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Title edited I thank მამუკა ჯიბლაძე and Corbennick for their suggestion on the title of this question. I changed the title based on the suggestion of Corbennick.

What is an example of a manifold $M$ which does not admit an atlas $\mathcal{A}$ with the following property?:

For every two charts $(\phi,U)$ and $(\psi,V)$ in $\mathcal{A}$, $\psi \circ \phi^{-1}$ is a polynomial map.(Its components are polynomial functions).

Motivation: The motivation for this question comes from the concept "Affine manifolds". For learning this concept and its related problem, Chern conjecture, I am indebted to Mike Cocos.

Title edited I thank მამუკა ჯიბლაძე and Corbennick for their suggestion on the title of this question. I changed the title based on the suggestion of Corbennick.

What is an example of a manifold $M$ which does not admit an atlas $\mathcal{A}$ with the following property?:

For every two charts $(\phi,U)$ and $(\psi,V)$ in $\mathcal{A}$, $\psi \circ \phi^{-1}$ is a polynomial map.(Its components are polynomial functions).

Motivation: The motivation for this question comes from the concept "Affine manifolds". I am indebted to Mike Cocos, for learning this concept and its related problem, that is Chern conjecture.

Title edited I thank მამუკა ჯიბლაძე and Corbennick for their suggestion on the title of this question. I changed the title based on the suggestion of Corbennick.

What is an example of a manifold $M$ which does not admit an atlas $\mathcal{A}$ with the following property?:

For every two charts $(\phi,U)$ and $(\psi,V)$ in $\mathcal{A}$, $\psi \circ \phi^{-1}$ is a polynomial map.(Its components are polynomial functions).

Motivation: The motivation for this question comes from the concept "Affine manifolds". For learning this concept and its related problem, Chern conjecture, I am indebted to Mike Cocos.

Title edited I thank მამუკა ჯიბლაძე and Corbennick for their suggestion on the title of this question. I changed the title based on the suggestion of Corbennick.

What is an example of a manifold $M$ which does not admit an atlas $\mathcal{A}$ with the following property?:

For every two charts $(\phi,U)$ and $(\psi,V)$ in $\mathcal{A}$, $\psi \circ \phi^{-1}$ is a polynomial map.(Its components are polynomial functions).

Motivation: The motivation for this question comes from the concept "Affine manifolds". For learning this concept and its related problem, Chern conjecture, I am indebted to Mike Cocos.

Title edited I thank მამუკა ჯიბლაძე and Corbennick for their suggestion on the title of this question. I changed the title based on the suggestion of Corbennick.

What is an example of a manifold $M$ which does not admit an atlas $\mathcal{A}$ with the following property?:

For every two charts $(\phi,U)$ and $(\psi,V)$ in $\mathcal{A}$, $\psi \circ \phi^{-1}$ is a polynomial map.(Its components are polynomial functions).

Title edited I thank მამუკა ჯიბლაძე and Corbennick for their suggestion on the title of this question. I changed the title based on the suggestion of Corbennick.

What is an example of a manifold $M$ which does not admit an atlas $\mathcal{A}$ with the following property?:

For every two charts $(\phi,U)$ and $(\psi,V)$ in $\mathcal{A}$, $\psi \circ \phi^{-1}$ is a polynomial map.(Its components are polynomial functions).

Motivation: The motivation for this question comes from the concept "Affine manifolds". For learning this concept and its related problem, Chern conjecture, I am indebted to Mike Cocos.