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I am working through a course in Riemannian Geometry and am confused with curves in the context of geodesics on a manifold.

A manifold is made up of local coordinate patches, so how can you define curves on a manifold which do not only exist locally? Is it possible? You need a coordinate system to be able to describe some parameterised curve, so is it not possible to describe a curve more globally than just in a local coordinate patch?

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A curve on a manifold $M$ is simply a map $\gamma$ from an interval $J \subset \Bbb R$ to $M$; we generally require that $\gamma$ be continuous and (when the manifold has, e.g., a smooth structure) often also require some differentiability condition. This definition makes no reference to particular coordinate patches.

Of course, given a coordinate chart $(U, \phi)$ on $M$, we can describe the coordinate representation $\hat\gamma = (\hat\gamma^1, \ldots, \hat\gamma^n)$ of $\gamma$ in this chart. This is just the map $\gamma^{-1}(U) \to \Bbb R^n$, $t \mapsto (\phi \circ \gamma)(t)$.

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  • $\begingroup$ So how would you find the length of some curve which exists in multiple coordinate charts without some coordinate representation which is valid in multiple charts? $\endgroup$ – J.Main Jun 12 '17 at 15:37
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The point is that the role of the atlas is to describe how local coordinate patches interact, so as to allow global phenomena to take place. It is helpful to think of an equivalent definition of a Riemannian manifold via isometric embedding in a sufficiently high-dimensional Euclidean space: $M^n\to \mathbb{R}^N$. This viewpoint is clearer for accomodating global behavior of curves.

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  • $\begingroup$ So how would you find the length of some curve which exists in multiple coordinate charts without some coordinate representation which is valid in multiple charts? $\endgroup$ – J.Main Jun 12 '17 at 15:37
  • $\begingroup$ You subdivide the curve into subarcs that are short enough so that each is contained in a suitable chart. Then you calculate the length of each arc in its chart. Then you sum the lengths of all the subarcs to get the total length of the curve. @J.Main $\endgroup$ – Mikhail Katz Jun 12 '17 at 15:52

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