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A'jk affine connection analytic angle arc length assume auto-parallel called Christoffel symbols circle class C2 components conformal mappings const constant curvature coordinate system covariant derivative covariant differentiation curvature tensor defined developable surface differential equations differential geometry direction element of arc ellipsoid Euclidean space Exercise exists following result formulas function fundamental Gauss Gaussian curvature geodesic curvature geodesic curves geodesic parallelism gik(x given hence Hint holonomy group indicatrix initial conditions integral intrinsic geometry invariant lemma linear connection lines of curvature minimal surfaces moving trihedron obtain orthogonal parallel displacement parallel vectors parameter transformation Proof properties quantities Riemannian curvature Riemannian geometry Riemannian metric Riemannian space rigid motions satisfy similarity mapping solution space curve spaces of constant spherical straight lines surfaces of revolution symmetric tangent plane tangent space tangent vectors Theorem theory of surfaces torsion total energy trajectories unit vector vanishes vector field vector space