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The Fundamental Equations for Hypersurfaces
Elements of the Theory of Surfaces in
A Compendium of Surfaces
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1-forms affine maps affine normal direction angle apolarity conditions asymptotic curve basis Chapter choose closed curve Codazzi-Mainardi equations compact oriented component connected consider constant curvature convex coordinate system Corollary corresponding cosh cylinder define derivative diffeomorphism everywhere fact follows function Gaussian curvature geometry graph hence homeomorphic hyperbolic imbedding immersion f inner product integral curves intersect inversion isometry Lemma linearly independent lines of curvature M C]R map f matrix moving frame neighborhood normal map obtain ordinary surface theory orientation preserving orthogonal orthonormal moving frame parabolic points parallel parameter curves parameterized by arclength perpendicular planar point positively oriented principal curvatures Problem Proof Proposition quadratic result Riemannian manifold Riemannian metric ruled surface satisfies second fundamental form shows simply sphere straight lines submanifold subset Suppose tangent developable tangent plane tangent vector tensor Theorem torus umbilics unit vector field vector field z-axis zero