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Parallel translation of a vector along a geodesic in a surface is characterized by the following three properties:

  1. The vector being transported moves continuously.
  2. It has constant norm.
  3. It maintains a constant angle with the geodesic.

In V.I. Arnold's book on mechanics he uses this starting point to define parallel translation along other curves. He does this by approximating a general curve by a concatenation of small geodesic segments.

Another way of formalizing Arnold's idea is to calculate for each tangent vector $(p,\dot{p}) \in TS$ ($S$ a Riemannian surface) and each vector $v \in T_pS$ the derivative $H(p,\dot{p})(v)$ of the parallel transport of $v$ along the geodesic with initial condition $(p,\dot{p})$. Then we say a curve $t \mapsto (p(t),v(t))$ is parallel if: $v'(t) = H(p(t),p'(t))(v(t))$ for all $t$.

My question is the following: Is there a conceptual proof that $H(p,\dot{p})(v)$ depends linearly on $\dot{p}$?

In other words: Why does parallel transport define a Horizontal linear subspace in $T_{(p,v)}TS$?

Of course one can verify this by calculating $H$ explicitly, but that's not what I'm looking for.

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1 Answer 1

The easiest way I see is to define parallel translation the usual way. (This angle property as well as linearity is evident.) Then show that angle property alone plus approximation give the same translation.

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