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Curves in the Plane and in Space
special and proper affine motions
What they knew about Surfaces before Gauss
14 other sections not shown
1-forms Addendum affine motion arbitrary bundle of frames calculation Cartan connection Chapter choose Christoffel symbols circle classical clearly components Consequently consider constant curvature convex coordinate system Corollary covariant derivative curvature tensor define definition denote diffeomorphism Ehresmann connection ellipsoid equivalent Euclidean motion expression extended manifold follows formula function f Gauss Gaussian curvature geodesic geometry given hence horizontal identity implies inner product invariant Koszul connection Lemma Levi-Civita connection line element linear locally isometric matrix metric relations moving frame neighborhood numbers obtain parallel translation parameterized by arclength perpendicular points c(s position principal bundle Proof Proposition prove result Riemann Riemannian manifold Riemannian metric Riemannian normal coordinate satisfy shows spanned structural equation surface tangent space tangent vector tensor of type Theorem tion torsion transformation unique usual Riemannian metric vector field vector space vertical volume write