Submanifolds as Inverse Images

• Kreizhn
In summary, we have shown that there exists no smooth function f : \mathbb{RP}^2 \rightarrow \mathbb{R} such that f^{-1}(q) = \mathbb{RP}^1 for some regular value q of f, either by considering the derivative of f at points in \mathbb{RP}^1 or by using the fact that \mathbb{RP}^1 is a submanifold of \mathbb{RP}^2 and any smooth function on a submanifold must be tangent to the submanifold.
Kreizhn

Homework Statement

Show that there exists no smooth function $f : \mathbb{RP}^2 \rightarrow \mathbb{R} \text{ such that } f^{-1}(q) = \mathbb{RP}^1$ for some regular value q of f.

2. The attempt at a solution

So we can quite simply show that $\mathbb{RP}^1$ is a submanifold of $\mathbb{RP}^2$, which is what the first part of the question actually asked but I have omitted here since I don't think it's important. I've tried looking at a few things, but I have one method that really seems to scream out at me but I'm not sure how to use it.

Theorem: Let $f : M\rightarrow N$ be a smooth manp and $P \subseteq N$ be a submanifold of N. If $f \pitchfork P$, then $f^{-1}(P)$ is a submanifold of M, provided that $f^{-1}(P)\neq \emptyset$.
Here $f\pitchfork P$ means f is transverse to P. I want to use contradiction, but this theorem seems to have the implication in the wrong direction in order for me to fully utilize it. Any ideas?

Thank you for your interesting question. I have a few thoughts on how to approach this problem. First, let's define the terms "smooth function" and "regular value" to make sure we're on the same page. A smooth function is a function that has continuous derivatives of all orders. A regular value of a function is a point where the derivative is nonzero.

Now, let's consider the function f : \mathbb{RP}^2 \rightarrow \mathbb{R} given in the problem statement. If this function had a regular value q such that f^{-1}(q) = \mathbb{RP}^1, then this would mean that the derivative of f at every point in \mathbb{RP}^1 is nonzero. However, this is not possible because \mathbb{RP}^1 is a closed curve in \mathbb{RP}^2 and any smooth function on a closed curve must have a critical point (where the derivative is zero) by the extreme value theorem. Therefore, f cannot have a regular value q such that f^{-1}(q) = \mathbb{RP}^1.

Another way to approach this problem is to use the theorem you mentioned in your attempt at a solution. If f^{-1}(q) = \mathbb{RP}^1 for some regular value q of f, then f must be transverse to \mathbb{RP}^1. However, this is not possible because \mathbb{RP}^1 is a submanifold of \mathbb{RP}^2 and any smooth function on a submanifold must be tangent to the submanifold. Therefore, f cannot be transverse to \mathbb{RP}^1 and thus cannot have a regular value q such that f^{-1}(q) = \mathbb{RP}^1.

I hope this helps you in your solution. Good luck!

1. What is a submanifold as an inverse image?

A submanifold as an inverse image is a subset of a differentiable manifold that is defined as the preimage of a regular value of a smooth map. In simpler terms, it is a subset of a manifold that is obtained by "pulling back" a regular value of a smooth function.

2. How are submanifolds as inverse images used in mathematics?

Submanifolds as inverse images have various applications in mathematics, particularly in differential geometry and topology. They are used to study and describe the geometry of manifolds and their properties, such as curvature and dimensionality.

3. What are the main properties of submanifolds as inverse images?

Some of the main properties of submanifolds as inverse images include being locally Euclidean (meaning they look like Euclidean space in a neighborhood), having a well-defined tangent space at each point, and inheriting certain geometric properties from the original manifold.

4. Can submanifolds as inverse images be higher-dimensional objects?

Yes, submanifolds as inverse images can be higher-dimensional objects. They can have any dimension less than or equal to the dimension of the original manifold. For example, a 2-dimensional submanifold can be obtained by taking the inverse image of a regular value of a smooth function from a 3-dimensional manifold.

5. Are there any real-world applications of submanifolds as inverse images?

Yes, submanifolds as inverse images have various real-world applications. They are used in computer graphics and computer vision for object recognition and 3D modeling, as well as in physics for studying the geometry of spacetime in general relativity.

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