Integrable Geodesic Flows on Two-dimensional Surfaces
A.V. Bolsinov, A.T. Fomenko · 2004
Chapter 11 Integrable Geodesic Flows on Two-dimensional Surfaces 11.1. STATEMENT OF THE PROBLEM Let M n be a smooth Riemannian manifold with a Riemannian metric g ij (x). Recall that geodesics of the given metric are dened as smooth parameterized curves (t) = (x (t); : : : ; x n (t)) that are solutions to the system of dierential equations r _ _ = 0 ; where _ = d dt is the velocity vector of the curve , and r is the covariant derivation operator related to the symmetric connection associated with the metric g ij . In local coordinates, these equations can be rewritten in the form d x i dt + X i jk dx j dt dx k dt = 0 ; where i jk (x) are smooth functions called the Christoel symbols of the connec- tion r and dened by the following explicit formulas: i jk (x) = X g is @g sj @x k + @g ks @x j @g kj @x s : The geodesics can be interpreted as trajectories of a single mass point that moves on the manifold without any external action, i.e., by inertia. Indeed, the equation CH. 11.