Constraints on Phase Space This question here motivated me to record to the following fact: Consider a $2n$ dimensional phase space with coordinates $q_1,...,q_n,p_1,...,p_n$. Consider the constraint $C(\vec q)=0$. What is the constraint on the momenta which, together with $C=0$, reduces the phase space dimension to $2n-2$?
 A: The answer is $\sum_i C_i(\vec q)p_i=0$, where $C_i\equiv \frac{\partial C}{\partial q_i}$.
In the quantum theory, this can be imposed as a constraint on the Hilbert space to select 'physical states': $$\sum_i C_i(\vec q)p_i~\psi_{\text{phys}}(\vec q)=0$$ where $\psi_{\text{phys}}(\vec q)$ is the wavefunction. This condition guarantees that physical wavefunctions are constant in the direction orthogonal to the constraint surface $C(\vec q)=0$, which means that physical wavefunctions can be projected onto the surface without ambiguity.
In the path integral, delta-functions of the constraints have to be imposed in integrals over $\vec q$ and $\vec p$. The invariant way of doing this is to insert:
\begin{equation}
\delta(C)~\delta\left(\frac{C_ip_i}{|C|^2}\right) \label{yo}
\end{equation}
where $|C|^2\equiv C_iC_i$. This is invariant in the sense that if the constraint is expressed differently (i.e. $C\neq \tilde C$ but $C=0\leftrightarrow\tilde C=0$) then the product of delta functions remains invariant under the substitution $C\to\tilde C$.
A: OP is considering a Hamiltonian system with a single primary constraint $C\approx 0$. Presumably since OP does not mention secondary constraints, there are none. Then  $C\approx 0$ is actually a first class constraint. By the Dirac conjecture, the HVF $$\delta =\{C,\cdot\}\tag{1}$$ is a gauge symmetry. The gauge-fixing condition $\chi\approx 0$ should break the gauge symmetry $$\delta\chi ~=~\{C,\chi\}~\neq~ 0.\tag{2}$$  It should be stressed that the physical system does not depend on the gauge-fixing condition $\chi\approx 0$, cf. OP's question. The $2n\!-\!2$ dimensional phase space with the 2 constraints $C\approx 0\approx\chi$ is the physical/reduced phase space.
