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A necessary and sufficient condition for (, /,,) to form a fiber bundle is that the mapping admits local cross-sections (Steenrod 1951, §7). The most general conditions under which the quotient map will admit local cross-sections are not known, although if G {\displaystyle G} is a Lie group and H {\displaystyle H} a closed subgroup (and thus a ...
A plane containing a cross-section of the solid may be referred to as a cutting plane. The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid. For instance, while all the cross-sections of a ball are disks, [2] the cross-sections of a cube depend on how the cutting plane is related to the ...
Sections are studied in homotopy theory and algebraic topology, where one of the main goals is to account for the existence or non-existence of global sections. An obstruction denies the existence of global sections since the space is too "twisted". More precisely, obstructions "obstruct" the possibility of extending a local section to a global ...
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations.This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (projective space) and a selective set of basic geometric concepts.
It is common in mathematics publications that define the Borromean rings to do so as a link diagram, a drawing of curves in the plane with crossings marked to indicate which curve or part of a curve passes above or below at each crossing. Such a drawing can be transformed into a system of curves in three-dimensional space by embedding the plane ...
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In arithmetic geometry, the Cox–Zucker machine is an algorithm created by David A. Cox and Steven Zucker.This algorithm determines whether a given set of sections [further explanation needed] provides a basis (up to torsion) for the Mordell–Weil group of an elliptic surface E → S, where S is isomorphic to the projective line.
A principal -bundle, where denotes any topological group, is a fiber bundle: together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism.
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