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Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.

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Lorentzian metrics on the torus up to continuos deformations

Yes those are already all different metrics. Since the tangent bundle of the 2-torus $\mathbb{T}^2$ is trivial you have a correspondence between the set of homotopy classes of maps $\mathbb{T}^2\to\ma …
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