The Geometry Of Geodesics
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Author |
: Herbert Busemann |
Publisher |
: Courier Corporation |
Total Pages |
: 434 |
Release |
: 2012-07-12 |
ISBN-10 |
: 9780486154626 |
ISBN-13 |
: 0486154629 |
Rating |
: 4/5 (26 Downloads) |
Synopsis The Geometry of Geodesics by : Herbert Busemann
A comprehensive approach to qualitative problems in intrinsic differential geometry, this text examines Desarguesian spaces, perpendiculars and parallels, covering spaces, the influence of the sign of the curvature on geodesics, more. 1955 edition. Includes 66 figures.
Author |
: Barrett O'Neill |
Publisher |
: Courier Corporation |
Total Pages |
: 404 |
Release |
: 2014-01-15 |
ISBN-10 |
: 9780486783116 |
ISBN-13 |
: 0486783111 |
Rating |
: 4/5 (16 Downloads) |
Synopsis The Geometry of Kerr Black Holes by : Barrett O'Neill
Suitable for advanced undergraduates and graduate students of mathematics as well as for physicists, this unique monograph and self-contained treatment constitutes an introduction to modern techniques in differential geometry. 1995 edition.
Author |
: A.N. Pressley |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 336 |
Release |
: 2013-11-11 |
ISBN-10 |
: 9781447136965 |
ISBN-13 |
: 1447136969 |
Rating |
: 4/5 (65 Downloads) |
Synopsis Elementary Differential Geometry by : A.N. Pressley
Pressley assumes the reader knows the main results of multivariate calculus and concentrates on the theory of the study of surfaces. Used for courses on surface geometry, it includes intersting and in-depth examples and goes into the subject in great detail and vigour. The book will cover three-dimensional Euclidean space only, and takes the whole book to cover the material and treat it as a subject in its own right.
Author |
: Gabriel P. Paternain |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 160 |
Release |
: 2012-12-06 |
ISBN-10 |
: 9781461216001 |
ISBN-13 |
: 1461216001 |
Rating |
: 4/5 (01 Downloads) |
Synopsis Geodesic Flows by : Gabriel P. Paternain
The aim of this book is to present the fundamental concepts and properties of the geodesic flow of a closed Riemannian manifold. The topics covered are close to my research interests. An important goal here is to describe properties of the geodesic flow which do not require curvature assumptions. A typical example of such a property and a central result in this work is Mane's formula that relates the topological entropy of the geodesic flow with the exponential growth rate of the average numbers of geodesic arcs between two points in the manifold. The material here can be reasonably covered in a one-semester course. I have in mind an audience with prior exposure to the fundamentals of Riemannian geometry and dynamical systems. I am very grateful for the assistance and criticism of several people in preparing the text. In particular, I wish to thank Leonardo Macarini and Nelson Moller who helped me with the writing of the first two chapters and the figures. Gonzalo Tomaria caught several errors and contributed with helpful suggestions. Pablo Spallanzani wrote solutions to several of the exercises. I have used his solutions to write many of the hints and answers. I also wish to thank the referee for a very careful reading of the manuscript and for a large number of comments with corrections and suggestions for improvement.
Author |
: M. Abate |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 407 |
Release |
: 2012-06-11 |
ISBN-10 |
: 9788847019416 |
ISBN-13 |
: 8847019419 |
Rating |
: 4/5 (16 Downloads) |
Synopsis Curves and Surfaces by : M. Abate
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.
Author |
: A. L. Besse |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 271 |
Release |
: 2012-12-06 |
ISBN-10 |
: 9783642618765 |
ISBN-13 |
: 3642618766 |
Rating |
: 4/5 (65 Downloads) |
Synopsis Manifolds all of whose Geodesics are Closed by : A. L. Besse
X 1 O S R Cher lecteur, J'entre bien tard dans la sphere etroite des ecrivains au double alphabet, moi qui, il y a plus de quarante ans deja, avais accueilli sur mes terres un general epris de mathematiques. JI m'avait parle de ses projets grandioses en promettant d'ailleurs de m'envoyer ses ouvrages de geometrie. Je suis entiche de geometrie et c'est d'elle dontje voudrais vous parler, oh! certes pas de toute la geometrie, mais de celle que fait l'artisan qui taille, burine, amene, gauchit, peaufine les formes. Mon interet pour le probleme dont je veux vous entretenir ici, je le dois a un ami ebeniste. En effet comme je rendais un jour visite il cet ami, je le trouvai dans son atelier affaire a un tour. Il se retourna bientot, puis, rayonnant, me tendit une sorte de toupie et me dit: {laquo}Monsieur Besse, vous qui calculez les formes avec vos grimoires, que pensez-vous de ceci?)) Je le regardai interloque. Il poursuivit: {laquo}Regardez! Si vous prenez ce collier de laine et si vous le maintenez fermement avec un doigt place n'importe ou sur la toupie, eh bien! la toupie passera toujours juste en son interieur, sans laisser le moindre espace.)) Je rentrai chez moi, fort etonne, car sa toupie etait loin d'etre une boule. Je me mis alors au travail ...
Author |
: John Oprea |
Publisher |
: MAA |
Total Pages |
: 508 |
Release |
: 2007-09-06 |
ISBN-10 |
: 0883857480 |
ISBN-13 |
: 9780883857489 |
Rating |
: 4/5 (80 Downloads) |
Synopsis Differential Geometry and Its Applications by : John Oprea
This book studies the differential geometry of surfaces and its relevance to engineering and the sciences.
Author |
: Edward S. Popko |
Publisher |
: CRC Press |
Total Pages |
: 525 |
Release |
: 2012-07-30 |
ISBN-10 |
: 9781466504301 |
ISBN-13 |
: 1466504307 |
Rating |
: 4/5 (01 Downloads) |
Synopsis Divided Spheres by : Edward S. Popko
This well-illustrated book-in color throughout-presents a thorough introduction to the mathematics of Buckminster Fuller's invention of the geodesic dome, which paved the way for a flood of practical applications as diverse as weather forecasting and fish farms. The author explains the principles of spherical design and the three main categories of
Author |
: Wilhelm P.A. Klingenberg |
Publisher |
: Walter de Gruyter |
Total Pages |
: 421 |
Release |
: 2011-05-03 |
ISBN-10 |
: 9783110905120 |
ISBN-13 |
: 3110905124 |
Rating |
: 4/5 (20 Downloads) |
Synopsis Riemannian Geometry by : Wilhelm P.A. Klingenberg
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob. Titles in planning include Mark M. Meerschaert, Alla Sikorskii, and Mohsen Zayernouri, Stochastic Models for Fractional Calculus, second edition (2018) Flavia Smarazzo and Alberto Tesei, Measure Theory: Radon Measures, Young Measures and Applications to Parabolic Problems (2019) Elena Cordero and Luigi Rodino, Time-Frequency Analysis of Operators (2019) Kezheng Li, Group Schemes and Their Actions (2019; together with Tsinghua University Press) Kai Liu, Ilpo Laine, and Lianzhong Yang, Complex Differential-Difference Equations (2021) Rajendra Vasant Gurjar, Kayo Masuda, and Masayoshi Miyanishi, Affine Space Fibrations (2022)
Author |
: Joel W. Robbin |
Publisher |
: Springer Nature |
Total Pages |
: 426 |
Release |
: 2022-01-12 |
ISBN-10 |
: 9783662643402 |
ISBN-13 |
: 3662643405 |
Rating |
: 4/5 (02 Downloads) |
Synopsis Introduction to Differential Geometry by : Joel W. Robbin
This textbook is suitable for a one semester lecture course on differential geometry for students of mathematics or STEM disciplines with a working knowledge of analysis, linear algebra, complex analysis, and point set topology. The book treats the subject both from an extrinsic and an intrinsic view point. The first chapters give a historical overview of the field and contain an introduction to basic concepts such as manifolds and smooth maps, vector fields and flows, and Lie groups, leading up to the theorem of Frobenius. Subsequent chapters deal with the Levi-Civita connection, geodesics, the Riemann curvature tensor, a proof of the Cartan-Ambrose-Hicks theorem, as well as applications to flat spaces, symmetric spaces, and constant curvature manifolds. Also included are sections about manifolds with nonpositive sectional curvature, the Ricci tensor, the scalar curvature, and the Weyl tensor. An additional chapter goes beyond the scope of a one semester lecture course and deals with subjects such as conjugate points and the Morse index, the injectivity radius, the group of isometries and the Myers-Steenrod theorem, and Donaldson's differential geometric approach to Lie algebra theory.