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...

(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....

In mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is locally a product space, but globally...

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...

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...

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...

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...

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...

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...

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...

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...

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...

→ ∞ {\displaystyle n\to \infty } . This gives a fiber bundle (called the universal circle bundle) S 1 ↪ S ∞ ↠ C P ∞ {\displaystyle S^{1}\hookrightarrow...

cohomology. U ( 1 ) {\displaystyle U(1)} -principal bundles over a space M {\displaystyle M} (see circle bundle) are geometrical realizations of 1-classes in...

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:...

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...

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...

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...

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...

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...

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...

Circle bundle – Principal fiber bundle Quasicircle Circle-related theory Apollonius' problem – Geometry problem about finding touching circles Limiting...

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X × G {\displaystyle...

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...

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...

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...

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...

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 (...

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...

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...