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Let $(T_1,g_1)$ and $(T_2,g_2)$ be two flat tori of dimension 2 such that their geodesic flows are $C^0$-conjugated, is there an isometry between $(T_1,g_1)$ and $(T_2,g_2)$ ?

I emphasize the fact that I ask a $C^0$-conjugacy. In the paper of Croke mentioned below in the answer it is a $C^1$-conjugacy.

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Yes, this is theorem C in:

Croke, Christopher B.(1-PA) Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv. 65 (1990), no. 1, 150–169.

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  • $\begingroup$ Thanks, but in this paper, it is a $C^1$-conjugacy and not a $C^0$-conjugacy, as I asked for. $\endgroup$
    – sydji
    Commented Jan 16, 2015 at 22:22

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