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I have a question. Consider a differentiable manifold. This structure is imposed by requiring differentiability of the transition functions between charts of the atlas. Does requiring on top of that, linearity or affinity of the transition functions, result in any specific extra structure on the manifold ? (I would be tempted to call it an "affine manifold", but I think the name is already taken). Clearly a flat affine space would be such kind of manifold, and also things like S1. I'm not sure about S^2 though.

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# Linear transition maps on manifolds

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