Geodesic
Separation of variables in the~Hamilton--Jacobi equation for geodesics in two and three dimensions
Teoretičeskaâ i matematičeskaâ fizika, Tome 218 (2024) no. 2, pp. 306-319

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On (pseudo)Riemannian manifolds of two and three dimensions, we list all metrics that admit a complete separation of variables in the Hamilton–Jacobi equation for geodesics. There are three different classes of separable metrics on two-dimensional surfaces. Three-dimensional manifolds admit six classes of separable metrics. Within each class, metrics are related by canonical transformations and a nondegenerate transformation of parameters that does not depend on coordinates.
Keywords: Hamilton–Jacobi equation, separation of variables, geodesic. 
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     author = {M. O. Katanaev},
     title = {Separation of variables in {the~Hamilton--Jacobi} equation for geodesics in two and three dimensions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {306--319},
     publisher = {mathdoc},
     volume = {218},
     number = {2},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2024_218_2_a5/}
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M. O. Katanaev. Separation of variables in the~Hamilton--Jacobi equation for geodesics in two and three dimensions. Teoretičeskaâ i matematičeskaâ fizika, Tome 218 (2024) no. 2, pp. 306-319. http://geodesic.mathdoc.fr/item/TMF_2024_218_2_a5/