$(p-1)-$dimensional manifold of fixed but arbitrary topology. The momentum canoniclly conjugate to $X^\mu$ is given by \[2.2\] \_(y):=’=- (\[(X\^)(y)\]\^[-1]{})\^[t]{}\_(X(y)) X\^\_[,]{}(y) Introducing temporal and spatial coordinates $y=(t,x)$ we have the elementary Poisson brackets \[2.2a\] {X\^(t,x),X\^(t,x’)}={\_(t,x),\_(t,x’)}=0, {\_(t,x),X\^(t,x’)}=’ \_\^(x,x’) In the sense of Dirac’s analysis of constrained systems the Legendre transform (\[2.2\]) is not onto, the Langrangean in (\[2.1\]) is singular and we arrive at the $p$ primary constraints \[2.3\] D\_a &:=& \_X\^\_[,a]{}=0\
C &:=& \[\^ \_\_+ (q)\]=0 where \[2.4\] q\_[ab]{}(y)=\[X\^\]\_[ab]{}(y) and $a,b,..=1,..,p-1$. Notice that $\eta$ maybe an arbitrary curved metric. The constraint $D_a$ is often called spatial diffeomorphism constraint in the LQG literature because its Hamiltonian vector field generates spatial diffeomorphisms of $\sigma$. Likewise, the constraint $C$ is called the Hamiltonian constraint because, on the solutions to the equations of motion, it generates canonically temporal reparameterizations. See our companion paper [@11] for a more detailed elaboration on this point.

These $p$ constraints are in fact first class so that there are no secondary constraints. Indeed we find the [**Hypersurface Deformation Algebra $\mathfrak{H}$**]{} \[2.5\] {D(),D(’)} &=& ’ D([L]{}\_’)\
{D(),C(N’)} &=& ’ C([L]{}\_N’)\
{C(N),C(N’)} &=& ’\_ d\^[p-1]{}x (N\_[,a]{} N’-N N\_[,a]{}) \[(q) q\^[ab]{}\] D\_b where $\cal L$ denotes the Lie derivative and where we have 