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Question. Let $M$ be a smooth manifold admitting an atlas $\mathcal{A}$ (i.e. a collection of coordinate charts that covers the whole manifold) with the following property:

for every pair $(U, \mathbf{x}),\ (U', \mathbf{x}')\in \mathcal{A}$, the transition function $$\mathbf{x'}\circ\mathbf{x}^{-1}\colon \mathbf{x}(U\cap U')\subset \mathbb{R}^n\to \mathbf{x}'(U\cap U')\subset\mathbb{R}^n $$ is (the restriction of) an isometry of $\mathbb{R}^n$ onto itself.

Is $M$ diffeomorphic to $\mathbb{R}^n$?

Motivation for this conjecture comes from the fact that in such a manifold one can define a Riemannian inner product in exactly the same way as one does in Euclidean space: $$\text{if }\mathbf{v}=v^\alpha\partial_\alpha,\ \mathbf{w}=w^\beta\partial_\beta,\quad \text{then } g(\mathbf{v}, \mathbf{w})=\sum_\alpha v^\alpha w^\alpha.$$ This definition is well posed because it is independent of the chosen coordinate chart in the atlas $\mathcal{A}$. The Riemannian manifold $(M, g)$ is flat and locally isometric to $\mathbb{R}^n$. I don't know if this extends to a global diffeomorphism, though.


Secondary question. What if the transition functions $\mathbf{x}'\circ\mathbf{x}^{-1}$ are merely affine transformations of $\mathbb{R}^n$ onto itself?

In this case, the previous assignment for a Riemannian metric $g$ is ill-posed. However, one can still define a flat affine connection as follows: $$\nabla_\mathbf{w} \mathbf{v}= w^\alpha\frac{\partial v^\beta}{\partial x^\alpha} \frac{\partial}{\partial x^\beta}.$$ It would be interesting to know if this suffices to have a global diffeomorphism onto $\mathbb{R}^n$.

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No. The torus can be written with an atlas of 9 charts (a 3 x 3 grid) with the transition functions all being translation.

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  • $\begingroup$ Are you referring to the 2-dimensional torus? (P.S.: I guess that this should hold in any dimension, actually) $\endgroup$ Commented Jul 14, 2014 at 15:43
  • $\begingroup$ Yes, although the same claim (with more charts!) is almost certainly true in every dimension. Do you want me to write out the neighborhoods/transition functions? $\endgroup$ Commented Jul 14, 2014 at 15:47
  • $\begingroup$ If you added "simply connected" to your question, I think it might get more interesting, but I'm pretty sure that there are simply connected affine manifolds that aren't euclidean space. Bill Goldman spent a bunch of time on this stuff back in the 1980s, I think. $\endgroup$ Commented Jul 14, 2014 at 15:48
  • $\begingroup$ Don't bother with explicit transition functions, it is OK. Even the $\mathbb{S}^1$ example is fine for me. The only question I have is: is there any relationship between the Riemannian metric $g(v, w)=\sum_\alpha v^\alpha w^\alpha$ defined above and the actual Riemannian metric of the manifold (which is the one coming from the embedding into Euclidean space)? Take the case of $\mathbb{S}^1$. $\endgroup$ Commented Jul 14, 2014 at 16:08
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    $\begingroup$ Almost makes you doubt the definition of "flat" for a moment, doesn't it? :) $\endgroup$ Commented Jul 14, 2014 at 21:45

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