I have what I think is a basic question. Say I have a manifold and a metric. How do I write down the most general curve for some arbitrary parameter?(adsbygoogle = window.adsbygoogle || []).push({});

For example in [itex]\mathbb{R}^2[/itex] with the Euclidean metric, I think I should write [itex]\gamma(\lambda) = x(\lambda)\hat{x} + y(\lambda)\hat{y}[/itex]

But what about the case of the [itex]S^{2}[/itex] parameterized by [itex](\theta,\phi)[/itex] given the metric [itex]\mathrm{diag}(1,\sin^{2}{\theta})[/itex]?

Is it [itex]\gamma(\lambda) = \theta(\lambda)\hat{\theta} + \phi(\lambda)\hat{\phi}[/itex]

or something like [itex]\gamma(\lambda) = \theta(\lambda)\hat{\theta} + \sin(\theta)\phi(\lambda)\hat{\phi}[/itex]?

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# Writing a general curve on a manifold given a metric

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