= Geodesic intersection from reflection symmetry
Suppose reflection in a plane is an <isometry> of a smooth <surface>, and its intersection with the plane is a regular curve. That curve is a <geodesic> when parametrized by arclength. At any intersection point, reflection fixes both the point and every tangent vector to the intersection. It carries the <geodesic> with these initial data to one with the same data. Local uniqueness of the <geodesic equations> forces it to remain in the fixed plane. This is the one-dimensional instance of fixed-point sets of Riemannian isometries being totally <geodesic>.
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