Is a flat space a affine space?

Question

An affine space of dimension n on $\mathbb R$ is defined to be a non-empty set $E$ such that there exists a vector space $V$ of dimension n on $\mathbb R$ and a mapping

$\phi:E \times E \rightarrow V,\space\space\space (A,B) \mapsto \phi(A,B):=\vec {AB}$

that obeys the following properties:

(i) For any point $O \in E$, the function

$\phi_O: E \rightarrow V,\space\space\space M \mapsto \vec {OM}$

is bijective.

(ii) For any triplet $(A,B,C)$ of elements of $E$, the following relation holds:

$\vec {AB} + \vec {BC} = \vec {AC}.$

My question is do all flat manifold can be considered an affine space?

Answer 1

Consider a right circular cylinder in $\Bbb R^3$.