What is the group of bundle?

The group G is called the structure group of the bundle; the analogous term in physics is gauge group. In the smooth category, a G-bundle is a smooth fiber bundle where G is a Lie group and the corresponding action on F is smooth and the transition functions are all smooth maps.

Are vector bundles principal bundles?

According this definition vector bundle is a special principal bundle, because vector space with vector addition as group operation is a topological group.

What is the rank of a vector bundle?

If kx is equal to a constant k on all of X, then k is called the rank of the vector bundle, and E is said to be a vector bundle of rank k. Often the definition of a vector bundle includes that the rank is well defined, so that kx is constant.

What is a trivial vector bundle?

An isomorphism of vector bundles over X of the form. E⟶X×ℝn. is called a trivialization of E. If E admits such an isomorphism, then it is called a trivializable vector bundle.

What is the collective noun of bundle?

Bundle’ is a term used for ‘a group of things that are fastened, tied or wrapped together. ‘ So, ‘bundle’ is a collective noun. ‘Sticks’ is a common noun.

What is a bundle in geometry?

From Wikipedia, the free encyclopedia. In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topology. Without this requirement, more general objects can be considered bundles.

Is the tangent bundle a vector space?

The tangent bundle of the sphere is the union of all these tangent spaces, regarded as a topological bundle of vector space (a vector bundle) over the 2-sphere. A tangent vector on X at x∈X is an element of TxX.

Is tangent bundle a manifold?

The collection of open sets on TM defined above does indeed form a topology. Moreover, if M is Hausdorff and second countable, so is TM. We conclude that if M is an n-dimensional then its tangent bundle TM is a 2n-dimensional manifold.

What is rank of module?

The rank of a free module M over an arbitrary ring R( cf. Free module) is defined as the number of its free generators. A sufficient condition for the rank of a free module over a ring R to be uniquely defined is the existence of a homomorphism ϕ:R→k into a skew-field k.

Which type of noun is bundle?

bundle used as a noun: A group of objects held together by wrapping or tying. A package wrapped or tied up for carrying. A cluster of closely bound muscle or nerve fibres. A large amount, especially of money.

Which collective noun is used for pearls?

The collective noun for pearls is a rope, a string, necklace, cluster, or group.

How do you bundle in math?

Bundling is also called grouping. This is a way to group numbers by putting the smaller units together to make a larger one. For instance, putting 10 ones together makes 1 ten. Putting 10 tens together makes 1 hundred.

What is the morphism of a principal fibre bundle?

A morphism of principal fibre bundles is a morphism of the fibre bundles f: π G → π G ′ for which the mapping of the fibres f π G − 1 ( b) induces a homomorphism of groups: where ξ b ( g) = g x , π G ( x) = b . In particular, a morphism is called equivariant if θ b = θ is independent of b , so that g f ( x) = θ ( g) f ( x) for any x ∈ X , g ∈ G .

Which is the trivial G-principal bundle over a topological space?

The trivial G-principal bundle on a topological space X is the product space X × G equipped with the action of G on X × G by right multiplication of G on itself. Definition 0.4. A G – principal bundle over a topological space X is a topological space P equipped with

Which is an automorphism group for G over X?

Automorphism groups For G a group ( internal to some category, traditionally that of topological spaces) and X some other object, a G-principal bundle over X – also called a G – torsor over X – is a bundle P → X equipped with a G – action ρ: P × G → P on P over X, such that and / or / equivalently (depending on technical details, see below)

How are universal principal bundles used in nLab?

Notably the existence of universal principal bundles finds its fundamental “explanation” here, where they are seen to be but a presentation of the construction of the homotopy fiber functor, which establishes the equivalence of groupoids