Suppose $M$ is a two-dimensional manifold. Let $\sigma:M \rightarrow M$ be an isometry such that $\sigma^2=1$. Suppose that the fixed point set $\gamma=\{x \in M| \sigma(x)=x\}$ is a connected one-dimensional submanifold of $M$. The question asks to show that $\gamma$ is the image of a geodesic.
Check that a curve is a geodesic.
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differential-geometry