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...

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

(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....

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...

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...

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...

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

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...

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 over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space...

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

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...

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...

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...

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

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:...

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

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...

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...

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...

Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles. They were discovered and classified by the German topologist Friedhelm...

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...

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...

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...

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...

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...