Metric induced by euclidean vs. normal coordinates (Riemannian Geometry)

2011-04-17T14:29:11Z

Hey,

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.

Say I have a diff. Manifold $S^2$. Embedded in $R^3$ the euclidean Metric induces this metric on $S^2$.

$g_{ij} = \begin{pmatrix} 1 & 0 \newline 0 & \sin^2\theta \end{pmatrix}$

with $\theta$ being part of the polar coordinates.

If I introduce riemannian normal coordinates to $S^2$ the metric should look like this:

$g_{ij} = \begin{pmatrix} 1 & 0 \newline 0 & 1/r^2 \end{pmatrix}$

(with $r,\phi$ being riemannian normal coordinates) Or at least that is what I think it should.

Is my assumption correct? Those Metrics should still give the same result for 2 vectors, correct?

Metric induced by euclidean vs. normal coordinates (Riemannian Geometry) - MathOverflow [closed]