The theorem states that in a Riemannian connected finite dimensional manifold of 3 dimensions the existence of a 6-dimensional group of local isometries that determine infinitesimal rigid displacements implies that this manifold must possess a constant curvature(one can read the complete statement and proofs in Lie's "Theorie der Transformationsgruppen, Vol. 3").
I'm interested in the conclusions one can derive from this restriction, for instance does this theorem imply some sort of limitation on Riemannian manifolds of variable curvature with respect to the existence of stationary points of functionals(variational principles) on the manifold?
Added: To be more specific, if we use the language of principal G-bundles and consider a tangent frame bundle with the total space as the group of local isometries, the fiber G the point-stabilizer group and the base manifold that on wich the isometry group acts, does this theorem limit the maximum number of dimensions of the total space to 6 for a 3-dimensional base manifold and to a finite number for any finite-dimensional base manifold?
Also, do the infinitesimal rigid motions expressed by the local isometry group in a Riemannian manifold determine the maximum number of dimensions of the group of symmetries, defined as vanishing functional derivatives and corresponding Euler-Lagrange equations routinely used in multidimensional calculus of variations, of any objects defined on the manifold that undergo such rigid motions?
The claim you are quoting is simply false in dimensions $>3$, just take the Riemannian direct product of, say, hyperbolic 3-space with, say, Euclidean space of dimension $>0$.
The correct result which holds in all dimensions $n$ is:
Theorem. Suppose that $M$ is an $n$-dimensional connected Riemannian manifold such that every point in $M$ has a neighborhood which admits a local isometry group of dimension $d=\frac{n(n+1)}{2}$. Then $M$ has constant sectional curvature.
Specializing to the case $n=3$ we get $d=6$.
Edit: As for the proof, it is absolutely straightforward: Check that the local isometry group acts locally transitively on $M$ with point-stabilizers locally isomorphic to $O(n)$. From this conclude that the action is locally transitive on the bundle of tangent $2$-planes over $M$. Hence, the sectional curvature is locally constant, thus globally constant by connectivity of $M$.
As for "limitations" imposed on the Riemannian metric on $M$, this theorem simply imposes limitations on the dimension of the group of local isometries of a Riemannian manifold. The dimension $n(n+1)/2$ is maximal. In case $n=3$, the group of local isometries cannot be 5-dimensional, but can be 4-dimensional and 6-dimensional. I do not think you can get anything else from this result. One can get sharper estimates on the dimension of the local isometry group in higher dimensions. As for "stationary points of functionals", you have to be much more specific for this to be answerable.