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What does it mean for a vector field X(t) to be parallel along a curve, gamma(t)?

and how can we show that if X(t) is parallel along gamma(t), then |X(t)| is constant?


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up vote 7 down vote accepted

On a Riemannian manifold $M$, one has a notion of "parallel transport" defined along any curve $\gamma: [a,b] \to M$: given any tangent vector $X(a)$ based at the point $\gamma(a)$, one obtains a family of tangent vectors $X(t)$ ($t \in [a,b]$) with $X(t)$ based at $\gamma(t)$. Intuitively, the idea is that $X(t)$ is "the same vector" as $X(a)$, but moved from $\gamma(a)$ to $\gamma(b)$. (So $X(t)$ is "parallel" to $X(a)$, whence the name.) The actual definition is somewhat involved, using the concept of the Levi--Civitta connection, as briefly discussed in Berci's answer.

One property of parallel transport is that, for any value of $t$, the parallel transport map $X(a) \mapsto X(t)$ from the tangent space at $\gamma(a)$ to the tangent space at $\gamma(t)$ is an isometry of inner product spaces, and in particular preserves lengths. Thus $|X(t)|$ is constant along the curve $\gamma$.

If all this is unfamiliar to you, you will need to learn the basics of the Levi--Cevitta connection and related ideas. This is a topic that is notoriously complicated for a beginner to learn, since the presentations often emphasize techincal precision over intuitive clarity (and the wikipedia page on parallel transport linked to in Berci's answer doesn't seem to deviate from this general approach). For myself, I first learned this material from Spivak's differential geometry books (I think the second volume is the relevant one here); they are long, but I think he does a good job of emphasizing the intuitive meaning of things.

This answer of mine might also help with the intuition.

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thanks for the explanation – mary Oct 19 '12 at 7:59

This notion of 'Parallel transport' is defined by the means of a covariant derivative (or Levi-Civittá connection on Riemannian manifolds).

Practically, understanding the covariant derivatives geometrically might be easier using the parallel transport as a primitive concept, as it is sketched on the same wiki page here..

For your other question, is $\gamma$ specifically given, or anything more? I'm not sure the statement is true in this form..

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