Question Number 224938 by fkwow344 last updated on 13/Oct/25
$$\mathcal{S};\left({x},{y},{z}\right)\in\mathbb{R}^{\mathrm{3}} \mid{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$$\mathrm{Find}\:\mathrm{Geodesic}\:\mathrm{equation} \\ $$$$\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\centerdot\frac{\mathrm{d}{x}^{{i}} }{\mathrm{d}{t}}+\Gamma_{{jk}} ^{{i}} \frac{\mathrm{d}{x}^{{j}} }{\mathrm{d}{t}}\centerdot\frac{\mathrm{d}{x}^{{k}} }{\mathrm{d}{t}}=\mathrm{0}\:\:,\:\:\Gamma_{{jk}} ^{{i}} =\mathrm{g}^{{il}} \left(\frac{\partial\mathrm{g}_{{kl}} }{\partial{x}_{{j}} }−\frac{\partial\mathrm{g}_{{jl}} }{\partial{x}_{{k}} }+\frac{\partial\mathrm{g}_{{jk}} }{\partial{x}_{{l}} }\right) \\ $$