Download or read book Lectures on Closed Geodesics written by W. Klingenberg and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: The question of existence of c10sed geodesics on a Riemannian manifold and the properties of the corresponding periodic orbits in the geodesic flow has been the object of intensive investigations since the beginning of global differential geo metry during the last century. The simplest case occurs for c10sed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic c10sed curve can be deformed into a c10sed curve having minimallength in its free homotopy c1ass. This minimal curve is, up to the parameterization, uniquely determined and represents a c10sed geodesic. The question of existence of a c10sed geodesic on a simply connected c10sed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem ofthe existence of periodic orbits in the restricted three body problem. Poincare [l.c.] outlined a proof that on an analytic convex surface which does not differ too much from the standard sphere there always exists at least one c10sed geodesic of elliptic type, i. e., the corres ponding periodic orbit in the geodesic flow is infinitesimally stable.

Download or read book The Geometry of Geodesics written by Herbert Busemann and published by Courier Corporation. This book was released on 2012-07-12 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Geodesics Retracts and the Norm Preserving Extension Property in the Symmetrized Bidisc written by Jim Agler and published by American Mathematical Soc.. This book was released on 2019-04-10 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: A set V in a domain U in Cn has the norm-preserving extension property if every bounded holomorphic function on V has a holomorphic extension to U with the same supremum norm. We prove that an algebraic subset of the symmetrized bidisc

Download or read book Manifolds all of whose Geodesics are Closed written by A. L. Besse and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 271 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 ...

Download or read book Geodesics and Ends in Certain Surfaces without Conjugate Points written by Patrick Eberlein and published by American Mathematical Soc.. This book was released on 1978 with total page 117 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we study the geodesics and ends of compact surfaces satisfying the "uniform visibility" axiom. We are primarily though not exclusively interested in finitely connected surfaces, which topologically are compact Riemann surfaces with a finite number of punctures.

Download or read book Behavior of Distant Maximal Geodesics in Finitely Connected Complete 2 dimensional Riemannian Manifolds written by Takashi Shioya and published by American Mathematical Soc.. This book was released on 1994 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph studies the topological shapes of geodesics outside a large compact set in a finitely connected, complete, and noncompact surface admitting total curvature. When the surface is homeomorphic to a plane, all such geodesics behave like those of a flat cone. In particular, the rotation numbers of the geodesics are controlled by the total curvature. Accessible to beginners in differential geometry, but also of interest to specialists, this monograph features many illustrations that enhance understanding of the main ideas.

Download or read book Spheroidal Geodesics Reference Systems Local Geometry written by Paul D. Thomas and published by . This book was released on 1970 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Riemannian Manifolds and Homogeneous Geodesics written by Valerii Berestovskii and published by Springer Nature. This book was released on 2020-11-05 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to Killing vector fields and the one-parameter isometry groups of Riemannian manifolds generated by them. It also provides a detailed introduction to homogeneous geodesics, that is, geodesics that are integral curves of Killing vector fields, presenting both classical and modern results, some very recent, many of which are due to the authors. The main focus is on the class of Riemannian manifolds with homogeneous geodesics and on some of its important subclasses. To keep the exposition self-contained the book also includes useful general results not only on geodesic orbit manifolds, but also on smooth and Riemannian manifolds, Lie groups and Lie algebras, homogeneous Riemannian manifolds, and compact homogeneous Riemannian spaces. The intended audience is graduate students and researchers whose work involves differential geometry and transformation groups.

Download or read book Geodesics on Surfaces of Genus Zero with Knobs written by Donald Everett Richmond and published by . This book was released on 1926 with total page 82 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Geodesic Math and How to Use It written by Hugh Kenner and published by Univ of California Press. This book was released on 2003-10-20 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 1976 literary critic Hugh Kenner published this fully-illustrated practical manual for the construction of geodesic domes, which had been invented 25 years previously by R. Buckminster Fuller. Now returned to print for the first time since 1990.

Download or read book Mirzakhani s Curve Counting and Geodesic Currents written by Viveka Erlandsson and published by Springer Nature. This book was released on 2022-09-20 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents an approachable proof of Mirzakhani’s curve counting theorem, both for simple and non-simple curves. Designed to welcome readers to the area, the presentation builds intuition with elementary examples before progressing to rigorous proofs. This approach illuminates new and established results alike, and produces versatile tools for studying the geometry of hyperbolic surfaces, Teichmüller theory, and mapping class groups. Beginning with the preliminaries of curves and arcs on surfaces, the authors go on to present the theory of geodesic currents in detail. Highlights include a treatment of cusped surfaces and surfaces with boundary, along with a comprehensive discussion of the action of the mapping class group on the space of geodesic currents. A user-friendly account of train tracks follows, providing the foundation for radallas, an immersed variation. From here, the authors apply these tools to great effect, offering simplified proofs of existing results and a new, more general proof of Mirzakhani’s curve counting theorem. Further applications include counting square-tiled surfaces and mapping class group orbits, and investigating random geometric structures. Mirzakhani’s Curve Counting and Geodesic Currents introduces readers to powerful counting techniques for the study of surfaces. Ideal for graduate students and researchers new to the area, the pedagogical approach, conversational style, and illuminating illustrations bring this exciting field to life. Exercises offer opportunities to engage with the material throughout. Basic familiarity with 2-dimensional topology and hyperbolic geometry, measured laminations, and the mapping class group is assumed.

Download or read book Curves and Surfaces written by M. Abate and published by Springer Science & Business Media. This book was released on 2012-06-11 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book The Variational Theory of Geodesics written by M. M. Postnikov and published by Courier Dover Publications. This book was released on 2019-11-13 with total page 211 pages. Available in PDF, EPUB and Kindle. Book excerpt: Compact, self-contained text by a noted theorist presents essentials of modern differential geometry and basic tools for study of Morse theory. Advanced treatment emphasizes Morse theory's analytical rather than topological aspects. 1967 edition.

Download or read book Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems written by Vesselin M. Petkov and published by John Wiley & Sons. This book was released on 2017-01-30 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a new edition of a title originally published in1992. No other book has been published that treats inverse spectral and inverse scattering results by using the so called Poisson summation formula and the related study of singularities. This book presents these in a closed and comprehensive form, and the exposition is based on a combination of different tools and results from dynamical systems, microlocal analysis, spectral and scattering theory. The content of the first edition is still relevant, however the new edition will include several new results established after 1992; new text will comprise about a third of the content of the new edition. The main chapters in the first edition in combination with the new chapters will provide a better and more comprehensive presentation of importance for the applications inverse results. These results are obtained by modern mathematical techniques which will be presented together in order to give the readers the opportunity to completely understand them. Moreover, some basic generic properties established by the authors after the publication of the first edition establishing the wide range of applicability of the Poison relation will be presented for first time in the new edition of the book.

Download or read book Empirical Measures Geodesic Lengths and a Variational Formula in First Passage Percolation written by Erik Bates and published by American Mathematical Society. This book was released on 2024-02-01 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Download or read book Solid Geometry written by Percival Frost and published by . This book was released on 1886 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Lectures on Closed Geodesics written by W Klingenberg and published by . This book was released on 1978-01-01 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: