MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.