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On a $d$-dimensional Riemannian manifold $M$ one can obtain geometric quantities that does not change under the action of the translation group $g_a(x) = exp(a_\mu(x) \partial^\mu)$ which translates each point $x$ locally by the vector $a_\mu$. An example is the scalar curvature $R$.

Are there other manifolds, where local translation invariance does not hold?

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    Why do you think that Ricci scalar is invariant under translations? – 2017-01-17
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    The group of translations does not in general act on a manifold; the formula that you wrote is not well defined. The meaning of the question you are asking in the end is totally unclear. Voting to close. – 2017-01-18

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