I know Klein bottle is a $2$-d smooth manifold with self intersection and has $2$-d tangent space at each point. I can image this for points where the manifold does not intersect itself. My trouble is I can't image a $2$-d tangent space at a self-intersection point.

In my view, the neighborhood of a self-intersection point locally looks like two $2$-d planes intersect. Then the tangent vectors at this point cloud be classified to be two classes; one class of vectors are tangent to one of the intersecting planes. That is, the tangent space at this point consists of two $2$-d planes. But if so, the tangent space in whole is not a vector space any more. What's wrong with my view ?

Here I adopt this definition of tangent vector:

Tangent vectors at $p$ are the derivatives of smooth paths in the manifold passing $p$.

  • $\begingroup$ The Klein bottle does not have self intersection! You may have seen pictures of it where it looks like it's intersecting itself, but that's only because it's impossible for an artist to draw it any other way in 3d. $\endgroup$ – Kenny Wong Oct 15 '17 at 13:15
  • $\begingroup$ @KennyWong I know Klein bottle is not possible to be embedded in $\mathbb{R}^3$ so it has self-intersections in $\mathbb{R}^3$, and it can be embeded in $\mathbb{R}^4$ without self-intersections. So how do we define self-intersections ? I'm not sure I understand self-intersections correctly. Could you explain a little bit ? Thanks $\endgroup$ – Hua Oct 15 '17 at 15:50
  • $\begingroup$ I also not sure how to give a good definition, seeing that I don't use this concept. Maybe something along the lines of looking at an open neighbourhood around the self-intersection point, and noticing that the tangent space looks like two intersecting planes? $\endgroup$ – Kenny Wong Oct 15 '17 at 17:45
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    $\begingroup$ When you're looking at the picture of a Klein bottle in $\Bbb R^3$, you're looking at the image of an immersion. (This is somewhat analogous to folding a circle into a figure-8 in the plane.) Along the circle of self-intersection, each of the the two "branches" has perfectly nice tangent planes, but the image is not an embedded manifold, and there is no overall tangent plane at those points on the circle. ... But keep in mind that the Klein bottle is a perfectly nice $2$-dimensional differentiable manifold unto itself, and it embeds nicely in $\Bbb R^4$. $\endgroup$ – Ted Shifrin Oct 15 '17 at 21:24
  • $\begingroup$ @TedShifrin Thanks for bringing up the concept of immersion. I guess I understand it more in light of this concept. $\endgroup$ – Hua Oct 16 '17 at 7:43

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