Euclidean space as a manifold
Let ${\mathbb{E}}^{n}$ be $n$dimensional Euclidean space^{}, and let $(\mathbb{V},\u27e8\cdot ,\cdot \u27e9)$ be the corresponding $n$dimensional inner product space of translation^{} isometries^{}. Alternatively, we can consider Euclidean space as an inner product space that has forgotten which point is its origin. Forgetting even more information, we have the structure of ${\mathbb{E}}^{n}$ as a differential manifold. We can obtain an atlas with just one coordinate chart, a Cartesian coordinate system $({x}^{1},\mathrm{\dots},{x}^{n})$ which gives us a bijection between ${\mathbb{E}}^{n}$ and ${\mathbb{R}}^{n}$. The tangent bundle^{} is trivial, with $\mathrm{T}{\mathbb{E}}^{n}\cong {\mathbb{E}}^{n}\times \mathbb{V}.$ Equivalently, every tangent space^{} ${\mathrm{T}}_{p}{\mathbb{E}}^{n},p\in {\mathbb{E}}^{n}$. is isomorphic to $\mathbb{V}$.
We can retain a bit more structure, and consider ${\mathbb{E}}^{n}$ as a Riemannian manifold^{} by equipping it with the metric tensor
$g$  $=$  $d{x}^{1}\otimes d{x}^{1}+\mathrm{\cdots}+d{x}^{n}\otimes d{x}^{n}$  
$=$  ${\delta}_{ij}d{x}^{i}\otimes d{x}^{j}.$ 
We can also describe $g$ in a coordinatefree fashion as
$$g(u,v)=\u27e8u,v\u27e9,u,v\in \mathbb{V}.$$ 
Properties

1.
Geodesics^{} are straight lines in ${\mathbb{R}}^{n}$.

2.
The Christoffel symbols^{} vanish identically.

3.
The Riemann curvature tensor^{} vanish identically.
Conversely, we can characterize Eucldiean space as a connected, complete Riemannian manifold with vanishing curvature and trivial fundamental group^{}.
Title  Euclidean space as a manifold 

Canonical name  EuclideanSpaceAsAManifold 
Date of creation  20130322 15:29:48 
Last modified on  20130322 15:29:48 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  9 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 53B21 
Classification  msc 53B20 