Geometrical Test for Immersion of Klein Bottle in Higher Dimensional Manifold

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In summary, the discussion focuses on finding a geometric test for the usual mapping of the Klein Bottle in ##\mathbb{R}^3## to be an immersion. A possible test is to check if the tangent planes at the self-intersection are distinct, but it is argued that this is not a reliable criterion. The example of the map ##x\rightarrow e^{ix}## is mentioned, which is an immersion of the real line into the complex plane onto the unit circle, with each point on the circle being hit infinitely many times by the same tangent line. It is noted that any manifold can be immersed in a higher dimensional manifold and that any covering transformation is an immersion.
  • #1
kent davidge
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I know that there exists non-geometrical 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?
59jKZVe.png
 

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  • #2
kent davidge said:
I know that there exists non-geometrical 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?
View attachment 227447
The tangent planes do not need to be distinct in an immersion. An immersion must be smooth so that there must be a well defined tangent plane in the image of any point.

The picture of the Klein bottle above is an immersion because along every part of the tube there is a tangent plane. At the intersection of the tube with itself there are two planes each coming from another part of the Klein bottle. But in theory these planes could be the same.

If the map had a crease or a cusp or crushed a line to a point then it would not be an immersion
 
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  • #3
lavinia said:
in theory these planes could be the same
do you mean we can rotate one of those planes I have drawn to the other's direction so that they become same? But wouldn't that make your statement
At the intersection of the tube with itself there are two planes each coming from another part of the Klein bottle
meaningless?
 
  • #4
kent davidge said:
do you mean we can rotate one of those planes I have drawn to the other's direction so that they become same? But wouldn't that make your statement

meaningless?
The two parts of the tube could kiss for a while before parting directions. At the points where they kiss the tangent planes would be the same. Since it must at some point pass through itself I imagine that the tangent planes must be distinct at some point along the intersection circle - but not sure.

The map ##x→e^{ix}## is an immersion of the real line into the complex plane onto the unit circle. Each point on the circle is hit infinitely many times ,each time with the same tangent line.
 
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  • #5
Kent, may be you are being confused about immersion and embedding.
 
  • #6
martinbn said:
Kent, may be you are being confused about immersion and embedding.
No, I'm really considering the immersion.
lavinia said:
Since it must at some point pass through itself I imagine that the tangent planes must be distinct at some point along the intersection circle - but not sure.
Yes, but I think this always happens, so cannot be used as a way to see if it's an immersion. The tangent planes being different after the intersection is a consequence of the end of the self intersection, i.e. the map is injective again, correct?
lavinia said:
The map ##x→e^{ix}## is an immersion of the real line into the complex plane onto the unit circle. Each point on the circle is hit infinitely many times ,each time with the same tangent line.
Unfortunately, I do not think this is a good example, because as far as I know anyone dimensional manifold can be immersed in a higher dimensional manifold. Also, any point of ##\mathbb{R}## has the same trivial tangent space, which is ##\mathbb{R}## itself. So the only way is the circle being hit infinitely many times by the same tangent line. :frown:

On the other hand, the Klein bottle is a 2-dimensional manifold, so I guess we would need a 2-dimensional known case to use as an example.
 
  • #7
kent davidge said:
Yes, but I think this always happens, so cannot be used as a way to see if it's an immersion. The tangent planes being different after the intersection is a consequence of the end of the self intersection, i.e. the map is injective again, correct?

It is the existence of tangent planes that makes it an immersion.

Unfortunately, I do not think this is a good example, because as far as I know anyone dimensional manifold can be immersed in a higher dimensional manifold. Also, any point of ##\mathbb{R}## has the same trivial tangent space, which is ##\mathbb{R}## itself. So the only way is the circle being hit infinitely many times by the same tangent line. :frown:

On the other hand, the Klein bottle is a 2-dimensional manifold, so I guess we would need a 2-dimensional known case to use as an example.
It is a good example. If you insist on a higher dimensional example consider the immersion of ##R^2## into ##R^4## onto the flat torus ##(x,y)→(e^{ix},e^{iy})##

BTW:
Any manifold can be immersed in some higher dimensional manifold.

Any covering transformation is an immersion. So there is an immersion of ##R^2## onto the Klein bottle. Each point on the Klein bottle is hit infinitely many times and always with the same tangent plane.
 
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1. What is a Klein Bottle immersion test?

A Klein Bottle immersion test is a scientific experiment used to measure the properties and characteristics of a Klein Bottle. This test involves submerging the Klein Bottle in a liquid to observe any changes in its shape, volume, and surface tension.

2. How is the Klein Bottle immersed in the liquid for the test?

The Klein Bottle is typically immersed in the liquid by placing it inside a larger container filled with the liquid. The container is then sealed to prevent any air from entering or escaping. The Klein Bottle is fully submerged in the liquid during the test.

3. What is the purpose of performing a Klein Bottle immersion test?

The purpose of this test is to study the unique properties and behavior of the Klein Bottle when submerged in a liquid. It can also provide insights into the underlying mathematical principles and theories behind the Klein Bottle's shape and structure.

4. What types of liquids are used for the Klein Bottle immersion test?

Any liquid can be used for this test, but typically, water or other common liquids with low viscosity are used. Some scientists may also use specialized liquids such as oils or alcohol to study the effects of different surface tensions on the Klein Bottle.

5. What are some potential applications of the Klein Bottle immersion test?

The results of this test can have various practical applications, such as in material sciences, fluid dynamics, and mathematical research. It can also help in the development of new technologies or products that utilize the unique properties of the Klein Bottle.

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