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

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...

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...

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...

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...

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

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

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...

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...

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

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

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...

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

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

a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or...

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...

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...

the fiber is a vector space and the tensor bundle is a special kind of vector bundle. The vector bundle is a natural idea of "vector space depending...

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...

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...

alike, unlike fiber bundles, where the fibers must all be isomorphic (in the case of vector bundles) and homeomorphic. A bundle is a triple (E, p, B)...

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...

the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics...

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...

example, to the notion of a (vector) bundle homomorphism between vector bundles, in which the fibers are vector spaces, and a bundle map φ is required to be...

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,...

setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are...