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](http://books.google.com.uy/books?id=Pd8-s6rOt_cC&lpg=PP1&ots=uKnfvGEGPw&dq=V.I.%2520arnold&pg=PA301#v=onepage&q=V.I.%2520arnold&f=false) 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. 



Parallel translation on surfaces
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.


