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I was wondering: Given a variety over a finite field, say the projective plane or sphere over $\mathbb{F}_q$. Then I can try to define parallel transport along (geodesic) curves. In particular, I can compare the parallel transport along different paths and see if or if not they differ by an element in the point stabilizer, say $\mathrm{SO}_{2}(\mathbb{F}_q)$. This should somehow mean curvature. Any literature tips? I only found parallel transport for $p$-adic varieties.

More explicitly I have the following problem: I want to pick for every two elements (suitable) $x,y$ on a projective plane or sphere an element in $\mathrm{SO}_{3}(\mathbb{F}_q)$ sending $x\mapsto y$; say the rotation along the circle connecting $x,y$. Then I want to understand how much this assignment is non-multiplicative. My guess is that for composing $x\mapsto y \mapsto z$ I can get for different $y$ the same number of times each possible results (differing by the point stabilizer), and those giving the same result as directly $x \mapsto z$ come precisely for the $y$ laying on the circle through $x$, $z$. But I see no way to prove this for general $q$.

This seems like an elementary question about spheres over finite fields, but I am not successful in finding results on it. My intuition puts it in the corner of parallel transport, but maybe the right way to deal with it comes from a complete different direction.

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    $\begingroup$ What is the definition of a sphere over a finite field? $\endgroup$
    – LSpice
    Commented Jul 9, 2018 at 18:43
  • $\begingroup$ @LSpice I presume it is the variety $\sum^n_{i=0} x_i^2 - 1 = 0$ inside $\mathbf{A}^{n+1}_{\mathbf{F}_q}$. I don't really understand the question, but it seems like the question is asking something about the non-abelian-ness of $\mathrm{SO}_3(\mathbf{F}_q)$. Regarding curvature in general: the Kodaira dimension of a smooth variety is a rough analogue of curvature in algebraic geometry. $\endgroup$
    – skd
    Commented Jul 9, 2018 at 22:59
  • $\begingroup$ Yes, with "sphere" I meant those varieties. @skd: I agree that onncommutativity of $SO_3\mathbb({F}_q)$ shows but my question is: I want additional structure on the variety (metric, connection or simliar thing) that produce a CHOICE of say an $SO_3\mathbb({F}_q)$-element for each pair of point $x\mapsto y$. $\endgroup$
    – Simon Lentner
    Commented Aug 25, 2018 at 7:55

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