Transition matrix of a local frame

The matrix-valued function describing how two local frames are related on an overlap.

Transition matrix of a local frame

Let $\pi:E\to M$ be a rank $r$ smooth vector bundle, and let $(e_1,\dots,e_r)$ and $(e'_1,\dots,e'_r)$ be two local frames of $E$ defined on open sets $U$ and $V$ , respectively. On the overlap $U\cap V$ , there is a unique map

g:U\cap V\to \mathrm{GL}(r,\mathbb F)

that is smooth and satisfies, for each $x\in U\cap V$ and each $j=1,\dots,r$ ,

e'_j(x)=\sum_{i=1}^r e_i(x)\,g_{ij}(x).

In matrix notation (with frames viewed as column vectors of sections), this reads

e'(x)=e(x)\,g(x).

The map $g$ is called the transition matrix from the frame $e$ to the frame $e'$ on $U\cap V$ .

If three frames $e,e',e''$ are defined on mutually overlapping sets, their transition matrices satisfy the cocycle identity on triple overlaps:

g_{e,e''}=g_{e,e'}\,g_{e',e''}.

Examples

Trivial bundle. For $E=M\times \mathbb F^r$ , any two frames over $U$ differ by a smooth $g:U\to \mathrm{GL}(r,\mathbb F)$ ; the transition matrix is exactly that $g$ .
Möbius line bundle. The Möbius real line bundle over $S^1$ can be described by two trivializations whose overlap transition function is the constant map $g(\theta)=-1\in \mathrm{GL}(1,\mathbb R)$ . This single nontrivial transition function encodes the twist.
Orientation check. For a real bundle, on $U\cap V$ the sign of $\det g(x)$ detects whether the two frames determine the same orientation (positive determinant) or the opposite one (negative determinant).