Editing Connection (vector bundle) (section)

=== Change of local trivialisation ===

Suppose <math>\mathbf{e'}</math> is another choice of local frame over the same trivialising set <math>U</math>, so that there is a matrix <math>g=(g_i^{\ j})</math> of smooth functions relating <math>\mathbf{e}</math> and <math>\mathbf{e'}</math>, defined by
:<math>e_i = \sum_{j=1}^k g_i^{\ j} e'_j.</math>
Tracing through the construction of the local connection form <math>A</math> for the frame <math>\mathbf{e}</math>, one finds that the connection one-form <math>A'</math> for <math>\mathbf{e'}</math> is given by
:<math>{A'}_i^{\ j} = \sum_{p,q=1}^k g_p^{\ j} A_q^{\ p} {(g^{-1})}_i^{\ q} - \sum_{p=1}^k (dg)_p^{\ j} {(g^{-1})}_i^{\ p}</math>
where <math>g^{-1} = \left({(g^{-1})}_i^{\ j}\right)</math> denotes the inverse matrix to <math>g</math>. In matrix notation this may be written
:<math>A' = g A g^{-1} - (dg)g^{-1}</math>

where <math>dg</math> is the matrix of one-forms given by taking the exterior derivative of the matrix <math>g</math> component-by-component.

In the case where <math>E=TM</math> is the tangent bundle and <math>g</math> is the Jacobian of a coordinate transformation of <math>M</math>, the lengthy formulae for the transformation of the Christoffel symbols of the Levi-Civita connection can be recovered from the more succinct transformation laws of the connection form above.