Category: Vector bundles

Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles.

For instance, Vector bundles always have a zero section whether they are trivial or not and sphere bundles may admit many global sections without being trivial.

Category: Vector bundles

Vector bundles are locally free.

Category: Vector bundles

# REDIRECT Vector bundle # Operations on vector bundles

Category: Vector bundles

Category: Vector bundles

Professor Hector ( Player 1 ) and Professor Vector ( Player 2 ) must collect all of the bundles of dynamite in each of 40 successive levels.

Category: Vector bundles

Category: Vector bundles

# Vector bundles: In a manner similar to the previous examples the projections ( p: V → S ) of real ( complex ) vector bundles to their base spaces form a category Vect < sub > R </ sub > ( Vect < sub > C </ sub >) over Top ( morphisms of vector bundles respecting the vector space structure of the fibres ).

Category: Vector bundles

Vector calculus plays an important role in differential geometry and in the study of partial differential equations.

Vector images are created mathematically ( using formulas, as in geometry ).

Vector images resulted from mathematical geometry ( vector ).

In simple cases, it relates l-adic representations of the étale fundamental group of an algebraic curve to objects of the derived category of l-adic sheaves on the moduli stack of vector bundles over the curve.

Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties — those that become trivial after pullback by a finite étale map — are important.

There are also a number other closely related theorems: an equivalent formulation of this theorem using line bundles and a generalization of the theorem to algebraic curves.

In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential operators between two vector bundles.

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated to complex vector bundles.

The generalized Chern classes in algebraic geometry can be defined for vector bundles ( or more precisely, locally free sheaves ) over any nonsingular variety.

In algebraic geometry, this classification of ( isomorphism classes of ) complex line bundles by the first Chern class is a crude approximation to the classification of ( isomorphism classes of ) holomorphic line bundles by linear equivalence classes of divisors.

Yang – Mills instantons have been explicitly constructed in many cases by means of twistor theory, which relates them to algebraic vector bundles on algebraic surfaces, and via the ADHM construction, or hyperkähler reduction ( see hyperkähler manifold ), a sophisticated linear algebra procedure.

This has many consequences in algebraic geometry, for example, the characterization of certain symmetric spaces, Chern number inequalities for stable bundles, and the restriction of the fundamental groups of a Kähler manifold.

Yau and Karen Uhlenbeck proved the existence and uniqueness of Hermitian – Einstein metrics ( or equivalently Hermitian Yang – Mills connections ) for stable bundles on any compact Kähler manifold, extending an earlier result of Donaldson for projective algebraic surfaces, and M. S. Narasimhan and C. S. Seshadri for algebraic curves.

This thesis firmly established in algebraic topology the use of spectral sequences, and clearly separated the notions of fiber bundles and fibrations from the notion of sheaf ( both concepts together having been implicit in the pioneer treatment of Jean Leray ).

The GAGA says that the theory of holomorphic vector bundles ( more generally coherent analytic sheaves ) on X coincide with that of algebraic vector bundles.

In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n ( and in greater generality for vector bundles and further, for coherent sheaves ).

One of the most important line bundles in algebraic geometry is the tautological line bundle on projective space.

There are theories of holomorphic line bundles on complex manifolds, and invertible sheaves in algebraic geometry, that work out a line bundle theory in those areas.

In the mathematical fields of topology and K-theory, the Serre – Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: " projective modules over commutative rings are like vector bundles on compact spaces ".

The original theorem, as stated by Jean-Pierre Serre in 1955, is more algebraic in nature, and concerns vector bundles on an algebraic variety over an algebraically closed field ( of any characteristic ).

Ultimately Élie Cartan generalized Klein's homogeneous model spaces to ( Cartan ) connections on certain principal bundles, placing the problem in the framework of Riemannian geometry.

Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles.

Principal bundles have important applications in topology and differential geometry.

Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry.

In the 1940s, practitioners of differential geometry began introducing other notions of covariant differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold.

Important examples include: holonomy of the Levi-Civita connection in Riemannian geometry ( called Riemannian holonomy ), holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles.

As Hamiltonian mechanics is generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition in terms of cotangent bundles.

In the 1920s, the French mathematician Élie Cartan formulated Einstein's theory in the language of bundles and connections, a generalization of Riemann's geometry to which Cartan made important contributions.