I've been wrestling with this all day, and it's starting to drive me crazy; if you can help me out, please take just a few minutes to read and answer.(adsbygoogle = window.adsbygoogle || []).push({});

I'm looking at S^2 in R^3 (the two dimensional unit sphere in space), and at a large circle going through the north pole as my closed curve.

I've proved that parallel translation in T_p(S^2) is simply the reduction of parallel translation in T_p(R^3) [1]. I'll write how in a second, but my problem is that this gives me a stupid contradiction:

Let's look at the tangent vector (0,0,1) at the north pole. Parallel translation of that vector in R^3 along my curve gives me the constant vector field (0,0,1). But that can't be the parallel translation in S^2, seeing as this vector field has tangent vectors that aren't in T_p(S^2) (almost all of them, actually)!

Now, for the relatively easy proof (but obviously wrong) of [1]. Covariant derivative is the same in both, since an isometry phi respects the covariant derivative up to phi_*, and for phi=id, phi_*=id. Now if we look at parallel translation in S^2, since by the previous line it's also parallel in R^3, then by uniqueness it must be the parallel translation in R^3!

Am I going crazy? What did I do wrong?

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# Parallel Translation

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