Let G be a Lie group with multiplication * and mu: G x M -> M a free (left) action of G on a manifold M.  A right moving frame is a map rho: M -> G such that rho(mu(g, x)) = rho(x)*g^(- 1) for all g in G and x in M.

A cross-section to the action mu: G x M -> M is a submanifold K of M, with codim(K) = dim(G), which is transverse to the orbits of mu.  Thus, if k1, k2 in K and mu(g, k1) = mu(g, k2), then k1 = k2.

The Invariantization command will map any function on M to a G invariant function.

Further commands for working with moving frames will be added to subsequent releases of the DifferentialGeometry package.

The commands RightMovingFrame and Invariantization are part of the DifferentialGeometry:-GroupActions:-MovingFrames package.  They can be used in the forms RightMovingFrame(...) and Invariantization(...) only after executing the commands with(DifferentialGeometry), with(GroupActions), and with(MovingFrames), but can always be used by executing DifferentialGeometry:-GroupActions:-MovingFrames:-RightMovingFrame(...) and DifferentialGeometry:-GroupActions:-MovingFrames:-Invariantization(...).

References:

Warning, multiple moving frames