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Can any metrical or axiomatic geometry be shown to be isomorphic to a Klein geometry? Can any manifold with a (pseudo) Riemannian metric be defined in the Kleinian sense?

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    $\begingroup$ Usually the definition of a Klein geometry requires local homogeneity, so clearly not. $\endgroup$
    – Ben McKay
    Feb 2 at 21:02
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    $\begingroup$ The answer will depend on precisely how constrained your notion of "metrical or axiomatic geometry" is. You could tautologically define things in a way where the answer is yes, I suppose. But you probably won't get agreement on that being the class of "metrical or axiomatic geometry". $\endgroup$ Feb 2 at 23:37
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    $\begingroup$ Here is a kind of answer for dimension at least 3. The case of dimension 2 assuming Desargues' axiom was answered positively by Blaschke. The reference to Blaschke can be found in Sharpe's book. $\endgroup$
    – Kapil
    Feb 3 at 3:31
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    $\begingroup$ The answer might also depend on the meaning of "any" ("is every geometry also..." vs. "is some geometry also..."). Many times it is best to avoid the "any" quantifier. $\endgroup$ Feb 3 at 11:15

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