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affine connexion angle arbitrary arc length asymptotic lines called Cartesian Chapter compact surface components condition consider constant curvature contravariant vector coordinate neighbourhood coordinate system corresponding covariant differentiation covariant tensors covariant vector curvature tensor defined definition denote differentiable manifold differential equations differential geometry distance Euclidean space example Exercise follows formula function Gaussian curvature geodesic arc geodesic curvature given gives helicoid helix Hence identity implies integrable intrinsic isometric isomorphism linear lines of curvature mapping Math matrix metric tensor null obtained osculating plane parallel field parameter parametric curves position vector principal curvatures principal normal proof prove r-planes radius real numbers real-valued region relation respect Riemannian manifold Riemannian space satisfy scalar Show space curve sphere suffixes surface of revolution tangent plane tangent space tangent vector tensor field tensor of type torsion total curvature transformation uniquely unit vector values vector field vector space zero