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Let $F:S^{2}\rightarrow\mathbb{R}^{4}$ be the immersion defined as $(x^{2}-y^{2},xy,xz,yz)$ and consider the metric on $S^{2}$ induced by $F$. Find $g_{ij}(0,0)$ for the upper hemisphere parameterization $\phi(x,y)=(x,y,\sqrt{1-x^{2}-y^{2}})$

For the metric on $S^{2}$ induced by $F$, we can explicity determine $g_{ij}$ (this is just finding $(DF)^{T}(DF)$): $$ g_{ij}=\left(\begin{array}{cccc} 4x^{2}+4y^{2} & 0 & 2xz & 0\\ 0 & y^{2}+z^{2} & yz & xz\\ 2xz & yz & x^{2}+z^{2} & xy\\ -2yz & xz & xy & z^{2}+y^{2}\end{array}\right)$$

I am wondering for the parameterization $\phi$, would I just substitute $z$ by $\sqrt{1-x^{2}-y^{2}}$ and calculuate $g_{ij}(0,0)$?

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    $(g_{ij})$ cannot be that matrix: it is a $2\times 2$ matrix!2011-10-04
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    I essentially used the wikipedia article http://en.wikipedia.org/wiki/Metric_tensor under "induced metric" to compute the metric. Perhaps I am not understanding something here?2011-10-04
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    Indeed, you are not understanding something: the matrix you are trying to compute is the matrix of coefficients of an inner product defined on the tangent spaces to the spheres, which are of dimension two. You should try a more systematic textbook, really.2011-10-04
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    @kobebryant: I think you have your matrices the wrong way around. $(DF)$ should be a $4 \times 2$ matrix, so $(DF)^T (DF)$ should be a $2 \times 2$ matrix.2011-10-04
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    In that case, I presume that $z$'s are replaced by $\sqrt{1-x^{2}-y^{2}}$ to get a 4x2 matrix for $DF$?2011-10-04

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