fiberscoordinates

Fiber coordinates refer to the coordinates that parametrize points along the fibers of a fiber bundle or vector bundle, in addition to coordinates on the base manifold. Given a fiber bundle π: E → M with typical fiber F, a local trivialization over an open set U ⊂ M identifies π^{-1}(U) with U × F. In such a chart, a point e ∈ E near the chosen trivialization has coordinates (x^i, y^a), where x^i are coordinates on the base M and y^a are coordinates along the fiber. The y^a are the fiber coordinates; they vary smoothly with e and vanish in the base direction.

In a vector bundle, the fiber F is typically modeled by R^k, and the fiber coordinates y^a are linear coordinates along each fiber. Under a change of trivialization, the fiber coordinates transform by a linear action dependent on the base point: y'^a = g^a_b(x) y^b, where g(x) is a transition function valued in the structure group. These transformations reflect the bundle’s local product structure and the absence of a globally fixed fiber coordinate system.

The concept is widely used in differential geometry and its applications. On the tangent bundle TM, for example, coordinates take the form (x^i, v^i), with v^i representing the components of a tangent vector in the chosen frame. Fiber coordinates also appear in jet bundles and in the formulation of connections, where the horizontal and vertical decomposition of the tangent space to E depends on fiber coordinates.

Thus, fiber coordinates provide a convenient, local way to describe positions along the fibers, complementing the base coordinates that describe location on the base manifold.