Assume also that under this Poisson bracket, { f , f } = 0 (note that it's not true that { g , g } = 0 in general for this "extended Poisson bracket" anymore) and { f , H } = 0 on the submanifold of zeros of f (If these brackets also happen to be zero everywhere, then we say the constraints close off shell). It turns out the right invertibility condition and the commutativity of flows conditions are independent of the choice of connection. So, we can drop the connection provided we are working solely with the restricted subspace.