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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?

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closed as not a real question by Anton Petrunin, Deane Yang, Igor Rivin, Will Jagy, Bill Thurston Apr 17 '11 at 20:01

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

1  
You are confusing things. Given any metric $g$ on a manifold $M$, there are always local coordinated (called normal coordinates) centered at any point $p$ in which the metric $g_{ij}(p) = \delta_{ij}$ and also $\frac{\partial}{\partial x^k}g_{ij} (p) = 0$. Now on $S^2$, one can place the round metric (which you call the "Euclidean" metric) or one can place some other metric. Any metric will admit local normal coordinates centered at each point. –  Spiro Karigiannis Apr 17 '11 at 18:58
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Your second matrix $g_{ij}$ is not a metric in normal coordinates. If you have a metric $g$ and look at it in any two local coordinate systems (whether normal or not) then, yes, they will give the same result for the "same" two tangent vectors at a point $p$, but of course the expressions for these "same" two tangent vectors as $2$-tuples of real numbers will be different because you are using two different bases (of coordinate vector fields) for the tangent space $T_p M$. –  Spiro Karigiannis Apr 17 '11 at 19:01

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