Curves and Surfaces
Authors: Abate, M., Tovena, F.
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The book provides an introduction to Differential Geometry of Curves and Surfaces. The theory of curves starts with a discussion of possible definitions of the concept of curve, proving in particular the classification of 1dimensional manifolds. We then present the classical local theory of parametrized plane and space curves (curves in ndimensional space are discussed in the complementary material): curvature, torsion, Frenet’s formulas and the fundamental theorem of the local theory of curves. Then, after a selfcontained presentation of degree theory for continuous selfmaps of the circumference, we study the global theory of plane curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of class C2, and Hopf theorem on the rotation number of closed simple curves. The local theory of surfaces begins with a comparison of the concept of parametrized (i.e., immersed) surface with the concept of regular (i.e., embedded) surface. We then develop the basic differential geometry of surfaces in R3: definitions, examples, differentiable maps and functions, tangent vectors (presented both as vectors tangent to curves in the surface and as derivations on germs of differentiable functions; we shall consistently use both approaches in the whole book) and orientation. Next we study the several notions of curvature on a surface, stressing both the geometrical meaning of the objects introduced and the algebraic/analytical methods needed to study them via the Gauss map, up to the proof of Gauss’ Teorema Egregium. Then we introduce vector fields on a surface (flow, first integrals, integral curves) and geodesics (definition, basic properties, geodesic curvature, and, in the complementary material, a full proof of minimizing properties of geodesics and of the HopfRinow theorem for surfaces). Then we shall present a proof of the celebrated GaussBonnet theorem, both in its local and in its global form, using basic properties (fully proved in the complementary material) of triangulations of surfaces. As an application, we shall prove the PoincaréHopf theorem on zeroes of vector fields. Finally, the last chapter will be devoted to several important results on the global theory of surfaces, like for instance the characterization of surfaces with constant Gaussian curvature, and the orientability of compact surfaces in R3.
 Reviews

From the reviews:
“The authors’ goal in writing this book is to present the theory of curves and surfaces from the viewpoint of contemporary mathematics. … New concepts and new definitions are fully motivated, and illustrated by numerous examples. … the book is beautifully written, very well organized, and most of all it may serve as both a less advanced text and a more advanced text for readers interested in the theory of curves and surfaces.” (Andrew Bucki, Mathematical Reviews, June, 2013)
“It is dedicated to the study of curves and surfaces both from a local and global viewpoint. It is written and organised in such a way that it can be used by a large scope of students, not only for beginning, intermediate or advanced undergraduate courses in mathematics or physics, but also for engineering or computer science students. … the book can be useful for postgraduate students, too. The book is well written and includes many examples and figures.” (Raúl Oset Sinha, Zentralblatt MATH, Vol. 1238, 2012)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Curves and Surfaces
 Authors

 M. Abate
 F. Tovena
 Series Title
 La Matematica per il 3+2
 Copyright
 2012
 Publisher
 SpringerVerlag Mailand
 Copyright Holder
 SpringerVerlag Milan
 eBook ISBN
 9788847019416
 DOI
 10.1007/9788847019416
 Softcover ISBN
 9788847019409
 Series ISSN
 20385722
 Edition Number
 1
 Number of Pages
 XIII, 396
 Topics