In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, "glued together", form another topological space (or manifold or variety). A typical example is the tangent bundle of a differentiable manifold: to every point of the manifold we attach the tangent space of the manifold at that point. Or consider a smooth curve in R2, and attach to every point of the curve the line normal to the curve at that ...

Including:

• Vector bundle - Definition and first consequences
• Vector bundle - Vector bundle morphisms
• Vector bundle - Sections and locally free sheaves
• Vector bundle - Operations on vector bundles
• Vector bundle - Variants and generalizations

Read more here: » Vector bundle: Encyclopedia - Vector bundle

A real vector bundle is given by the following data: topological spaces X (the "base space") and E (the "total space") a continuous map π : E → X (the "projection") for every x in X, the structure of a real vector space on the fiber π−1({x}) satisfying the following compatibility condition: for every point in X there is an open neighborhood U, a natural number n, and a homeomorphism φ : U × See also:

Vector bundle, Vector bundle - Definition and first consequences, Vector bundle - Vector bundle morphisms, Vector bundle - Sections and locally free sheaves, Vector bundle - Operations on vector bundles, Vector bundle - Variants and generalizations

Read more here: » Vector bundle: Encyclopedia II - Vector bundle - Definition and first consequences

A morphism from the vector bundle π1 : E1 → X1 to the vector bundle π2 : E2 → X2 is given by a pair of continuous maps f : E1 → E2 and g : X1 → X2 such that gπ1 = π2f for every x in X1, the map π1−1({x}) → π2−1({g(x)}) induced by f ...

See also:

Vector bundle, Vector bundle - Definition and first consequences, Vector bundle - Vector bundle morphisms, Vector bundle - Sections and locally free sheaves, Vector bundle - Operations on vector bundles, Vector bundle - Variants and generalizations

Read more here: » Vector bundle: Encyclopedia II - Vector bundle - Vector bundle morphisms

Given a vector bundle π : E → X and an open subset U of X, we can consider sections of π on U, i.e. continuous functions s : U → E with πs = idU. Essentially, a section assigns to every point of U a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a di ...

See also:

Vector bundle, Vector bundle - Definition and first consequences, Vector bundle - Vector bundle morphisms, Vector bundle - Sections and locally free sheaves, Vector bundle - Operations on vector bundles, Vector bundle - Variants and generalizations

Read more here: » Vector bundle: Encyclopedia II - Vector bundle - Sections and locally free sheaves