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I don't know of an example of a Riemannian manifold that admits a (natural) group structure, but not a Lie group structure.

The manifolds that I do know are various types of matrix manifolds (all of which are Lie groups), the circle, sphere, etc. (all of which admit $SO(n)$), and lorentz space-time (which does not have a natural group structure that I know of, and is hence excluded from the question).

I admit, the question is somewhat soft in nature due to the "natural" group structure condition.

Siddharth Bhat
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1 Answers1

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The Riemannian metric is not really related to the question, so I will ignore it. As for the naturality it's not quite clear what you mean but I will always assume that you want topology and group structure to be compatible (in the sense that it is a topological group). Here are some references to possible interpretations of your question:

1) Given a (topological) manifold $X$, does there always exist a group structure that turns $X$ into a topological group? In general no - there are many obstructions like orientability, homogeneity, ... see the discussions here and here.

2) Given a topological group $X$ that happens to be a manifold, does $X$ always admit a differentiable structure that turns it into a Lie group? The answer is, even a unique one, see the discussion here.

Jan Bohr
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  • How do we know that the metric is not related to the question? Many thanks for the answer. My thinking is that the very existence of a metric might restrict the class of smooth manifolds? – Siddharth Bhat Oct 31 '19 at 17:11
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    Every smooth manifold admits a smooth Riemannian metric. The only way you would get more information would be to impose some restrictions on the metric, e.g. a curvature constraint. – Jan Bohr Oct 31 '19 at 17:16
  • @JanBohr so is there an example of a Riemannian manifold which is also a group, but is not a Lie group (since the group operation or the inversion operation isn't smooth) ? – PermQi Feb 19 '24 at 14:04