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I am trying to show that the geodesics of $\mathbb{S}^n$ are the great circles, as an exercise for my introductory Riemannian geometry class. I don't really know how to go about this. I suppose that using the geodesic equation would be too complicated so I am trying to use the fact that if $M$ is a submanifold of $\mathbb{R}^N$, $c$ a smooth curve on $M$, and $X$ a vector field along $c$, then the covariant derivative along $c$ $\frac{D}{dt}X=(\frac{dX}{dt})^T$, where $(\frac{dX}{dt})^T$ is the tangent part of the usual derivative in $\mathbb{R}^N$.

Could anyone provide a hint on how to proceed? Thanks

baltazar
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    The "usual geometric method" is to invoke symmetry (assuming you already know geodesics exist on a compact Riemannian manifold), see (e.g.) Shortest path on a sphere. Your idea of looking for curves whose acceleration is normal to the sphere also works; it helps to consider the Euclidean plane $P$ containing the center of the sphere and the initial velocity of the geodesic, and to show the entire geodesic stays in $P$. – Andrew D. Hwang Nov 01 '15 at 13:05

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