Curvatures, photon spheres, and black hole shadows

In a recent work PRD 106, L021501 (2022), new geometric approach is proposed to obtain the photon sphere (circular orbit) and black hole shadow radius. this approach, spheres radius are determined using geodesic curvature Gaussian in optical geometry of spacetimes. However, calculations (2022) only restricted subclass static spherically symmetric holes with spacetime metric $g_{tt} \cdot g_{rr}=-1$, $g_{\theta\theta}=r^{2}$ $g_{\phi\phi}=r^{2}\sin^{2}\theta$. work, we extend more general (with $ds^{2}=g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}$). Furthermore, it can be proved that our results from completely equivalent those conventional based on effective potentials test particles.