Constraints in General Relativity

I've got a little problem in trying to understand the constraints. In
particular I'm stumped as to how one can show that the momentum
constraint is a conserved quantity. I'm working from MTW, where the
momentum constraint is written as

H^i = D_j p^{ij}

where D is a covariant derivative and p^{ij} is the momenta conjugate
to the metric. If I expand the above using the definition of the
covariant derivative I get

H^i = @_j p^{ij} + C^i_{jm} p^{mj} + C^j_{jm} p^im

where @ denotes partial differentiation and C^i_{jk} are the
Christoffel symbols. The idea of the constraints is that they must be
preserved under evolution, right? Well, I calculate that the time
derivative of H^i is

(@/@t)H^i = D_j (@ p^{ij}/@t) + p^{mj}(@ C^i_{jm}/@t) + p^{im}(@
C^j_{jm}/@t).

My problem is that I can use this to get an expression for the time
derivative of H^i in terms of some expressions multiplied by the
constraints (which vanish since the constraints are zero) plus some
other terms involving the shift; it's these extra shift terms that I
can't seem to make vanish.

I seem to recall seeing someone derive (@/@t)H^i years ago and they
claimed that

(@/@t)H^i = @_j p^{ij} + C^i_{jm} p^{mj}

but I can't understand how they got this. The advantage of this
approach was that it resulted in an expression purely in terms of the
constraints, with no extra shift terms. Any ideas how to prove that the
momentum constraint is propagated in general relativity?

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