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I learned a few months ago that parallel transport or covariant derivative of vector along a close loop on Riemannian manifold just cause "rotation" of vector about some angle,but doesn't change it's magnitude (assuming no torsion).Now I already know a little bit more about lie group SO(2or 3) and their algebras when I deal with rigid body and it always has to do with rotation of vectors.My question is that if parallel transport or covariant derivative just causes "rotation" of vector when it moves,How can I express this Parallel transport or covariant derivative in terms of just SO(2 or 3) acting on vector on 2 or 3D reimanm manifold as covariant deriva. deal with just "rotation" of vector when it moves on manifold?

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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$
    – Qmechanic
    Commented Jun 27, 2018 at 5:42
  • $\begingroup$ The covariant derivative doesn't cause the rotation of the vector - it's path the tangent vector is following which causes the rotation of the tangent vector - and the rotation is relative to it's starting point. The path is typically either a triangle or a parallelogram. For instance, if the angles of the triangle don't add up to 180 degrees then the space is considered to be curved relative to Euclidean space. $\endgroup$ Commented Apr 1, 2020 at 5:50
  • $\begingroup$ The covariant derivative in essence defines a straight line on a curved manifold. $\endgroup$ Commented Apr 1, 2020 at 6:06

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For each point and a closed loop $\gamma :[0,1]\to M$ around the point, the parallel transport around $\gamma$ maps each tangent vector to that point to a tangent vector of the same point. In the case of the Levi-Civita connection, this map of the tangent space to itself is as you said an element of SO(N).

To calculate this map (called the Holonomy) $Hol(\gamma ,p,\nabla ):T_{p}M\to T_pM$ explicitly, you will need to solve the parallel transport ODE $\nabla _\dot \gamma V=0$ with all initial conditions $V(0)$. This is a linear ODE and you will get that the map $Hol=V(0)\to V(1)$ is in $SO(T_pM)$.

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