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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?

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    Only if the manifold is 3-dimensional. – 2017-02-16
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    The whole third volume of Lie's work on transformation groups is dedicated to the so called Helmholtzian "problem of space" referring generically to three dimensional space but in fact theorem 42 in section 99 of the above mentioned reference by Lie and Engel, specifies that it applies to 3 or more than three dimensions. – 2017-02-16

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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.

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    You are right,I'll edit the question again to restrict it to three dimensions. Apologies for the slip of mind. – 2017-02-17
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    Why the possibility of 4-dimensional group of local isometries? As for the relation with a variational principle it just appears to me that demanding that a functional be stationary for certain parameters in fields on a manifold has some relation with the rigidity of the local isometry group. It is quite evident that for instance the isometry group in Euclidean geometry conditions the existence o certain symmetries of the action in analytical mechanics that are formalized with the calculus of variations. – 2017-02-17
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    @bonif: Take for instance $S^2\times R$ with the product metric and the constant curvature metric on $S^2$. As for the rest of your remark, this is way too cryptic for me. You may want to elaborate in your question, maybe giving an example of what your have in mind. Is it a functional on the manifold itself? On the tangent bundle? On the space of Riemannian metrics on the manifold?... – 2017-02-17
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    Thanks @Moishe Cohen. I was thinking about functionals on the manifold itself. For instance in the one-parameter case geodesics as extremal curves come to mind, is there an analogy for the parametric group case? But I agree that I would need to make much clearer my question, otherwise this looks too much like shots in the dark. Also it might be maybe more appropriate for the physics SE. – 2017-02-17