Euclidean space as a manifold

Let ${𝔼}^n$ be $n$-dimensional Euclidean space, and let $(𝕍,⟨\cdot ,\cdot ⟩)$ be the corresponding $n$-dimensionalinner product space of translation isometries. Alternatively, we canconsider Euclidean space as an inner product space that has forgottenwhich point is its origin. Forgetting even more information, we havethe structure of ${𝔼}^n$ as a differential manifold. We can obtain anatlas with just one coordinate chart, a Cartesian coordinate system$(x^1,\mathrm{\dots },x^n)$ which gives us a bijection between ${𝔼}^n$ and ${ℝ}^n$. Thetangent bundle is trivial, with $\mathrm{T}{𝔼}^n\cong {𝔼}^n\times 𝕍.$Equivalently, every tangent space ${\mathrm{T}}_p{𝔼}^n,p\in {𝔼}^n$. is isomorphicto $𝕍$.

We can retain a bit more structure, and consider ${𝔼}^n$ as a Riemannianmanifold 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 coordinate-free fashion as

$$g(u,v)=⟨u,v⟩,u,v\in 𝕍.$$

Properties

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

2. The Christoffel symbols vanish identically.

3. The Riemann curvature tensor vanish identically.

Conversely, we cancharacterize Eucldiean space as a connected, complete Riemannianmanifold with vanishing curvature and trivial fundamental group.