.
To fix or fasten figuratively; -- with on or upon; as, eyes affixed upon the ground.
v. t.
To refine.
v. t.
To reduce to a fine, unmixed, or pure state; to free from impurities; to free from dross or alloy; to separate from extraneous matter; to purify; to defecate; as, to refine gold or silver; to refine iron; to refine wine or sugar.
a.
Of or pertaining to the Alps, or to any lofty mountain; as, Alpine snows; Alpine plants.
a.
Of, from, in, or pertaining to, the belly or the intestines; as, alvine discharges; alvine concretions.
v. t.
To perform, as the duties of an office; to discharge.
v. t.
To determine or clearly exhibit the boundaries of; to mark the limits of; as, to define the extent of a kingdom or country.
v. t.
To define.
v. i.
To pay a fine. See Fine, n., 3 (b).
n.
A special duty, trust, charge, or position, conferred by authority and for a public purpose; a position of trust or authority; as, an executive or judical office; a municipal office.
a.
Andean; as, Andine flora.
pl.
of Affix
n.
The place where a particular kind of business or service for others is transacted; a house or apartment in which public officers and others transact business; as, the register's office; a lawyer's office.
v. t.
To subjoin, annex, or add at the close or end; to append to; to fix to any part of; as, to affix a syllable to a word; to affix a seal to an instrument; to affix one's name to a writing.
v. t.
To attach, unite, or connect with; as, names affixed to ideas, or ideas affixed to things; to affix a stigma to a person; to affix ridicule or blame to any one.
a.
To make fine; to refine; to purify, to clarify; as, to fine gold.
n.
That part of the sea at a good distance from the shore, or where there is deep water and no need of a pilot; also, distance from the shore; as, the ship had ten miles offing; we saw a ship in the offing.
imp. & p. p.
of Affix