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Visualizing Immersions vs. Submersions

  1. Oct 22, 2009 #1
    What is the best way to intuitively and visually distinguish between immersion and submersions? For example, I understand that the standard picture of the Klein Bottle in R^3 is an immersion. How do I see this? (Obviously, it's not an embedding because the Klein Bottle self-intersects in R^3. But how do I see that the differential map is 1-to-1 but not onto?) What would a submersion look like?

    Also, can I visualize a copy of the real line in R^3 so that, it is an immersion but not a submersion nor an embedding? Also as a submersion but not an immersion?
  2. jcsd
  3. Oct 23, 2009 #2
    The only way something can be a submersion and an immersion is if it's a local diffeomorphism (e.g. a covering map, like the real line to S^1). It's difficult to tell by visual inspection if a map is an immersion - you just calculate the Jacobian. A submersion can most readily be seen by a map R^2->R^1 with no critical points (since then you can visualize it as a graph in R^3).
  4. Oct 23, 2009 #3
    it is am immersion because there is a tangent plane
    it is not onto because the normal does not lie on the tangent plane

    a line that spirals infinitely around a point and converges to that point is not embedded.
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