G-structure - Connections On G-structures

Connections On G-structures

Let Q be a G-structure on M. A principal connection on the principal bundle Q induces a connection on any associated vector bundle: in particular on the tangent bundle. A linear connection ∇ on TM arising in this way is said to be compatible with Q. Connections compatible with Q are also called adapted connections.

Concretely speaking, adapted connections can be understood in terms of a moving frame. Suppose that V_i is a basis of local sections of TM (i.e., a frame on M) which defines a section of Q. Any connection ∇ determines a system of basis-dependent 1-forms ω via

∇_X V_i = ω_ij(X)V_j

where, as a matrix of 1-forms, ω ∈ Ω1(M)⊗gl(n). An adapted connection is one for which ω takes its values in the Lie algebra g of G.