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Principal fiber bundle
In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S 1 {\displaystyle S^{1}} . Oriented circle bundles are also known as
Circle_bundle
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
(also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere.
Hopf_fibration
Continuous surjection satisfying a local triviality condition
In mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is locally a product space, but globally
Fiber_bundle
Vector bundle of rank 1
homotopy type of a circle. From the perspective of homotopy theory, a real line bundle therefore behaves much the same as a fiber bundle with a two-point
Line_bundle
Tangent spaces of a manifold
instance, in the case where the manifold is a Lie group. The tangent bundle of the unit circle is trivial because it is a Lie group (under multiplication and
Tangent_bundle
Topological space
decomposition as a disjoint union of circles. In other words, it is a S 1 {\displaystyle S^{1}} -bundle (circle bundle) over a 2-dimensional orbifold. Many
Seifert_fiber_space
space. A circle bundle is a special case of a sphere bundle. A sphere bundle that is a product space is orientable, as is any sphere bundle over a simply
Sphere_bundle
Special type of principal bundle
\operatorname {U} (1)} -bundles without their group action are in particular circle bundles. These are basically topological spaces with a circle glued to every
Principal_U(1)-bundle
Mathematical parametrization of vector spaces by another space
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X
Vector_bundle
obstruction to endow circle bundles over surfaces with a flat structure. It is named after John Milnor and John W. Wood. For linear bundles, flatness is defined
Milnor–Wood_inequality
Mathematical object
interesting action of the circle group T on S3 giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. If one thinks of S3
3-sphere
In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space
Surface bundle over the circle
Surface_bundle_over_the_circle
Group of real 2×2 matrices with unit determinant
PSL(2, R) can be described as the unit tangent bundle of the hyperbolic plane. It is a circle bundle, and has a natural contact structure induced by
SL2(R)
Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles. They were discovered and classified by the German topologist Friedhelm
Graph_manifold
then, by an observation of Shoshichi Kobayashi, the circle bundle S in its canonical line bundle admits a Sasaki–Einstein metric, in a manner that makes
Sasakian_manifold
Unified field theory
understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group
Kaluza–Klein_theory
Quotient of special unitary group by its center
from the long exact sequence for bundles and the above fact that SU(n) is a Z / n {\displaystyle \mathbb {Z} /n} bundle over PU(n). The cohomology in the
Projective_unitary_group
Mathematics concept
vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle π : E → B
Vertical and horizontal bundles
Vertical_and_horizontal_bundles
band, the only two possible I-bundles over the circle S 1 {\displaystyle S^{1}} . The annulus is a trivial or untwisted bundle because it corresponds to the
I-bundle
Mathematical concept
→ ∞ {\displaystyle n\to \infty } . This gives a fiber bundle (called the universal circle bundle) S 1 ↪ S ∞ ↠ C P ∞ {\displaystyle S^{1}\hookrightarrow
Complex_projective_space
Symmetry of physical laws under a charge-conjugation transformation
equations, can be interpreted as a structure on a U(1) fiber bundle, the so-called circle bundle. This provides a geometric interpretation of electromagnetism:
C-symmetry
In Euclidean geometry, the bundle theorem is a statement about six circles and eight points in the Euclidean plane. In general incidence geometry, it
Bundle_theorem
General relativity in M-theory
to the connection of the circle bundle and a 2-form field strength which is equal to the Chern class of the old circle bundle. One may then lift this theory
Higher-dimensional supergravity
Higher-dimensional_supergravity
Courting behavior
case established that bundling was a common practice in certain rural social circles at the time. By the 20th century, bundling seems to have disappeared
Bundling_(tradition)
Theory in theoretical physics
contains no topological strings. However topological M-theory on a circle bundle over a 6-manifold has been conjectured to be equivalent to the topological
Topological_string_theory
Mathematical construct of fiber bundles
the Mobius strip as a fiber bundle over the circle. The vertical bundle o*VE is still a Mobius strip, while the tangent bundle TM is the cylinder, so there
Solder_form
Fiber bundle whose fibers are group torsors
In the mathematical area of topology, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product
Principal_bundle
cohomology. U ( 1 ) {\displaystyle U(1)} -principal bundles over a space M {\displaystyle M} (see circle bundle) are geometrical realizations of 1-classes in
Bundle_gerbe
Application of Lagrangian mechanics to field theories
U(1)-fiber bundle. That is, classical electrodynamics, all of its effects and equations, can be completely understood in terms of a circle bundle over Minkowski
Lagrangian_(field_theory)
Topologically stable solution of a partial differential equation
1 ) ≃ S 1 {\displaystyle U(1)\simeq S^{1}} is the circle; the mappings arise in the circle bundle. Such maps can be thought of as winding a string around
Topological_defect
A torus bundle, in the sub-field of geometric topology in mathematics, is a kind of surface bundle over the circle, which in turn is a class of three-manifolds
Torus_bundle
Set of topological invariants
that the vector bundle is not orientable. For example, the first Stiefel–Whitney class of the Möbius strip, as a line bundle over the circle, is not zero
Stiefel–Whitney_class
Principal bundle associated to a vector bundle
In mathematics, a frame bundle is a principal fiber bundle F ( E ) {\displaystyle F(E)} associated with any vector bundle E {\displaystyle E} . The fiber
Frame_bundle
Circle bundle – Principal fiber bundle Quasicircle Circle-related theory Apollonius' problem – Geometry problem about finding touching circles Limiting
List_of_circle_topics
American mathematician (1946–2012)
Morris Hirsch, with his thesis Foliations of Three-Manifolds which are Circle Bundles in 1972. After completing his Ph.D., Thurston spent a year at the Institute
William_Thurston
Vector bundle of cotangent spaces at every point in a manifold
mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold
Cotangent_bundle
It is a generalization of the Lie bracket from an operation on the tangent bundle
from an operation on the tangent bundle to an operation on the direct sum of the tangent bundle and the vector bundle of p-forms. The case p = 1 {\displaystyle
Courant_bracket
Characteristic class of oriented, real vector bundles
real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth
Euler_class
Topological space that locally resembles Euclidean space
of a normal bundle, but instead there is an intrinsic stable normal bundle. The n-sphere Sn is a generalisation of the idea of a circle (1-sphere) and
Manifold
Bundle in which the fiber is a surface
In mathematics, a surface bundle is a bundle in which the fiber is a surface. When the base space is a circle the total space is three-dimensional and
Surface_bundle
Type of differentiable manifold
{\displaystyle p} . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on
Parallelizable_manifold
American video streaming service
also announced a bundle including its other U.S. streaming services Hulu (ad-supported version) and ESPN+, marketed as The Disney Bundle, initially for
Disney+
Construction in group theory
fiber is C× ≅ S1, so up to homotopy, GL → PGL is a circle bundle. The higher homotopy groups of the circle vanish, so the homotopy groups of GL(n, C) and
Projective_linear_group
Intrinsic geometric structures in mathematics
the frame or circle bundles of M. The definitions of the tangent bundle, the unit tangent bundle and the (oriented orthonormal) frame bundle F can be extended
Riemannian connection on a surface
Riemannian_connection_on_a_surface
Curve whose normals converge asymptotically
a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant curvature where all the
Horocycle
torus Σφ. Since φ is the identity on ∂Σ, ∂Σφ is the trivial circle bundle over a union of circles, that is, a union of tori; one torus for each boundary component
Open_book_decomposition
Study of vector bundles, principal bundles, and fibre bundles
theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused
Gauge_theory_(mathematics)
Intersection of a torus and a plane
In geometry, Villarceau circles (/viːlɑːrˈsoʊ/) are a pair of circles produced by cutting a torus obliquely through its center at a special angle. Given
Villarceau_circles
English actress (born 1997)
starred in the Netflix series Agatha Christie's Seven Dials as Lady Eileen "Bundle" Brent. McKenna-Bruce was born in 1997 and was raised in Chislehurst, Bromley
Mia_McKenna-Bruce
Concept in topological group theory
the quotient by the center. By Iwasawa decomposition, both groups are circle bundles over the complex upper half-plane, and their universal cover S L 2 (
Covering_group
Complex four-component spinor
electromagnetism, is a U(1) fiber bundle (the circle bundle), and the Aharonov–Bohm effect demonstrates the holonomy of that bundle. All this has no direct impact
Plane-wave solutions to the Dirac equation
Plane-wave_solutions_to_the_Dirac_equation
Handheld game console
Crossing: Happy Home Designer-themed bundle including the console, game software, two faceplates, and an Amiibo card. The bundle was priced at $219.99. A second
New_Nintendo_3DS
Quotient of a weakly contractible space by a free action
the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle E G → B G {\displaystyle EG\to BG}
Classifying_space
Symbol of the British Union of Fascists
The BUF was founded in 1932 and adopted the fasces as its emblem, whose bundle of sticks represents strength through unity, and whose axe represents the
Flash_and_circle
Exact homotopy case
group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by
Classifying_space_for_U(n)
Brazilian mathematician
Ciências. Vol. 37, no. 2. p. 167-184. Lyra, Carlos Benjamin de (1959). "On circle bundles over complex projective space". Anais da Academia Brasileira de Ciências
Carlos_Benjamin_de_Lyra
Straight path on a curved surface or a Riemannian manifold
double tangent bundle TTM into horizontal and vertical bundles: T T M = H ⊕ V . {\displaystyle TTM=H\oplus V.} The double tangent bundle can be visualized
Geodesic
Differentiable manifold
manifold with zero Webster torsion and zero Webster curvature. The unit circle bundle over compact Riemann surfaces with genus strictly greater than 1 also
CR_manifold
Mathematical space
I-bundles Knot and link complements Lens space Seifert fiber spaces, Circle bundles Spherical 3-manifold Surface bundles over the circle Torus bundle A
3-manifold
Differentiable function whose derivative is everywhere injective
normal bundles plus trivial bundles, and thus if the stable normal bundle has cohomological dimension k, it cannot come from an (unstable) normal bundle of
Immersion_(mathematics)
Scheduled passenger air journeys
ways, with the most common standard flight length measurement being great-circle distance, a formula that calculates the shortest distance across the curvature
Longest_flights
Smartphone sold from 2014-2016
device runs a modified version of Android called SilentOS that comes with a bundle of security-minded tools. Versions of the Blackphone were sold from 2014
Blackphone
System of moving vectors in differential geometry
affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along
Parallel_transport
Constructs a fiber bundle from a base space, fiber and a set of transition functions
In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundles with a structure group from a given base space, fiber
Fiber bundle construction theorem
Fiber_bundle_construction_theorem
Four-dimensional analog of the icosahedron
once. It is a different bundle of fibers than the bundle of Clifford parallel polygon great circles, but the two fiber bundles describe the same discrete
600-cell
Branch of mathematics
considerable interest in physics. The apparatus of vector bundles, principal bundles, and connections on bundles plays an extraordinarily important role in modern
Differential_geometry
Lie group of complex numbers of unit modulus; topologically a circle
theory of bundles associated to the circle group, which are primary objects of study in homotopy theory and algebraic topology. Elements of the circle group
Circle_group
Bound bundle of wooden rods, sometimes with an axe
plurale tantum, from the Latin word fascis, meaning 'bundle'; Italian: fascio littorio) is a bound bundle of wooden rods, often, but not always, including
Fasces
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
frame bundle associated to the tautological bundle on a Grassmannian. When one passes to the n → ∞ {\displaystyle n\to \infty } limit, these bundles become
Stiefel_manifold
Proposed ideological bundle
"TESCREAL" in 2023, first using it in a draft of a paper titled "The TESCREAL bundle: Eugenics and the promise of utopia through artificial general intelligence"
TESCREAL
American folk singer-songwriter (born 1985)
guitar, and in 2006 she debuted her first self-released album A Bundled Bundle of Bundles. A year later, in 2007, she released her first studio album Faces
Mariee_Siou
American mathematician
(1998), no. 3, 363–381. C L Epstein, CR-structures on three-dimensional circle bundles. Invent. Math. 109 (1992), no. 2, 351–403. C L Epstein A relative index
Charles Epstein (mathematician)
Charles_Epstein_(mathematician)
Fiber bundle Principal bundle Frame bundle Hopf bundle Associated bundle Vector bundle Tangent bundle Cotangent bundle Line bundle Jet bundle Sheaf (mathematics)
List of differential geometry topics
List_of_differential_geometry_topics
Theorem in geometry
is not an interval bundle over a surface and M is an atoroidal then the skinning map has a fixed point. (If N is an interval bundle then the skinning map
Hyperbolization_theorem
Concept in differential geometry
bundles Spin(n) → SO(n) and Spin(2) → SO(2) respectively. This makes the SpinC group both a bundle over the circle with fibre Spin(n), and a bundle over
Spin_structure
Geometric space whose points represent algebro-geometric objects of some fixed kind
{\displaystyle \mathbf {P} _{\mathbb {Z} }^{n}} , the embedding is given by a line bundle L → X {\displaystyle {\mathcal {L}}\to X} and n + 1 {\displaystyle n+1}
Moduli_space
2025 animated television series
sending Devil May Cry rocketing up the Steam top sellers in a $20 franchise bundle". PC Gamer. Retrieved November 27, 2025. "Integrated Report 2025" (PDF)
Devil_May_Cry_(TV_series)
Doughnut-shaped surface of revolution
manner (with two degenerate circles), a fact that is important in the study of S3 as a fiber bundle over S2 (the Hopf bundle). The surface described above
Torus
2026 studio album by Kanye West
options for the album including multi-colored vinyl, CD, cassette, and bundles; the cassette and CD images contained the track listing. On January 30
Bully_(album)
American actor, producer and director (born 1937)
Title Role Notes 1960 Maverick Ticket Taker / Chuck / Brazos 3 episodes — "Bundle from Britain" — "A Bullet for the Teacher" — "Kiz" 1960–1961 Surfside 6
Max_Baer_Jr.
Line of gamepads developed by Sony
a secondary peripheral for the first PlayStation console. The console's bundle was updated to include DualShock, and phase out the original PlayStation
DualShock
Function in mathematics
manifold) Connection (principal bundle) Connection (vector bundle) Connection (affine bundle) Connection (composite bundle) Connection (algebraic framework)
Connection_(mathematics)
M_{f}={\frac {(I\times X)}{(1,x)\sim (0,f(x))}}} The result is a fiber bundle whose base is a circle and whose fiber is the original space X. If X is a manifold
Mapping_torus
Graphical method for displaying the inter-relationships between a set of entities
inter-relationships between data in a matrix. The data are arranged radially around a circle with the relationships between the data points typically drawn as arcs connecting
Chord diagram (information visualization)
Chord_diagram_(information_visualization)
Simplest non-trivial closed knot with three crossings
meaning that its complement in S 3 {\displaystyle S^{3}} is a fiber bundle over the circle S 1 {\displaystyle S^{1}} . The trefoil K may be viewed as the set
Trefoil_knot
checks in with a/v specialist Steve Hayes as he pulls "futureproof" wiring bundles through the walls and back to a central switching box. Finally, custom
List of This Old House episodes (seasons 11–20)
List_of_This_Old_House_episodes_(seasons_11–20)
Tiles used in mahjong game
of circles. The 1 Circle is generally a large circle of multiple colors, while the rest of the circle tiles consist of smaller circles, each circle being
Mahjong_tiles
Type of Kac–Moody algebras
extensions of the affine Lie group by a single generator are topologically circle bundles over this free loop group, which are classified by a two-class known
Affine_Lie_algebra
Concerns 3 circles through triples of points on the vertices and sides of a triangle
the edges of the tetrahedron intersect in a common point. Bundle theorem Clifford's circle theorems Miquel configuration A high school teacher in the
Miquel's_theorem
2009 video game
It was re-released as a cross-platform game in the third Humble Indie Bundle in 2011, at which time the underlying engine known as "Haaf's Game Engine"
Hammerfight
2001 video game
Castlevania: Circle of the Moon, titled Castlevania in PAL regions, is a 2001 action role-playing game developed and published by Konami for the Game Boy
Castlevania: Circle of the Moon
Castlevania:_Circle_of_the_Moon
2024 single by Coldplay featuring Little Simz, Burna Boy, Elyanna and Tini
23-08-2024)". EarOne (in Italian). Retrieved 23 August 2024. "We Pray - Vinyl & CD Bundle". Coldplay Store UK. 8 August 2024. Archived from the original on 8 August
We_Pray
Studio Liverpool Sony Computer Entertainment 2009-06-19EU Without Warning Circle Studio Capcom 2005-10-28EU Wizardry Empire III: Haoh no Keifu Starfish SD
List of PlayStation 2 games (L–Z)
List_of_PlayStation_2_games_(L–Z)
1303 mathematical monograph by Zhu Shijie
Jade Mirror of the Four Unknowns, Siyuan yujian (simplified Chinese: 四元玉鉴; traditional Chinese: 四元玉鑒), also referred to as Jade Mirror of the Four Origins
Jade Mirror of the Four Unknowns
Jade_Mirror_of_the_Four_Unknowns
Block of Unicode symbols
Unicode block of 96 symbols at code point range U+25A0–25FF. The BLACK CIRCLE is displayed when typing in a password field, in order to hide characters
Geometric Shapes (Unicode block)
Geometric_Shapes_(Unicode_block)
American author and novelist
Heart (2017) Between You and Me (June 26, 2018) The Oysterville Sewing Circle (2019) Welcome to Beachtown (2023) The Twelve Dogs of Christmas (2023) Wayward
Susan_Wiggs
2025 single by Tame Impala
May 2026. Retrieved 1 May 2026. "Jennie Announces New "Dracula (Remix)" Bundle: Here's When It Drops". inMusic. 29 April 2026. Archived from the original
Dracula_(song)
Political party in the United States
. The traditional centre right of the postwar decades could do so by "bundling" moderate social conservatism (moderate by the standards of its day, at
Republican Party (United States)
Republican_Party_(United_States)
(January 26, 2026). "Regular Show Fortnite Skins Revealed: Release Date, Bundle & Cosmetics Confirmed". Vice. Retrieved April 3, 2026. "The Wonderfully
2026_in_animation
Games for the Sony PlayStation / PS1 / PSone
Entertainment Unreleased 2000 October 15, 2000 Mary-Kate and Ashley: Winner's Circle Tantalus Software Acclaim Entertainment Unreleased May 25, 2001 March 1
List of PlayStation (console) games (M–Z)
List_of_PlayStation_(console)_games_(M–Z)
CIRCLE BUNDLE
CIRCLE BUNDLE
Female
Slovene
Feminine form of Slovene Ciril, CIRILA means "lord."
Surname or Lastname
English
English : variant spelling of Cordell.Possibly an Americanized spelling of German Kördel, a pet form of an old German personal name, formed with kuoni ‘daring’. Compare Conrad.
Boy/Male
English
From the bird hill.
Boy/Male
Christian, Hindu, Indian
Bright Circle
Girl/Female
Latin
Circle of light.
Male
Slovene
Slovene form of Greek Kyrillos, CIRIL means "lord."
Girl/Female
Latin
Circle of light.
Boy/Male
English
Birch.
Girl/Female
Greek Latin
A witch.
Boy/Male
French Israeli
The circle.
Female
English
English name derived from the vocabulary word, from Latin miraculum, MIRACLE means "marvel, wonder."
Girl/Female
Latin
Circle of light.
Male
Celtic
, sea circle.
Girl/Female
British, English
Botanical Name; The Myrtle is a Dark Green Shrub with Pink or White Blossoms
Girl/Female
Bengali, Indian
Circle; Normal
Female
French
French form of Latin Carola, CAROLE means "man."
Boy/Male
Spanish Greek
noble.
Female
Yiddish
(מִירל) Yiddish form of Hebrew Miryam, MIRELE means "obstinacy, rebelliousness" or "their rebellion."Â
Girl/Female
Japanese
Ball; circle.
Girl/Female
French American
The french form of the English Carol, a dimunitive of Charles meaning strong.
CIRCLE BUNDLE
CIRCLE BUNDLE
Boy/Male
Indian, Sanskrit
Truth
Boy/Male
Egyptian
Light skin.
Surname or Lastname
English
English : unexplained. There are four farms so named in Warwickshire, one in Oxfordshire, and one in Worcestershire, and the surname is most probably derived from one of these.
Girl/Female
Arabic, Muslim
Friend; Righteous
Boy/Male
Tamil
Lord Shiva
Girl/Female
Hindu, Indian
Taste
Boy/Male
Tamil
Winner
Boy/Male
Hindu, Indian
Name of a Tree
Boy/Male
Greek
Boy/Male
Biblical
The work of the Lord.
CIRCLE BUNDLE
CIRCLE BUNDLE
CIRCLE BUNDLE
CIRCLE BUNDLE
CIRCLE BUNDLE
n.
An imaginary circle or orbit in the heavens; one of the celestial spheres.
n.
A circle; a circus; a circular erection or arrangement of objects.
v. t.
To girdle; to encircle.
n.
A miracle play.
a.
Having the form of a circle; round.
v. t.
To form a circle about; to inclose within a circle or ring; to surround; as, to encircle one in the arms; the army encircled the city.
imp. & p. p.
of Circle
n.
A little circle; esp., an ornament for the person, having the form of a circle; that which encircles, as a ring, a bracelet, or a headband.
n.
An amphitheatrical circle for sports; a circus.
v. i.
To change into curd; to coagulate; as, rennet causes milk to curdle.
n.
An instrument of observation, the graduated limb of which consists of an entire circle.
a.
Having the nature, properties, or qualities, of an adult man; characteristic of developed manhood; hence, masterful; forceful; specifically, capable of begetting; -- opposed to womanly, feminine, and puerile; as, virile age, virile power, virile organs.
n.
A circlet.
n.
A circle.
n.
To encompass, as by a circle; to surround; to inclose; to encircle.
p. pr. & vb. n.
of Circle
v. i.
To move circularly; to form a circle; to circulate.
n.
Alt. of Corcule
v. t.
See Encircle.
n.
One entire round in a circle or a spire; as, a cycle or set of leaves.