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I know that there exists nongeometrical proofs that the usual mapping of the Klein Bottle in ##\mathbb{R}^3## is an immersion. But I would like to see an actual geometrical 'test'. I was thinking if saying that on the self intersection, which I circled in red below, the map being injective can be justified by saying that it's because there are two distinct tangent spaces, which is drawn in the right picture?
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