Differential Geometry of Curves and Surfaces: Revised and Updated Second EditionOne of the most widely used texts in its field, this volume introduces the differential geometry of curves and surfaces in both local and global aspects. The presentation departs from the traditional approach with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. Many examples and exercises enhance the clear, well-written exposition, along with hints and answers to some of the problems. The treatment begins with a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. Suitable for advanced undergraduates and graduate students of mathematics, this text's prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables. For this second edition, the author has corrected, revised, and updated the entire volume. |
Other editions - View all
Common terms and phrases
angle arc length arcwise connected Assume asymptotic curves axis closed curve coefficients compact complete surface conjugate const constant contained convex coordinate curves coordinate neighborhood CR³ cylinder defined definition denote diffeomorphism differentiable function differentiable map equations Example Exercise exists a neighborhood fact Figure follows fundamental form Gauss Gauss map Gauss-Bonnet theorem Gaussian curvature geodesic given hence homeomorphism intersection isometry Jacobi field Lemma local isometry normal vector obtain open set orientation orthogonal p₁ parallel transport parametrized by arc parametrized curve parametrized surface plane curve proof Prop PROPOSITION prove regular curve regular parametrized regular surface ruled surface Show surface of revolution t₁ T₁(S tangent plane tangent vector theorem Tp(S V₁ vector field W₁ zero θυ ди дх