## A First Course in Differential Geometry |

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algebraic angle arbitrary asymptotic lines called compact components Compute const constant coordinates corresponding covariant derivative DEFINITION denote diffeomorphism differentiable function differentiable manifold differentiable mapping differential geometry domain ds ds elementary euclidean space Exercise follows Frenet formulas Frenet frame function f fundamental form Gauss geodesic line geometric global Hence Hint immersed manifolds implicit equations inflection point integral intersection invariant Lemma linear lines of curvature m-dimensional metric natural parameter normal curvature normal line normal vector notation obtain open neighborhood open subset orthogonal osculating plane parameterization parametric equations parametric lines path plane curve principal normal Proof properties Proposition Prove radius ruled surface scalar second fundamental tensor singular points sphere straight line surface of revolution tangent plane tangent space tangent vector tangent vector field tensor field theorem torsion total curvature u1 sin u2 vector field Weingarten transformation whence yields