In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space...

vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields...

tangent bundle of a differentiable manifold M {\displaystyle M} is a manifold T M {\displaystyle TM} which assembles all the tangent vectors in M {\displaystyle...

complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through...

information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under...

bundle I-bundle Natural bundle Principal bundle Projective bundle Pullback bundle Quasifibration Universal bundle Vector bundle Wu–Yang dictionary Seifert...

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and...

In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k {\displaystyle...

algebra. A vector bundle is a family of vector spaces parametrized continuously by a topological space X. More precisely, a vector bundle over X is a...

covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay...

classes. For example, a principal bundle has a global section if and only if it is trivial. On the other hand, a vector bundle always has a global section,...

principal bundle is the frame bundle F ( E ) {\displaystyle F(E)} of a vector bundle E {\displaystyle E} , which consists of all ordered bases of the vector space...

orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation...

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...

g ) , {\displaystyle (M,g),\,} one defines the spinor bundle to be the complex vector bundle π S : S → M {\displaystyle \pi _{\mathbf {S} }\colon {\mathbf...

gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused...

In mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent T E {\displaystyle...

theory of fiber bundles with a structure group G {\displaystyle G} (a topological group) allows an operation of creating an associated bundle, in which the...

invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney...

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...

tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of...

The pullback of a vector bundle is a vector bundle of the same rank. In particular, the pullback of a line bundle is a line bundle. (Briefly, the fiber...

canonical vector-valued 1-form on the frame bundle, the torsion Θ {\displaystyle \Theta } of the connection form ω {\displaystyle \omega } is the vector-valued...

In mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection. Let π :...

secondary vector bundle structure refers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smooth vector bundle...

representation: for each vector v in V, there is a finite-dimensional G-submodule of V that contains v. A definition is simpler for a vector bundle (i.e., a variety...

any fiber bundle associated to P {\displaystyle P} via the associated bundle construction. In particular, on any associated vector bundle the principal...

vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may...

{\displaystyle GL(n)} -bundle, the frame bundle. In particular, every smooth manifold has a canonical vector bundle, the tangent bundle. For a Lie group G...

mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology...