Metric induced by euclidean vs. normal coordinates (Riemannian Geometry) - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-25T16:52:29Z http://mathoverflow.net/feeds/question/62039 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62039/metric-induced-by-euclidean-vs-normal-coordinates-riemannian-geometry Metric induced by euclidean vs. normal coordinates (Riemannian Geometry) Kay 2011-04-17T14:29:11Z 2011-04-17T14:29:11Z <p>Hey,</p> <p>I've just started my way in Differential Geometry and been having alot of questions but my first has to do with the difference in metrics.</p> <p>Say I have a diff. Manifold $S^2$. Embedded in $R^3$ the euclidean Metric induces this metric on $S^2$. </p> <p>$g_{ij} = \begin{pmatrix} 1 &amp; 0 \newline 0 &amp; \sin^2\theta \end{pmatrix}$</p> <p>with $\theta$ being part of the polar coordinates. </p> <p>If I introduce riemannian normal coordinates to $S^2$ the metric should look like this:</p> <p>$g_{ij} = \begin{pmatrix} 1 &amp; 0 \newline 0 &amp; 1/r^2 \end{pmatrix}$</p> <p>(with $r,\phi$ being riemannian normal coordinates) Or at least that is what I think it should.</p> <p>Is my assumption correct? Those Metrics should still give the same result for 2 vectors, correct?</p>