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From my previous post I have learnt that spherical triangles can have different interior angle sums. Is this enough to argue that the triangles are not isomorphic? I am not sure how isomorphism works on a sphere. Thanks in advance.

(I only know isomorphism in the context of algebra where it is a bijective homomorphism...)

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    @Henning Yes, just like under general affine transformations of $\mathbb{R}^2$, all triangles are equivalent. But the subgroup of translations, rotations, reflections, and dilations give the usual notion of symmetry. So what I meant was, maybe some subgroup of general great-circle-preserving transformations would provide some kind of interesting equivalence.2012-02-10

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