Define a map $\mathbb{S}^n \to \mathbb{RP}^n$ given by $x \mapsto \{ x,-x\}.$ Clearly this is not a diffeomorphism, but how can one show that it is an immersion and submersion?
2026-03-31 19:18:20.1774984700
antipodal map from sphere to projective space is immersion and submersion?
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This is a local diffeomorphism, and the result follows.