Proving Compatibility of Charts in a Union of Atlases

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The discussion centers on proving that the union of two atlases, A and B, on a manifold M forms another atlas. The exercise from "A Course in Modern Mathematical Physics" by Peter Szekeres prompts the user to demonstrate compatibility between charts C from A and D from B. The user expresses confusion regarding the compatibility proof and suggests that the exercise may contain a typo, proposing an alternative statement regarding the transitivity of chart compatibility. The consensus indicates that the original proposition is indeed valid, and the compatibility of charts in the union can be established through differentiable functions.

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  • Understanding of manifold theory and atlases
  • Familiarity with differentiable functions and their properties
  • Knowledge of chart compatibility in differential geometry
  • Basic concepts of transitivity in mathematical proofs
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  • Study the concept of chart compatibility in differential geometry
  • Learn about differentiable functions and their implications in manifold theory
  • Research the properties of maximal atlases and their extensions
  • Explore transitivity in mathematical proofs, particularly in the context of atlases
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Mathematicians, students of differential geometry, and anyone interested in the foundational aspects of manifold theory and atlas compatibility.

jojo12345
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One of the exercises in the text I'm using for self-study asks to prove that the union of a pair of atlases A and B on a manifold is another atlas. However, I don't see any way to show that two charts C,D in A\cup B with C\in A~,~D\in B are compatible. Could anyone give me a bit of help? Maybe just a hint?

The book is A Course in Modern Mathematical Physics by Peter Szekeres. The exercise is the first on in chapter 15.
 
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The wording of the exercise is as follows:

If A and B are two atlases on a manifold M, then their union is another atlas. Prove this statement. [Hint: A differentiable function of a differentiable function is always differentiable]

This is really verbatim from the text. However, after reading the line in the text after the exercise, I get the impression that the author made a typo. The next line:

"Any atlas can this be extended to a maximal atlas by adding to it all charts that are compatible with the charts of the atlas."

To me this suggests two things. First, the proposition in the exercise is false. Second, what the author actually wanted to ask was the following:

Show that if (1) A,B, and C are atlases on M, (2) the charts in A are compatible with the charts in B, and (3) the charts in B are compatible with the charts in C, then the charts in A are compatible with the charts in C (transitivity).

What do people think?
 

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