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I'm self-studying differential geometetry from Taubes' textbook on Differential Geometry and I found a lot of unclear things in the following passage: round metric on S^n

Taubes gives a formula for $\Gamma^i_{jk}$ but doesn't give the derivation yet and expects us to use it. $g^{ij}$ denotes the i,jth entry in the inverse matrix of $g$. [Christoffel symbols]2]

I have a few questions:

  1. How does he calculate the pullback of $g_{ij}$? Because I got a different answer $g_{ij} = \delta_{ij} + y_i y_j (1-||y||^2)^{-1}$
  2. How does he calculate the $\Gamma^i_{jk}$? I can't even begin to calculate it, as it requires me to find the inverse matrix of $g$, a task which seems far too complicated in this case.
  3. How come the equation with the $\mathcal{O}(|y|^2)$ is actually equivalent to the formula he gave before? His argument is incomprehensible to me.
  4. How does the formula imply that $t \to x_j(t)$ lies on a plane?

These questions are enough for now.

wilkersmon
  • 1,046
  • I did not check his or your calculations but what he is doing is morally wrong: One should be using the stereographic projection as coordinates on the sphere. Then the pull-back metric on $R^n$ is conformally-flat (a scalar function times $\delta_{ij}$) and the computation of Christoffel symbols becomes quite easy. – Moishe Kohan Feb 14 '18 at 10:32
  • @MoisheCohen where could I read more about this approach? – wilkersmon Feb 14 '18 at 11:13
  • Take a look here: https://math.stackexchange.com/questions/1905392/curvature-tensor-of-a-conformally-flat-manifold (I especially like Gunnar's comment). – Moishe Kohan Feb 14 '18 at 11:31
  • Also here for the computation of the pull-back metric (although the computation in the solution is more painful than it should be): https://math.stackexchange.com/questions/1199628/metric-on-n-sphere-in-terms-of-stereographic-projection-coordinates – Moishe Kohan Feb 14 '18 at 11:47

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