Euclidean space as a manifold

Let 𝔼n be n-dimensional Euclidean spaceMathworldPlanetmath, and let (𝕍,,) be the corresponding n-dimensional inner product space of translationPlanetmathPlanetmath isometriesMathworldPlanetmath. 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 𝔼n as a differential manifold. We can obtain an atlas with just one coordinate chart, a Cartesian coordinate system (x1,,xn) which gives us a bijection between 𝔼n and n. The tangent bundleMathworldPlanetmath is trivial, with T𝔼n𝔼n×𝕍. Equivalently, every tangent spaceMathworldPlanetmath Tp𝔼n,p𝔼n. is isomorphic to 𝕍.

We can retain a bit more structure, and consider 𝔼n as a Riemannian manifoldMathworldPlanetmath by equipping it with the metric tensor

g = dx1dx1++dxndxn
= δijdxidxj.

We can also describe g in a coordinate-free fashion as



  1. 1.

    GeodesicsMathworldPlanetmath are straight lines in n.

  2. 2.

    The Christoffel symbolsMathworldPlanetmathPlanetmath vanish identically.

  3. 3.

    The Riemann curvature tensorMathworldPlanetmath vanish identically.

Conversely, we can characterize Eucldiean space as a connected, complete Riemannian manifold with vanishing curvature and trivial fundamental groupPlanetmathPlanetmath.

Title Euclidean space as a manifold
Canonical name EuclideanSpaceAsAManifold
Date of creation 2013-03-22 15:29:48
Last modified on 2013-03-22 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