g x for the action, so that the defining properties become e x = x and g (h x) = (gh) x. Otherwise put, the action associates to g E G a smooth map eg : M --+ M, defined by eg(x) = £(g, x) such that ee = idM and eg 0 eh = eghIn particular, each eg is a diffeomorphism with inverse £g-1, so we can view the action as a homomorphism from G into the group of diffeomorphisms of M.

Such a section immediately gives rise to a fiber bundle chart p-1(U) -* U x H by mapping g E p-1(U) to (p(g), o,(p(g))-Ig) with inverse given by (x, h) H u(x)h. The corresponding transition functions are given by (x, h) '---* Q,,,o(x)h, where Qao IQo(x). Unt3 -p H is given by Q,,o(x) = Given a general fiber bundle p : P -+ M with standard fiber a Lie group H, we define a principal bundle atlas to consist of charts which are compatible in the sense that the transition functions are given by (x, h) i-+ 0n3 (x)h for smooth functions H.

CARTAN GEOMETRIES 26 0a : p-I(UD) Ua x S which are fibered morphisms. Each of the pairs (U,,, 0,,) is called a fiber bundle chart. , 0a) and (U0, 00) such that U(,o UJ n Up # 0, one has the transition function Sao : U,,o x S -+ S defined by (gyp I (x, y)) = (x, Qo (x, y)) Assume next, that we have given a fiber bundle p : Y -+ M whose standard fiber is a finite-dimensional vector space V. Then two fiber bundle charts are called compatible if the corresponding transition function is linear in the second variable.