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I'm trying to understand basic differential geometry (my background is in mathematical logic), and I'm having a bit of difficulty with a basic point. Frequently we want to consider the set of metrics on a given manifold modulo some equivalence relation (generally, isometry), but I can't visualize metrics very easily. Is there some example of a reasonably simple manifold, and two "very different" metrics on it, that would help me understand what spaces of possible metrics can look like?

Thanks in advance

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Consider $\mathbb R$. You could for instance pull back the (usual) metric from $(-\pi/2,\pi/2)$ via $\arctan: \mathbb R \to (-\pi/2,\pi/2)$. This will induce the distance $d(x,y) = |\arctan(x)-\arctan(y)|$. The resulting space can't be isomorphic to $\mathbb R$, since it is not a complete metric space. But I'm not sure that's what you're looking for. – Sam Feb 16 '12 at 7:59
Yes, ideally I am interested in complete metrics (i.e., metrics under which the space is a complete metric space). – user13568 Feb 16 '12 at 17:22
The torus $S^1 \times S^1$ can be given the "obvious" round metric coming from an embedding into $\mathbb{R}^3$, or a flat metric coming from a covering by $\mathbb{R}^2$. These are manifestly not isometric, as one has non-trivial curvature and the other is flat. – Zhen Lin Feb 16 '12 at 17:39
One very simple example: the usual metric, $d(x,y)=|x-y|$, on $\mathbb{R}$, and the bounded metric $$d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}=\frac{|x-y|}{1+|x-y|}\;.$$ – Brian M. Scott Feb 18 '12 at 1:09

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