MathbbRnAsARiemannianManifold 2005-09-04 05:40:23 2006-06-11 13:05:10 Definition Euclidean space % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bbm} \newcommand{\En}{\mathbbmss{E}^n} \newcommand{\V}{\mathbbmss{V}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\T}{\operatorname{T}} Let $\En$ be $n$-dimensional Euclidean space, and let $(\V,\langle \cdot, \cdot\rangle)$ 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 $\En$ as a differential manifold. We can obtain an atlas with just one coordinate chart, a Cartesian coordinate system $(x^1,\ldots,x^n)$ which gives us a bijection between $\En$ and $\R^n$. The tangent bundle is trivial, with $\T\En \cong \En \times \V.$ Equivalently, every tangent space $\T_p\En,\; p\in \En$. is isomorphic to $\V$. We can retain a bit more structure, and consider $\En$ as a Riemannian manifold by equipping it with the metric tensor \begin{eqnarray*} g &=& dx^1 \otimes dx^1 + \cdots + dx^n \otimes dx^n \\ &=& \delta_{ij} dx^i \otimes dx^j. \end{eqnarray*} We can also describe $g$ in a coordinate-free fashion as \[ g(u,v) = \langle u,v\rangle,\quad u,v\in \V.\] \subsubsection*{Properties} \begin{enumerate} \item Geodesics are straight lines in $\R^n$. \item The Christoffel symbols vanish identically. \item The Riemann curvature tensor vanish identically. \end{enumerate} Conversely, we can characterize Eucldiean space as a connected, complete Riemannian manifold with vanishing curvature and trivial fundamental group.