Umfang:
Online-Ressource (XIII, 396 p, digital)
ISBN:
9788847019416
Serie:
Collana Unitext - La Matematica per il 3+2
Inhalt:
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 1-dimensional manifolds. We then present the classical local theory of parametrized plane and space curves (curves in n-dimensional 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 self-contained presentation of degree theory for continuous self-maps 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 Hopf-Rinow theorem for surfaces). Then we shall present a proof of the celebrated Gauss-Bonnet 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 Poincaré-Hopf 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.
Anmerkung:
Description based upon print version of record
,
""Title Page""; ""Copyright Page""; ""Preface""; ""Table of Contents""; ""1 Local theory of curves""; ""1.1 How to define a curve""; ""1.2 Arc length""; ""1.3 Curvature and torsion""; ""Guided problems""; ""Exercises""; ""PARAMETRIZATIONS AND CURVES""; ""LENGTH AND RECTIFIABLE CURVES""; ""REGULAR AND BIREGULAR CURVES""; ""CURVATURE AND TORSION""; ""FRENET FRAME AND OSCULATING PLANE""; ""FRENET-SERRET FORMULAS""; ""FUNDAMENTAL THEOREM OF THE LOCAL THEORY OF CURVES""; ""EVOLUTE AND INVOLUTE""; ""SPHERICAL INDICATRICES""; ""Supplementary material""; ""1.4 The local canonical form""
,
""Exercises""""ORDER OF CONTACT""; ""1.5 Whitney�s Theorem""; ""1.6 Classification of 1-submanifolds""; ""Exercises""; ""1.7 Frenet-Serret formulas in higher dimensions""; ""Exercises""; ""2 Global theory of plane curves""; ""2.1 The degree of curves in S1""; ""2.2 Tubular neighborhoods""; ""2.3 The Jordan curve theorem""; ""2.4 The turning tangents theorem""; ""Guided problems""; ""Exercises""; ""HOMOTOPIES""; ""TUBULAR NEIGHBORHOOD""; ""WINDING NUMBER AND THE JORDAN CURVE THEOREM""; ""ROTATION INDEX AND THE TURNING TANGENTS THEOREM""; ""Supplementary material""; ""2.5 Convex curves""
,
""Exercises""""2.6 The four-vertex theorem""; ""2.7 Isoperimetric inequality""; ""2.8 Schönflies� theorem""; ""3 Local theory of surfaces""; ""3.1 How to define a surface""; ""3.2 Smooth functions""; ""3.3 Tangent plane""; ""3.4 Tangent vectors and derivations""; ""Guided problems""; ""Exercises""; ""IMMERSED SURFACES AND REGULAR SURFACES""; ""SMOOTH FUNCTIONS""; ""TANGENT PLANE""; ""SMOOTH MAPS BETWEEN SURFACES""; ""Supplementary material""; ""3.5 Sard�s theorem""; ""3.6 Partitions of unity""; ""4 Curvatures""; ""4.1 The first fundamental form""; ""4.2 Area""; ""4.3 Orientability""
,
""4.4 Normal curvature and second fundamental form""""4.5 Principal, Gaussian and mean curvatures""; ""4.6 Gauss� Theorema egregium""; ""Guided problems""; ""Exercises""; ""FIRST FUNDAMENTAL FORM""; ""ISOMETRIES AND SIMILITUDES""; ""ORIENTABLE SURFACES""; ""SECOND FUNDAMENTAL FORM""; ""PRINCIPAL, GAUSSIAN AND MEAN CURVATURES""; ""LINES OF CURVATURE""; ""ISOMETRIES, AGAIN""; ""ASYMPTOTIC CURVES""; ""ELLIPTIC, HYPERBOLIC, PARABOLIC, PLANAR AND UMBILICAL POINTS""; ""GAUSS� THEOREMA EGREGIUM""; ""CONFORMAL MAPS""; ""RULED SURFACES""; ""MINIMAL SURFACES""; ""Supplementary material""
,
""4.7 Transversality""""Exercises""; ""4.8 Tubular neighborhoods""; ""4.9 The fundamental theorem of the local theory of surfaces""; ""5 Geodesics""; ""5.1 Geodesics and geodesic curvature""; ""5.2 Minimizing properties of geodesics""; ""5.3 Vector fields""; ""Guided problems""; ""Exercises""; ""GEODESICS AND GEODESIC CURVATURE""; ""VECTOR FIELDS""; ""LIE BRACKET""; ""Supplementary material""; ""5.4 The Hopf-Rinow theorem""; ""Exercises""; ""5.5 Locally isometric surfaces""; ""6 The Gauss-Bonnet theorem""; ""6.1 The local Gauss-Bonnet theorem""; ""6.2 Triangulations""
,
""6.3 The global Gauss-Bonnet theorem""
Weitere Ausg.:
ISBN 9788847019409
Weitere Ausg.:
Buchausg. u.d.T. Abate, Marco Curves and surfaces Milan [u.a.] : Springer, 2012 ISBN 9788847019409
Sprache:
Englisch
Fachgebiete:
Mathematik
Schlagwort(e):
Differentialgeometrie
;
Fläche
;
Kurve
DOI:
10.1007/978-88-470-1941-6
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Bookmarklink
