Is the Submersion Property Preserved in Maps Between R and S^1?

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In summary: Therefore, there is no submersion from S^1 to R.In summary, there is no submersion from S^1 to R, but there is a submersion from R to S^1. The definition of submersion should include smoothness in addition to surjectivity. To prove non-existence for a submersion from S^1 to R, one can use the fact that there is no maximum value for the image of a smooth map from S^1 to R.
  • #1
quasar987
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I think it's accepted to post HW type question in here.

"Is there a submersion from S^1 to R? From R to S^1?"

By a submersion from M to N, I mean a map f:M-->N whose tangent map is surjective.

I answered 'yes' to both questions, which I find dubious.

Take for an atlas of S^1, {({[itex]\exp(i\theta):0<\theta<2\pi[/itex]}, z-->arg(z)), ({[itex]\exp(i\theta):-\pi<\theta<\pi[/itex]}, z-->arg(z))}

Submersion from S^1 to R: Let x be in S^1 and f:S^1-->[0,2pi[ be f(z)=arg(z). Let r be a real number. I must show that there is a path y:]-e,e[-->S^1 such that y(0)=x and d/dt(f o y)(0)=r. Well such a path is y(t)=xexp(irt).

Submersion from R to S^1: Let x to be in R and f:R-->S^1 be f(y)=exp(ig(y)), where g:R-->[0,2pi[ is the "mod 2 pi" map. Let r be a real number. I must show that there is a path y:]-e,e[-->R such that y(0)=x and d/dt(p o f o y)(0)=r, for p a chart of S^1 around f(x). Well such a path is y(t)=x(t+1)^(r/x). I have constructed y so that y'(0)=r.

Yes, because by construction, p o f = g, and d/dt(g o y)(0)=g'(y(0))*y'(0)=1*r=r.
 
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  • #2
What you wrote is wrong and a bit difficult to read.

1.)A submersion from S^1 -> R does not exist.
2.)A submersion from R -> S^1 does exist.
3.)Your definition of submersion:

"By a submersion from M to N, I mean a map f:M-->N whose tangent map is surjective."

is missing a key element. Usually a submersion is required to be smooth in addition to having the surjectivity condition you mentioned.

4.)The argument map from S^1 -> R is not continuous. That is the problem with your first proof.

To write more clearly, I suggest the following.

1.) Open parenthesis are usually written using "(" and ")". Most readers like looking at these symbols as opposed to "]" and "[". Use the standard notation. Nonstandard notation requires a reader to visualize the "standard version". This reduces mental capacity of the reader to accommodate other errors, misunderstandings, and varying notational conventions.

2.)Make it clear what you are doing.

"I must show that there is a path y:]-e,e[-->S^1 such that y(0)=x and d/dt(f o y)(0)=r."

I eventually realized what you were doing with this step. However, it took more effort than it should have given that I was familiar with these concepts and definitions.

Instead of writing, "I must show that..."

Write, "In order to show f is surjective, ..."

This helps the reader immediately know what your goal is with that particular step.
 
  • #3
Thanks for correcting my definition.

But how would a proof of non-existence go? Do you have a hint for me?
 
  • #4
Let f:S^1-->R be any smooth map and consider Im(f) subset of R. Let m=max Im(f). At any point x in the inverse image of m, the map f will not be a submersion.

Take local coordinates c around the point x. Then,
f compose c:(-e,e)-->R is not a submersion. Hence, f is not a submersion.

The graph of f compose c subset of (-e,e)xR looks like "n". At the point where the tangent line is horizontal, the function f compose c is not a submersion.
 

1. What is "Submersion btw R and S^1"?

"Submersion btw R and S^1" refers to a mathematical concept where a map or function is continuously differentiable and maps a subset of the real numbers (R) to a circle (S^1). It is often used in topology and differential geometry to study the properties of smooth maps between different spaces.

2. What is the significance of studying submersion btw R and S^1?

The study of submersion btw R and S^1 is important because it helps us understand the behavior of smooth maps between different spaces, which has many applications in fields such as physics, engineering, and computer science. It also allows us to classify and distinguish different types of smooth maps based on their properties.

3. What are some examples of submersion btw R and S^1?

One example of submersion btw R and S^1 is the stereographic projection, which maps points on a sphere (S^2) to points on a plane (R^2). Another example is the winding number function, which maps a curve on a plane to the circle (S^1) of values it winds around.

4. How is submersion btw R and S^1 related to other mathematical concepts?

Submersion btw R and S^1 is closely related to the concepts of smooth maps, differentiability, and manifolds. It is also connected to the idea of fiber bundles, where the circle (S^1) serves as the fiber and the real numbers (R) serve as the base space.

5. What are some real-world applications of submersion btw R and S^1?

Submersion btw R and S^1 has many practical applications, such as in computer graphics, where it is used to map 3D objects onto 2D screens. It is also used in robotics and control theory to model and analyze the movements of mechanical systems. In physics, submersion btw R and S^1 is used to study the behavior of electromagnetic fields and to describe the topology of spacetime in general relativity.

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