Geodesic sphere ($S_r(p)=\{q:d_g(p,q)=r\}$)

A geodesic sphere is a level set of Riemannian distance from a point. For sufficiently small positive $r$, it is the exponential map image of the radius-$r$ tangent sphere, a smooth compact hypersurface. The Gauss lemma makes its tangent space orthogonal to the radial direction. Larger distance spheres need not be smooth at the cut locus.