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YCor
YCor
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What are the open subsets of $\mathbb{R}^n$ thatthat are diffeomorphic to $\mathbb$\mathbb{R}^n$

Hello,

I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it isto be diffeomorphic to $\mathbb{R}^n$ :

For example :

  1. Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$ ?

  2. Reciprocally, are all open subsets of $\mathbb{R}^n$ which are diffeomorphic to $\mathbb{R}^n$, star-shaped ?

Thank you for your answers and proofs

What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

Hello,

I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ :

For example :

  1. Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$ ?

  2. Reciprocally, are all open subsets of $\mathbb{R}^n$ which are diffeomorphic to $\mathbb{R}^n$, star-shaped ?

Thank you for your answers and proofs

What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ to be diffeomorphic to $\mathbb{R}^n$ :

For example :

  1. Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$ ?

  2. Reciprocally, are all open subsets of $\mathbb{R}^n$ which are diffeomorphic to $\mathbb{R}^n$, star-shaped ?

Thank you for your answers and proofs

retag
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Ryan Budney
Ryan Budney
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Hello,

I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ :

For example :

  1. Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$ ?

  2. Reciprocally, are all open subsets of $\mathbb{R}^n$ which are diffeomorphic to $\mathbb{R}^n$, star-shaped ?

Thank you for your answers and proofs

Hello,

I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ :

For example :

  1. Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$ ?

  2. Reciprocally, are all open subsets of $\mathbb{R}^n$ which are diffeomorphic to $\mathbb{R}^n$, star-shaped ?

Thank you for your answers and proofs

Hello,

I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ :

For example :

  1. Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$ ?

  2. Reciprocally, are all open subsets of $\mathbb{R}^n$ which are diffeomorphic to $\mathbb{R}^n$, star-shaped ?

Thank you for your answers and proofs

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Oliver
Oliver
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What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

Hello,

I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ :

For example :

  1. Are all open star-shaped subsets of $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$ ?

  2. Reciprocally, are all open subsets of $\mathbb{R}^n$ which are diffeomorphic to $\mathbb{R}^n$, star-shaped ?

Thank you for your answers and proofs