This first course in differential geometry presents the fundamentals of the metric differential geometry of curves and surfaces in a Euclidean space of three dimensions. Written by an outstanding teacher and mathematician, it explains the material in the most effective way, using vector notation and technique. It also provides an introduction to the study of Riemannian geometry.
Suitable for advanced undergraduates and graduate students, the text presupposes a knowledge of calculus. The first nine chapters focus on the theory, treating the basic properties of curves and surfaces, the mapping of surfaces, and the absolute geometry of a surface. The final chapter considers the applications of the theory to certain important classes of surfaces: surfaces of revolution, ruled surfaces, translation surfaces, and minimal surfaces. Nearly 200 problems appear throughout the text, offering ample reinforcement of every subject.
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absolute geometry according analytic functions asymptotic direction asymptotic lines binormal center of curvature circle components conjugate system const coordinates corresponding points cylinder deﬁned deﬁnition developable surface direction cosines Dupin indicatrix dv/du edge of regression edu2 envelope equal family of curves Fdudv ﬁnd ﬁrst ﬁxed follows formulas fundamental form gdv2 geodesic curvature geodesic parallels geodesic torsion given helicoid Hence identically inasmuch indicatrix inﬁnitely inﬁnitesimal isometric parameters isometric system isotropic curves linear element lines of curvature minimal surface necessary and suﬁicient obtained orthogonal system orthogonal trajectories osculating plane parametric curves parametric representation plane curve positive direction principal directions proper vector Prove regular point represented respect right helicoid rulings Show space curve spherical representation straight line suﬁicient condition surface of revolution system of curves tangent plane tangent surface THEOREM tion total curvature translation surface trihedral triple twisted curve u-curve unit vectors values variable zero