Can one always find, in a compact riemannian manifold, a closed geodesic isometric to a usual circle when endowed with the ambient distance ? For instance, in the usual flat torus, the only geodesics verifying this property are those of slope $0$, $1/2$ and $1$. 


The natural candidates are, of course, closed geodesics of minimal length...



Geodesic circles on riemannian manifolds
I'm not sure but I believe you are asking if there always exists a closed geodesic such that it gives a distance preserving embedding of $S^1$ with respect to the length metric on the circle and the ambient metric on the target manifold. If this is the question then the answer is no.


Balacheff, Croke and Katz in ["A Zoll counterexample to a geodesic length conjecture"](http://www.springerlink.com/content/ax605170431104q2/) have an example of a metric on $S^2$ where the length of the shortest closed geodesic is strictly bigger than twice the diameter of $S^2$. This of course gives a counterexample to the above question too.


