Topological, Differential and Conformal Geometry of Surfaces

Topological, Differential and Conformal Geometry of Surfaces

Author: Norbert A'Campo

Publisher: Springer Nature

Published: 2021-10-27

Total Pages: 282

ISBN-13: 3030890325

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This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes’ Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss–Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow’s Theorem on compact holomorphic submanifolds in complex projective spaces. Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.


Book Synopsis Topological, Differential and Conformal Geometry of Surfaces by : Norbert A'Campo

Download or read book Topological, Differential and Conformal Geometry of Surfaces written by Norbert A'Campo and published by Springer Nature. This book was released on 2021-10-27 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes’ Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss–Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow’s Theorem on compact holomorphic submanifolds in complex projective spaces. Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.

Topological, Differential and Conformal Geometry of Surfaces

Topological, Differential and Conformal Geometry of Surfaces

Author: Norbert A'Campo

Publisher:

Published: 2021

Total Pages: 0

ISBN-13: 9783030890339

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This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes' Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss-Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow's Theorem on compact holomorphic submanifolds in complex projective spaces. Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.


Book Synopsis Topological, Differential and Conformal Geometry of Surfaces by : Norbert A'Campo

Download or read book Topological, Differential and Conformal Geometry of Surfaces written by Norbert A'Campo and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes' Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss-Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow's Theorem on compact holomorphic submanifolds in complex projective spaces. Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.

Geometry and Topology of Manifolds: Surfaces and Beyond

Geometry and Topology of Manifolds: Surfaces and Beyond

Author: Vicente Muñoz

Publisher: American Mathematical Soc.

Published: 2020-10-21

Total Pages: 408

ISBN-13: 1470461323

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This book represents a novel approach to differential topology. Its main focus is to give a comprehensive introduction to the classification of manifolds, with special attention paid to the case of surfaces, for which the book provides a complete classification from many points of view: topological, smooth, constant curvature, complex, and conformal. Each chapter briefly revisits basic results usually known to graduate students from an alternative perspective, focusing on surfaces. We provide full proofs of some remarkable results that sometimes are missed in basic courses (e.g., the construction of triangulations on surfaces, the classification of surfaces, the Gauss-Bonnet theorem, the degree-genus formula for complex plane curves, the existence of constant curvature metrics on conformal surfaces), and we give hints to questions about higher dimensional manifolds. Many examples and remarks are scattered through the book. Each chapter ends with an exhaustive collection of problems and a list of topics for further study. The book is primarily addressed to graduate students who did take standard introductory courses on algebraic topology, differential and Riemannian geometry, or algebraic geometry, but have not seen their deep interconnections, which permeate a modern approach to geometry and topology of manifolds.


Book Synopsis Geometry and Topology of Manifolds: Surfaces and Beyond by : Vicente Muñoz

Download or read book Geometry and Topology of Manifolds: Surfaces and Beyond written by Vicente Muñoz and published by American Mathematical Soc.. This book was released on 2020-10-21 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book represents a novel approach to differential topology. Its main focus is to give a comprehensive introduction to the classification of manifolds, with special attention paid to the case of surfaces, for which the book provides a complete classification from many points of view: topological, smooth, constant curvature, complex, and conformal. Each chapter briefly revisits basic results usually known to graduate students from an alternative perspective, focusing on surfaces. We provide full proofs of some remarkable results that sometimes are missed in basic courses (e.g., the construction of triangulations on surfaces, the classification of surfaces, the Gauss-Bonnet theorem, the degree-genus formula for complex plane curves, the existence of constant curvature metrics on conformal surfaces), and we give hints to questions about higher dimensional manifolds. Many examples and remarks are scattered through the book. Each chapter ends with an exhaustive collection of problems and a list of topics for further study. The book is primarily addressed to graduate students who did take standard introductory courses on algebraic topology, differential and Riemannian geometry, or algebraic geometry, but have not seen their deep interconnections, which permeate a modern approach to geometry and topology of manifolds.

Classical and Discrete Differential Geometry

Classical and Discrete Differential Geometry

Author: David Xianfeng Gu

Publisher: CRC Press

Published: 2023-01-31

Total Pages: 690

ISBN-13: 1000804461

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This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks. With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation. The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.


Book Synopsis Classical and Discrete Differential Geometry by : David Xianfeng Gu

Download or read book Classical and Discrete Differential Geometry written by David Xianfeng Gu and published by CRC Press. This book was released on 2023-01-31 with total page 690 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks. With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation. The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.

Computational Conformal Geometry

Computational Conformal Geometry

Author: Xianfeng David Gu

Publisher:

Published: 2008

Total Pages: 324

ISBN-13:

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Book Synopsis Computational Conformal Geometry by : Xianfeng David Gu

Download or read book Computational Conformal Geometry written by Xianfeng David Gu and published by . This book was released on 2008 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Differential Geometry and Integrable Systems

Differential Geometry and Integrable Systems

Author: Martin A. Guest

Publisher: American Mathematical Soc.

Published: 2002

Total Pages: 370

ISBN-13: 0821829386

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Ideas and techniques from the theory of integrable systems are playing an increasingly important role in geometry. Thanks to the development of tools from Lie theory, algebraic geometry, symplectic geometry, and topology, classical problems are investigated more systematically. New problems are also arising in mathematical physics. A major international conference was held at the University of Tokyo in July 2000. It brought together scientists in all of the areas influenced byintegrable systems. This book is the first of three collections of expository and research articles. This volume focuses on differential geometry. It is remarkable that many classical objects in surface theory and submanifold theory are described as integrable systems. Having such a description generallyreveals previously unnoticed symmetries and can lead to surprisingly explicit solutions. Surfaces of constant curvature in Euclidean space, harmonic maps from surfaces to symmetric spaces, and analogous structures on higher-dimensional manifolds are some of the examples that have broadened the horizons of differential geometry, bringing a rich supply of concrete examples into the theory of integrable systems. Many of the articles in this volume are written by prominent researchers and willserve as introductions to the topics. It is intended for graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics. The second volume from this conference also available from the AMS is Integrable Systems,Topology, and Physics, Volume 309 CONM/309in the Contemporary Mathematics series. The forthcoming third volume will be published by the Mathematical Society of Japan and will be available outside of Japan from the AMS in the Advanced Studies in Pure Mathematics series.


Book Synopsis Differential Geometry and Integrable Systems by : Martin A. Guest

Download or read book Differential Geometry and Integrable Systems written by Martin A. Guest and published by American Mathematical Soc.. This book was released on 2002 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ideas and techniques from the theory of integrable systems are playing an increasingly important role in geometry. Thanks to the development of tools from Lie theory, algebraic geometry, symplectic geometry, and topology, classical problems are investigated more systematically. New problems are also arising in mathematical physics. A major international conference was held at the University of Tokyo in July 2000. It brought together scientists in all of the areas influenced byintegrable systems. This book is the first of three collections of expository and research articles. This volume focuses on differential geometry. It is remarkable that many classical objects in surface theory and submanifold theory are described as integrable systems. Having such a description generallyreveals previously unnoticed symmetries and can lead to surprisingly explicit solutions. Surfaces of constant curvature in Euclidean space, harmonic maps from surfaces to symmetric spaces, and analogous structures on higher-dimensional manifolds are some of the examples that have broadened the horizons of differential geometry, bringing a rich supply of concrete examples into the theory of integrable systems. Many of the articles in this volume are written by prominent researchers and willserve as introductions to the topics. It is intended for graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics. The second volume from this conference also available from the AMS is Integrable Systems,Topology, and Physics, Volume 309 CONM/309in the Contemporary Mathematics series. The forthcoming third volume will be published by the Mathematical Society of Japan and will be available outside of Japan from the AMS in the Advanced Studies in Pure Mathematics series.

Conformal Geometry of Surfaces in S4 and Quaternions

Conformal Geometry of Surfaces in S4 and Quaternions

Author: Francis E. Burstall

Publisher: Springer

Published: 2004-10-20

Total Pages: 96

ISBN-13: 3540453016

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The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bäcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.


Book Synopsis Conformal Geometry of Surfaces in S4 and Quaternions by : Francis E. Burstall

Download or read book Conformal Geometry of Surfaces in S4 and Quaternions written by Francis E. Burstall and published by Springer. This book was released on 2004-10-20 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bäcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.

Geometry of Surfaces

Geometry of Surfaces

Author: John Stillwell

Publisher:

Published: 1992

Total Pages: 238

ISBN-13:

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"Geometry of Surfaces explores the interplay between geometry and topology in the simplest nontrivial case : the surfaces of constant curvature. As such, it provides a concise introduction to modern geometry for a wide audience. Requiring only a little prior knowledge of undergraduate mathematics, the book begins by discussing the three simplest surfaces : the Euclidean plane (zero curvature), the sphere (positive curvature), and the hyperbolic plane (negative curvature). Using the efficient machinery of isometry grouops, the author extends the discussion to all surfaces of constant curvature, which are typically obtained from the simplest ones by suitable isometries. The book then turns to the classification of the finitely many Euclidean and spherical surfaces and to a study of some remarkable hyperbolic surfaces. The general problem of classification is then considered from a topological and group-theoretic viewpoint. Because the theory of surfaces of constant curvature is intimately connected with the rest of modern mathematics, this book is an ideal starting point for students learning geometry, providing the simplest possible introduction to curvature, group actions, and covering spaces. The concepts developed here are, historically, the source of many concepts of complex analysis, differential geometry, topology, and combinatorial group theory, as well as such hot topics as fractal geometry and string theory. The prerequisites are modest, including only a little linear algebra, calculus, basic group theory, and basic topology. The formal coverage is extended by exercises and informal discussions throughout the text."--taken from back cover.


Book Synopsis Geometry of Surfaces by : John Stillwell

Download or read book Geometry of Surfaces written by John Stillwell and published by . This book was released on 1992 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Geometry of Surfaces explores the interplay between geometry and topology in the simplest nontrivial case : the surfaces of constant curvature. As such, it provides a concise introduction to modern geometry for a wide audience. Requiring only a little prior knowledge of undergraduate mathematics, the book begins by discussing the three simplest surfaces : the Euclidean plane (zero curvature), the sphere (positive curvature), and the hyperbolic plane (negative curvature). Using the efficient machinery of isometry grouops, the author extends the discussion to all surfaces of constant curvature, which are typically obtained from the simplest ones by suitable isometries. The book then turns to the classification of the finitely many Euclidean and spherical surfaces and to a study of some remarkable hyperbolic surfaces. The general problem of classification is then considered from a topological and group-theoretic viewpoint. Because the theory of surfaces of constant curvature is intimately connected with the rest of modern mathematics, this book is an ideal starting point for students learning geometry, providing the simplest possible introduction to curvature, group actions, and covering spaces. The concepts developed here are, historically, the source of many concepts of complex analysis, differential geometry, topology, and combinatorial group theory, as well as such hot topics as fractal geometry and string theory. The prerequisites are modest, including only a little linear algebra, calculus, basic group theory, and basic topology. The formal coverage is extended by exercises and informal discussions throughout the text."--taken from back cover.

Introduction to Differential Geometry

Introduction to Differential Geometry

Author: Joel W. Robbin

Publisher: Springer Nature

Published: 2022-01-12

Total Pages: 426

ISBN-13: 3662643405

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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.


Book Synopsis Introduction to Differential Geometry by : Joel W. Robbin

Download or read book Introduction to Differential Geometry written by Joel W. Robbin and published by Springer Nature. This book was released on 2022-01-12 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Mostly Surfaces

Mostly Surfaces

Author: Richard Evan Schwartz

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 330

ISBN-13: 0821853686

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The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.


Book Synopsis Mostly Surfaces by : Richard Evan Schwartz

Download or read book Mostly Surfaces written by Richard Evan Schwartz and published by American Mathematical Soc.. This book was released on 2011 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.