A Zorich proposition about local charts of smooth surfaces

In summary, the theorem states that if a mapping (12.1) is a map from a k-dimensional unit cube to a neighborhood of a generic point on a surface S, then it can be extended (locally) to a diffeomorphism between two suitable neighborhoods in ##\mathbb{R}^n##. This is proven by using the inverse function theorem and extending the mapping to all of ##\mathbb{R}^n##. The significance of this theorem is that it allows for the extension of a mapping to a higher dimensional space.
  • #1
Unconscious
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From Zorich, Mathematical Analysis II, 1st ed., pag.163:

kc0W8.png


where the referred mapping (12.1) is a map ##\varphi:I_k\to U_S(x_0)##, in which:

1. ##I_k\subset\mathbb{R}^n## is the k-dimensional unit cube,
2. ##x_0## is a generic point on the surface ##S## and ##U_S(x_0)## is a neighborhood of ##x_0## of points of ##S##.

I think that there is a deep significance in this theorem that I can't see yet, but anyway, looking brutally at the proof:

W70oc.png


we can see that up to this point the author uses the inverse function theorem to say that, if we consider only the first k coordinates both in the domain and in the range, ## \varphi ## is actually a diffeomorphism between two suitable neighborhoods ## U_{ \mathbb {R}^k_t} (0) ## e ## U _ {\mathbb {R} ^ k_x} (\varphi (0)) ##.
Then, the proof continues in this way:

yHB59.png


and here I don't understand why the whole ## f: U _ {\mathbb {R} ^ n_x} (\varphi (0)) \to U_ {\mathbb {R} ^ n_t} (0) ## (##n## instead of ##k##) should be a diffeomorphism. Indeed, it seems to me that ## t ^ {k + 1}, ..., t ^ n ## are always zero, as ##x## varies.
 
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  • #2
Is your claim that ##f^l(x_1,...,x_n)=x^l## for ##l >k##? That seems unlikely (or you would have had something that was diffeomorphic on the whole space to begin with).

EDIT: sorry, I was too hasty. ##f^l## is only a function of ##x_1,...,x_k##, so could not possibly be equal to ##x_l## for ##l>k##.

To address your other possible confusion (though maybe you get the point and you're just not sure why it matters), the point of the theorem is if you have a diffeomorphism mapping to some lower dimensional manifold of ##\mathbb{R}^n## you can extend it (at least locally) to a diffeomorphism mapping to all of ##\mathbb{R}^n##.

It might help to consider an example. ##\phi(x,y)=(x,y,\sqrt{1-x^2-y^2})## is a function from a small square around the origin in ##\mathbb{R}^2## to a 2 dimensional manifold in ##\mathbb{R}^3##. All the proof is doing is saying you can turn this into ##\phi(x,y,z)=(x,y,z+\sqrt{1-x^2-y2})## and now it's a map from a small region around the origin in ##\mathbb{R}^3## to a 3 d submanifold of ##\mathbb{R}^3##.
 
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  • #3
Thank you, clear.
 

Related to A Zorich proposition about local charts of smooth surfaces

1. What is the Zorich proposition about local charts of smooth surfaces?

The Zorich proposition states that for any smooth surface, there exists a finite number of local charts that can be used to describe the surface. These charts are coordinate systems that map points on the surface to points in a 2-dimensional plane.

2. How does the Zorich proposition relate to differential geometry?

The Zorich proposition is a fundamental concept in differential geometry, which studies the properties of smooth surfaces and their local charts. It provides a framework for understanding the local behavior of surfaces and how they can be described using coordinate systems.

3. Can the Zorich proposition be extended to higher dimensions?

Yes, the Zorich proposition can be extended to higher dimensions. In fact, it is a key concept in the study of smooth manifolds, which are higher-dimensional spaces that can be described using local charts.

4. How does the Zorich proposition impact the study of surfaces in mathematics?

The Zorich proposition is a fundamental tool in the study of surfaces in mathematics. It allows mathematicians to describe and analyze surfaces using local charts, making it easier to understand their properties and relationships with other surfaces.

5. Are there any practical applications of the Zorich proposition?

Yes, the Zorich proposition has practical applications in fields such as computer graphics and computer-aided design. It is also used in physics and engineering to model and analyze the behavior of surfaces in real-world systems.

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