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...Geometry of curves and surfaces weiyi zhang mathematics institute university warwick september contents course description abit preparation dierentiation methods describing a curve fixed coordinates moving frames parametrized intrinsic way coordinate free in rn arclength parametrization curvature orthonormal frame frenet serret equations plane more results for space taylor expansion fundamental theorem the local theory isoperimetric inequality four vertex r denitions examples compact level sets first form length angle area isometry conformal equiareal second normals orientability gauss map curvatures rst properties calculation gaussian mean principal s theorema egregium special cases constant parallel transport covariant derivative geodesics general facts on revolution shortest paths geodesic half model hyperbolic bonnet polygons global vector elds euler number...