Let M be a manifold and let chi(M) be the module (over the ring C(M) of all smooth functions on M) of vector fields on M.  Then a linear connection nabla on the tangent bundle of M is a mapping chi(M) x chi(M) -> chi(M) which is C(M) linear in its first argument and a derivation on its second argument.  If vector fields X_1, X_2, ..., X_n define a local frame on M, then the coefficients Gamma^k_{ij} of nabla with respect to this frame are defined by nabla_{X_i} X_j = Gamma^k_{ji} X_k.

More generally, let E -> M be a vector bundle and let Sigma(M) be the module (over the ring C(M) of all smooth functions on M) of sections of E.  Then a connection nabla on E is a mapping chi(M) x Sigma(M) -> Sigma(M) which is linear in its first argument and a derivation on it second argument.  If vector fields X_1, X_2, ..., X_n define a local frame on M and Z_1, Z_2, ..., Z_r a local basis for the sections of E, then the coefficients Gamma^a_{bi} of nabla with respect to these frames are defined by nabla_{X_i} Z_b = Gamma^a_{bi} Z_a.

Within the DifferentialGeometry package, connections are displayed using the tensor notation Gamma^k_{ji} omega^j X_k omega^i or Gamma^a_{bi} eta^b Z_a omega^i, where the omega^j are the dual coframe to the X_i and the eta^b are the dual coframe to the Z_a.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form Connection(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-Connection.