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Apr 8, 2010 at 13:59 comment added Benoît Kloeckner Beware that a flat torus can be $C^1$ embedded into $\mathbb{R}^3$. Since one can define geodesics as locally minimizing curves, the notion still makes sense in the $C^1$ regularity.
Nov 6, 2009 at 15:57 history answered TerronaBell CC BY-SA 2.5