## A course in differential geometry |

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angle arc length assume basis bijection Bp(p called change of variables circle closed geodesic compact surface conjugate points consider convex coordinate system curvature functions curve c(t defined denote diffeomorphism differentiable function differentiable mappings differential equations differential geometry distinguished Frenet-frame equal equivalent Euclidean space ex(t example expp follows Frenet equations Gauss curvature geodesic coordinates given grad implies injective inner product integral inverse isometry Jacobi field Lemma Let c(t line element linear map linearly independent Ma)aeA Math minimal geodesic minimal surfaces neighborhood open set oriented surface osculating ovaloid parallel translation parameterized by arc plane curve polygon positively oriented principal curvature principal direction Proof Proposition prove radius Remark Riemannian metric satisfies second fundamental form sphere subset sufficiently small surface of revolution surface with Riemannian tangent space tangent vector tangential vector field theorem topology torus TuR2 unit-speed geodesic vector field