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Differential Geometry and Complex Analysis / Isaac Chavel ; SpringerLink (Online service) ; Hershel M. Farkas / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (1985)

Titre : Differential Geometry and Complex Analysis : A Volume Dedicated to the Memory of Harry Ernest Rauch Type de document : document électronique Auteurs : Isaac Chavel ; SpringerLink (Online service) ; Hershel M. Farkas Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 1985 Importance : XIV, 224 p Présentation : online resource ISBN/ISSN/EAN : 978-3-642-69828-6 Langues : Anglais ( eng)Tags : Mathematics Mathematical analysis Analysis (Mathematics) Differential geometry Differential Geometry Analysis Résumé : This volume is dedicated to the memory of Harry Ernest Rauch, who died suddenly on June 18, 1979. In organizing the volume we solicited: (i) articles summarizing Rauch's own work in differential geometry, complex analysis and theta functions (ii) articles which would give the reader an idea of the depth and breadth of Rauch's researches, interests, and influence, in the fields he investigated, and (iii) articles of high scientific quality which would be of general interest. In each of the areas to which Rauch made significant contribution - pinching theorems, teichmiiller theory, and theta functions as they apply to Riemann surfaces - there has been substantial progress. Our hope is that the volume conveys the originality of Rauch's own work, the continuing vitality of the fields he influenced, and the enduring respect for, and tribute to, him and his accom plishments in the mathematical community. Finally, it is a pleasure to thank the Department of Mathematics, of the Grad uate School of the City University of New York, for their logistical support, James Rauch who helped us with the biography, and Springer-Verlag for all their efforts in producing this volume. Isaac Chavel . Hershel M. Farkas Contents Harry Ernest Rauch - Biographical Sketch. . . . . . . . VII Bibliography of the Publications of H. E. Rauch. . . . . . X Ph.D. Theses Written under the Supervision of H. E. Rauch. XIII H. E. Rauch, Geometre Differentiel (by M. Berger) . . . . . . . Note de contenu : H. E. Rauch, Géomètre Différentiel -- H. E. Rauch, Function Theorist -- H. E. Rauch, Theta Function Practitioner -- Some loci in Teichmüller Space for Genus Six Defined by Vanishing Thetanulls -- Möbius transformations and Clifford Numbers -- Polynomial Approximation in Quasidisks -- An Inequality for Riemann Surfaces -- Extremal Kähler Metrics -- On the Characteristic Numbers of Complete Manifolds of Bounded Curvature and Finite Volume -- Deformation of Surfaces Preserving Principal Curvatures -- One-dimensional Metric Foliations in Constant Curvature Spaces -- The Existence of Three Short Closed Geodesics -- On Lifting Kleinian Groups to SL (2, ?) -- On the Ends of Trajectories -- An Integrability Condition for Simple Lie Groups -- Uniqueness in the Cauchy Problem for a Degenerate Elliptic Second Order Equation -- On the Structure of Complete Manifolds with Positive Scalar Curvature Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=128927 Differential Geometry and Complex Analysis : A Volume Dedicated to the Memory of Harry Ernest Rauch [document électronique] / Isaac Chavel ; SpringerLink (Online service) ; Hershel M. Farkas . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 1985 . - XIV, 224 p : online resource.ISBN: 978-3-642-69828-6

Langues : Anglais (eng)

Tags : Mathematics Mathematical analysis Analysis (Mathematics) Differential geometry Differential Geometry Analysis Résumé : This volume is dedicated to the memory of Harry Ernest Rauch, who died suddenly on June 18, 1979. In organizing the volume we solicited: (i) articles summarizing Rauch's own work in differential geometry, complex analysis and theta functions (ii) articles which would give the reader an idea of the depth and breadth of Rauch's researches, interests, and influence, in the fields he investigated, and (iii) articles of high scientific quality which would be of general interest. In each of the areas to which Rauch made significant contribution - pinching theorems, teichmiiller theory, and theta functions as they apply to Riemann surfaces - there has been substantial progress. Our hope is that the volume conveys the originality of Rauch's own work, the continuing vitality of the fields he influenced, and the enduring respect for, and tribute to, him and his accom plishments in the mathematical community. Finally, it is a pleasure to thank the Department of Mathematics, of the Grad uate School of the City University of New York, for their logistical support, James Rauch who helped us with the biography, and Springer-Verlag for all their efforts in producing this volume. Isaac Chavel . Hershel M. Farkas Contents Harry Ernest Rauch - Biographical Sketch. . . . . . . . VII Bibliography of the Publications of H. E. Rauch. . . . . . X Ph.D. Theses Written under the Supervision of H. E. Rauch. XIII H. E. Rauch, Geometre Differentiel (by M. Berger) . . . . . . . Note de contenu : H. E. Rauch, Géomètre Différentiel -- H. E. Rauch, Function Theorist -- H. E. Rauch, Theta Function Practitioner -- Some loci in Teichmüller Space for Genus Six Defined by Vanishing Thetanulls -- Möbius transformations and Clifford Numbers -- Polynomial Approximation in Quasidisks -- An Inequality for Riemann Surfaces -- Extremal Kähler Metrics -- On the Characteristic Numbers of Complete Manifolds of Bounded Curvature and Finite Volume -- Deformation of Surfaces Preserving Principal Curvatures -- One-dimensional Metric Foliations in Constant Curvature Spaces -- The Existence of Three Short Closed Geodesics -- On Lifting Kleinian Groups to SL (2, ?) -- On the Ends of Trajectories -- An Integrability Condition for Simple Lie Groups -- Uniqueness in the Cauchy Problem for a Degenerate Elliptic Second Order Equation -- On the Structure of Complete Manifolds with Positive Scalar Curvature Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=128927 Differential Geometry: Manifolds, Curves, and Surfaces / Marcel Berger / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (1988)

Titre : Differential Geometry: Manifolds, Curves, and Surfaces Type de document : document électronique Auteurs : Marcel Berger (1927-....) ; Bernard Gostiaux ; SpringerLink (Online service) Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 1988 Collection : Graduate Texts in Mathematics, ISSN 0072-5285 num. 115 Importance : XII, 476 p Présentation : online resource ISBN/ISSN/EAN : 978-1-4612-1033-7 Langues : Anglais ( eng)Tags : Mathematics Differential geometry Differential Geometry Résumé : This book consists of two parts, different in form but similar in spirit. The first, which comprises chapters 0 through 9, is a revised and somewhat enlarged version of the 1972 book Geometrie Differentielle. The second part, chapters 10 and 11, is an attempt to remedy the notorious absence in the original book of any treatment of surfaces in three-space, an omission all the more unforgivable in that surfaces are some of the most common geometrical objects, not only in mathematics but in many branches of physics. Geometrie Differentielle was based on a course I taught in Paris in 1969- 70 and again in 1970-71. In designing this course I was decisively influ enced by a conversation with Serge Lang, and I let myself be guided by three general ideas. First, to avoid making the statement and proof of Stokes' formula the climax of the course and running out of time before any of its applications could be discussed. Second, to illustrate each new notion with non-trivial examples, as soon as possible after its introduc tion. And finally, to familiarize geometry-oriented students with analysis and analysis-oriented students with geometry, at least in what concerns manifolds Note de contenu : 0. Background -- 0.0 Notation and Recap -- 0.1 Exterior Algebra -- 0.2 Differential Calculus -- 0.3 Differential Forms -- 0.4 Integration -- 0.5 Exercises -- 1. Differential Equations -- 1.1 Generalities -- 1.2 Equations with Constant Coefficients. Existence of Local Solutions -- 1.3 Global Uniqueness and Global Flows -- 1.4 Time- and Parameter-Dependent Vector Fields -- 1.5 Time-Dependent Vector Fields: Uniqueness And Global Flow -- 1.6 Cultural Digression -- 2. Differentiable Manifolds -- 2.1 Submanifolds of Rn -- 2.2 Abstract Manifolds -- 2.3 Differentiable Maps -- 2.4 Covering Maps and Quotients -- 2.5 Tangent Spaces -- 2.6 Submanifolds, Immersions, Submersions and Embeddings -- 2.7 Normal Bundles and Tubular Neighborhoods -- 2.8 Exercises -- 3. Partitions of Unity, Densities and Curves -- 3.1 Embeddings of Compact Manifolds -- 3.2 Partitions of Unity -- 3.3 Densities -- 3.4 Classification of Connected One-Dimensional Manifolds -- 3.5 Vector Fields and Differential Equations on Manifolds -- 3.6 Exercises -- 4. Critical Points -- 4.1 Definitions and Examples -- 4.2 Non-Degenerate Critical Points -- 4.3 Sard?s Theorem -- 4.4 Exercises -- 5. Differential Forms -- 5.1 The Bundle ?rT*X -- 5.2 Differential Forms on a Manifold -- 5.3 Volume Forms and Orientation -- 5.4 De Rham Groups -- 5.5 Lie Derivatives -- 5.6 Star-shaped Sets and Poincaré?s Lemma -- 5.7 De Rham Groups of Spheres and Projective Spaces -- 5.8 De Rham Groups of Tori -- 5.9 Exercises -- 6. Integration of Differential Forms -- 6.1 Integrating Forms of Maximal Degree -- 6.2 Stokes? Theorem -- 6.3 First Applications of Stokes? Theorem -- 6.4 Canonical Volume Forms -- 6.5 Volume of a Submanifold of Euclidean Space -- 6.6 Canonical Density on a Submanifold of Euclidean Space -- 6.7 Volume of Tubes I -- 6.8 Volume of Tubes II -- 6.9 Volume of Tubes III -- 6.10 Exercises -- 7. Degree Theory -- 7.1 Preliminary Lemmas -- 7.2 Calculation of Rd(X) -- 7.3 The Degree of a Map -- 7.4 Invariance under Homotopy. Applications -- 7.5 Volume of Tubes and the Gauss-Bonnet Formula -- 7.6 Self-Maps of the Circle -- 7.7 Index of Vector Fields on Abstract Manifolds -- 7.8 Exercises -- 8. Curves: The Local Theory -- 8.0 Introduction -- 8.1 Definitions -- 8.2 Affine Invariants: Tangent, Osculating Plan, Concavity -- 8.3 Arclength -- 8.4 Curvature -- 8.5 Signed Curvature of a Plane Curve -- 8.6 Torsion of Three-Dimensional Curves -- 8.7 Exercises -- 9. Plane Curves: The Global Theory -- 9.1 Definitions -- 9.2 Jordan?s Theorem -- 9.3 The Isoperimetric Inequality -- 9.4 The Turning Number -- 9.5 The Turning Tangent Theorem -- 9.6 Global Convexity -- 9.7 The Four-Vertex Theorem -- 9.8 The Fabricius-Bjerre-Halpern Formula -- 9.9 Exercises -- 10. A Guide to the Local Theory of Surfaces in R3 -- 10.1 Definitions -- 10.2 Examples -- 10.3 The Two Fundamental Forms -- 10.4 What the First Fundamental Form Is Good For -- 10.5 Gaussian Curvature -- 10.6 What the Second Fundamental Form Is Good For -- 10.7 Links Between the two Fundamental Forms -- 10.8 A Word about Hypersurfaces in Rn+1 -- 11. A Guide to the Global Theory of Surfaces -- 11.1 Shortest Paths -- 11.2 Surfaces of Constant Curvature -- 11.3 The Two Variation Formulas -- 11.4 Shortest Paths and the Injectivity Radius -- 11.5 Manifolds with Curvature Bounded Below -- 11.6 Manifolds with Curvature Bounded Above -- 11.7 The Gauss-Bonnet and Hopf Formulas -- 11.8 The Isoperimetric Inequality on Surfaces -- 11.9 Closed Geodesics and Isosystolic Inequalities -- 11.10 Surfaces AU of Whose Geodesics Are Closed -- 11.11 Transition: Embedding and Immersion Problems -- 11.12 Surfaces of Zero Curvature -- 11.13 Surfaces of Non-Negative Curvature -- 11.14 Uniqueness and Rigidity Results -- 11.15 Surfaces of Negative Curvature -- 11.16 Minimal Surfaces -- 11.17 Surfaces of Constant Mean Curvature, or Soap Bubbles -- 11.18 Weingarten Surfaces -- 11.19 Envelopes of Families of Planes -- 11.20 Isoperimetric Inequalities for Surfaces -- 11.21 A Pot-pourri of Characteristic Properties -- Index of Symbols and Notations Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=122273 Differential Geometry: Manifolds, Curves, and Surfaces [document électronique] / Marcel Berger (1927-....) ; Bernard Gostiaux ; SpringerLink (Online service) . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 1988 . - XII, 476 p : online resource. - (Graduate Texts in Mathematics, ISSN 0072-5285; 115) .ISBN: 978-1-4612-1033-7

Langues : Anglais (eng)

Tags : Mathematics Differential geometry Differential Geometry Résumé : This book consists of two parts, different in form but similar in spirit. The first, which comprises chapters 0 through 9, is a revised and somewhat enlarged version of the 1972 book Geometrie Differentielle. The second part, chapters 10 and 11, is an attempt to remedy the notorious absence in the original book of any treatment of surfaces in three-space, an omission all the more unforgivable in that surfaces are some of the most common geometrical objects, not only in mathematics but in many branches of physics. Geometrie Differentielle was based on a course I taught in Paris in 1969- 70 and again in 1970-71. In designing this course I was decisively influ enced by a conversation with Serge Lang, and I let myself be guided by three general ideas. First, to avoid making the statement and proof of Stokes' formula the climax of the course and running out of time before any of its applications could be discussed. Second, to illustrate each new notion with non-trivial examples, as soon as possible after its introduc tion. And finally, to familiarize geometry-oriented students with analysis and analysis-oriented students with geometry, at least in what concerns manifolds Note de contenu : 0. Background -- 0.0 Notation and Recap -- 0.1 Exterior Algebra -- 0.2 Differential Calculus -- 0.3 Differential Forms -- 0.4 Integration -- 0.5 Exercises -- 1. Differential Equations -- 1.1 Generalities -- 1.2 Equations with Constant Coefficients. Existence of Local Solutions -- 1.3 Global Uniqueness and Global Flows -- 1.4 Time- and Parameter-Dependent Vector Fields -- 1.5 Time-Dependent Vector Fields: Uniqueness And Global Flow -- 1.6 Cultural Digression -- 2. Differentiable Manifolds -- 2.1 Submanifolds of Rn -- 2.2 Abstract Manifolds -- 2.3 Differentiable Maps -- 2.4 Covering Maps and Quotients -- 2.5 Tangent Spaces -- 2.6 Submanifolds, Immersions, Submersions and Embeddings -- 2.7 Normal Bundles and Tubular Neighborhoods -- 2.8 Exercises -- 3. Partitions of Unity, Densities and Curves -- 3.1 Embeddings of Compact Manifolds -- 3.2 Partitions of Unity -- 3.3 Densities -- 3.4 Classification of Connected One-Dimensional Manifolds -- 3.5 Vector Fields and Differential Equations on Manifolds -- 3.6 Exercises -- 4. Critical Points -- 4.1 Definitions and Examples -- 4.2 Non-Degenerate Critical Points -- 4.3 Sard?s Theorem -- 4.4 Exercises -- 5. Differential Forms -- 5.1 The Bundle ?rT*X -- 5.2 Differential Forms on a Manifold -- 5.3 Volume Forms and Orientation -- 5.4 De Rham Groups -- 5.5 Lie Derivatives -- 5.6 Star-shaped Sets and Poincaré?s Lemma -- 5.7 De Rham Groups of Spheres and Projective Spaces -- 5.8 De Rham Groups of Tori -- 5.9 Exercises -- 6. Integration of Differential Forms -- 6.1 Integrating Forms of Maximal Degree -- 6.2 Stokes? Theorem -- 6.3 First Applications of Stokes? Theorem -- 6.4 Canonical Volume Forms -- 6.5 Volume of a Submanifold of Euclidean Space -- 6.6 Canonical Density on a Submanifold of Euclidean Space -- 6.7 Volume of Tubes I -- 6.8 Volume of Tubes II -- 6.9 Volume of Tubes III -- 6.10 Exercises -- 7. Degree Theory -- 7.1 Preliminary Lemmas -- 7.2 Calculation of Rd(X) -- 7.3 The Degree of a Map -- 7.4 Invariance under Homotopy. Applications -- 7.5 Volume of Tubes and the Gauss-Bonnet Formula -- 7.6 Self-Maps of the Circle -- 7.7 Index of Vector Fields on Abstract Manifolds -- 7.8 Exercises -- 8. Curves: The Local Theory -- 8.0 Introduction -- 8.1 Definitions -- 8.2 Affine Invariants: Tangent, Osculating Plan, Concavity -- 8.3 Arclength -- 8.4 Curvature -- 8.5 Signed Curvature of a Plane Curve -- 8.6 Torsion of Three-Dimensional Curves -- 8.7 Exercises -- 9. Plane Curves: The Global Theory -- 9.1 Definitions -- 9.2 Jordan?s Theorem -- 9.3 The Isoperimetric Inequality -- 9.4 The Turning Number -- 9.5 The Turning Tangent Theorem -- 9.6 Global Convexity -- 9.7 The Four-Vertex Theorem -- 9.8 The Fabricius-Bjerre-Halpern Formula -- 9.9 Exercises -- 10. A Guide to the Local Theory of Surfaces in R3 -- 10.1 Definitions -- 10.2 Examples -- 10.3 The Two Fundamental Forms -- 10.4 What the First Fundamental Form Is Good For -- 10.5 Gaussian Curvature -- 10.6 What the Second Fundamental Form Is Good For -- 10.7 Links Between the two Fundamental Forms -- 10.8 A Word about Hypersurfaces in Rn+1 -- 11. A Guide to the Global Theory of Surfaces -- 11.1 Shortest Paths -- 11.2 Surfaces of Constant Curvature -- 11.3 The Two Variation Formulas -- 11.4 Shortest Paths and the Injectivity Radius -- 11.5 Manifolds with Curvature Bounded Below -- 11.6 Manifolds with Curvature Bounded Above -- 11.7 The Gauss-Bonnet and Hopf Formulas -- 11.8 The Isoperimetric Inequality on Surfaces -- 11.9 Closed Geodesics and Isosystolic Inequalities -- 11.10 Surfaces AU of Whose Geodesics Are Closed -- 11.11 Transition: Embedding and Immersion Problems -- 11.12 Surfaces of Zero Curvature -- 11.13 Surfaces of Non-Negative Curvature -- 11.14 Uniqueness and Rigidity Results -- 11.15 Surfaces of Negative Curvature -- 11.16 Minimal Surfaces -- 11.17 Surfaces of Constant Mean Curvature, or Soap Bubbles -- 11.18 Weingarten Surfaces -- 11.19 Envelopes of Families of Planes -- 11.20 Isoperimetric Inequalities for Surfaces -- 11.21 A Pot-pourri of Characteristic Properties -- Index of Symbols and Notations Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=122273 Differential Geometry of Curves and Surfaces / Tapp, Kristopher / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (2016)

Titre : Differential Geometry of Curves and Surfaces Type de document : document électronique Auteurs : Tapp, Kristopher ; SpringerLink (Online service) Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 2016 Collection : Undergraduate Texts in Mathematics, ISSN 0172-6056 Importance : VIII, 366 p. 186 illus. in color Présentation : online resource ISBN/ISSN/EAN : 978-3-319-39799-3 Langues : Anglais ( eng)Tags : Mathematics Differential geometry Differential Geometry Résumé : This is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging. Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships. Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut?s Theorem is presented as a conservation law for angular momentum. Green?s Theorem makes possible a drafting tool called a planimeter. Foucault?s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface. In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn?t work without general relativity, formalized through the language of differential geometry. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it. Note de contenu : Introduction -- Curves -- Additional topics in curves -- Surfaces -- The curvature of a surface -- Geodesics -- The Gauss?Bonnet theorem -- Appendix A: The topology of subsets of Rn -- Recommended excursions -- Index Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=123209 Differential Geometry of Curves and Surfaces [document électronique] / Tapp, Kristopher ; SpringerLink (Online service) . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 2016 . - VIII, 366 p. 186 illus. in color : online resource. - (Undergraduate Texts in Mathematics, ISSN 0172-6056) .ISBN: 978-3-319-39799-3

Langues : Anglais (eng)

Tags : Mathematics Differential geometry Differential Geometry Résumé : This is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging. Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships. Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut?s Theorem is presented as a conservation law for angular momentum. Green?s Theorem makes possible a drafting tool called a planimeter. Foucault?s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface. In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn?t work without general relativity, formalized through the language of differential geometry. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it. Note de contenu : Introduction -- Curves -- Additional topics in curves -- Surfaces -- The curvature of a surface -- Geodesics -- The Gauss?Bonnet theorem -- Appendix A: The topology of subsets of Rn -- Recommended excursions -- Index Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=123209 Differential Geometry of Foliations / Bruce L. Reinhart / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (1983)

Titre : Differential Geometry of Foliations : The Fundamental Integrability Problem Type de document : document électronique Auteurs : Bruce L. Reinhart ; SpringerLink (Online service) Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 1983 Collection : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematic [1983-....], ISSN 0071-1136 num. 99 Importance : X, 196 p Présentation : online resource ISBN/ISSN/EAN : 978-3-642-69015-0 Langues : Anglais ( eng)Tags : Mathematics Differential geometry Differential Geometry Résumé : Whoever you are! How can I but offer you divine leaves . . . ? Walt Whitman The object of study in modern differential geometry is a manifold with a differ ential structure, and usually some additional structure as well. Thus, one is given a topological space M and a family of homeomorphisms, called coordinate sys tems, between open subsets of the space and open subsets of a real vector space V. It is supposed that where two domains overlap, the images are related by a diffeomorphism, called a coordinate transformation, between open subsets of V. M has associated with it a tangent bundle, which is a vector bundle with fiber V and group the general linear group GL(V). The additional structures that occur include Riemannian metrics, connections, complex structures, foliations, and many more. Frequently there is associated to the structure a reduction of the group of the tangent bundle to some subgroup G of GL(V). It is particularly pleasant if one can choose the coordinate systems so that the Jacobian matrices of the coordinate transformations belong to G. A reduction to G is called a G-structure, which is called integrable (or flat) if the condition on the Jacobians is satisfied. The strength of the integrability hypothesis is well-illustrated by the case of the orthogonal group On. An On-structure is given by the choice of a Riemannian metric, and therefore exists on every smooth manifold Note de contenu : I. Differential Geometric Structures and Integrability -- 1. Pseudogroups and Groupoids -- 2. Foliations -- 3. The Integrability Problem -- 4. Vector Fields and Pfaffian Systems -- 5. Leaves and Holonomy -- 6. Examples of Foliations -- II. Prolongations, Connections, and Characteristic Classes -- 1. Truncated Polynomial Groups and Algebras -- 2. Prolongation of a Manifold -- 3. Higher Order Structures -- 4. Connections and Characteristic Classes -- 5. Foliations, Connections, and Secondary Classes -- III. Singular Foliations -- 1. The Classifying Space for a Topological Groupoid -- 2. Vector Fields and the Cohomology of Lie Algebras -- 3. Frobenius Structures -- IV. Metric and Measure Theoretic Properties of Foliations -- 1. Analytic Background -- 2. Measure, Volume, and Foliations -- 3. Foliations of a Riemannian Manifold -- 4. Riemannian Foliations -- 5. Foliations with a Few Derivatives -- Supplementary Bibliography -- Index of Terminology -- Index of Symbols Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=128923 Differential Geometry of Foliations : The Fundamental Integrability Problem [document électronique] / Bruce L. Reinhart ; SpringerLink (Online service) . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 1983 . - X, 196 p : online resource. - (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematic [1983-....], ISSN 0071-1136; 99) .ISBN: 978-3-642-69015-0

Langues : Anglais (eng)

Tags : Mathematics Differential geometry Differential Geometry Résumé : Whoever you are! How can I but offer you divine leaves . . . ? Walt Whitman The object of study in modern differential geometry is a manifold with a differ ential structure, and usually some additional structure as well. Thus, one is given a topological space M and a family of homeomorphisms, called coordinate sys tems, between open subsets of the space and open subsets of a real vector space V. It is supposed that where two domains overlap, the images are related by a diffeomorphism, called a coordinate transformation, between open subsets of V. M has associated with it a tangent bundle, which is a vector bundle with fiber V and group the general linear group GL(V). The additional structures that occur include Riemannian metrics, connections, complex structures, foliations, and many more. Frequently there is associated to the structure a reduction of the group of the tangent bundle to some subgroup G of GL(V). It is particularly pleasant if one can choose the coordinate systems so that the Jacobian matrices of the coordinate transformations belong to G. A reduction to G is called a G-structure, which is called integrable (or flat) if the condition on the Jacobians is satisfied. The strength of the integrability hypothesis is well-illustrated by the case of the orthogonal group On. An On-structure is given by the choice of a Riemannian metric, and therefore exists on every smooth manifold Note de contenu : I. Differential Geometric Structures and Integrability -- 1. Pseudogroups and Groupoids -- 2. Foliations -- 3. The Integrability Problem -- 4. Vector Fields and Pfaffian Systems -- 5. Leaves and Holonomy -- 6. Examples of Foliations -- II. Prolongations, Connections, and Characteristic Classes -- 1. Truncated Polynomial Groups and Algebras -- 2. Prolongation of a Manifold -- 3. Higher Order Structures -- 4. Connections and Characteristic Classes -- 5. Foliations, Connections, and Secondary Classes -- III. Singular Foliations -- 1. The Classifying Space for a Topological Groupoid -- 2. Vector Fields and the Cohomology of Lie Algebras -- 3. Frobenius Structures -- IV. Metric and Measure Theoretic Properties of Foliations -- 1. Analytic Background -- 2. Measure, Volume, and Foliations -- 3. Foliations of a Riemannian Manifold -- 4. Riemannian Foliations -- 5. Foliations with a Few Derivatives -- Supplementary Bibliography -- Index of Terminology -- Index of Symbols Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=128923 Differential Geometry of Frame Bundles / Luis A. Cordero / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (1989)

Titre : Differential Geometry of Frame Bundles Type de document : document électronique Auteurs : Luis A. Cordero ; Christopher Terence John Dodson ; SpringerLink (Online service) ; León, Manuel de Editeur : Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer Année de publication : 1989 Collection : Mathematics and Its Applications num. 47 Importance : X, 234 p Présentation : online resource ISBN/ISSN/EAN : 978-94-009-1265-6 Langues : Anglais ( eng)Tags : Mathematics Global analysis (Mathematics) Manifolds (Mathematics) Differential geometry Physics Differential Geometry Global Analysis and Analysis on Manifolds Mathematical and Computational Physics Résumé : It isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu!ik's The Chillese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics Note de contenu : 1 The Functor Jp1 -- 1.1 The Bundle Jp1M ? M -- 1.2 Jp1G for a Lie group G -- 1.3 Jp1V for a vector space V -- 1.4 The embedding jp -- 2 Prolongation of G-structures -- 2.1 Imbedding of Jn1FM into FFM -- 2.2 Prolongation of G-structures to FM -- 2.3 Integrability -- 2.4 Applications -- 3 Vector-valued differential forms -- 3.1 General Theory -- 3.2 Applications -- 4 Prolongation of linear connections -- 4.1 Forms with values in a Lie algebra -- 4.2 Prolongation of connections -- 4.3 Complete lift of linear connections -- 4.4 Connections adapted to G-structures -- 4.5 Geodesics of ?C -- 4.6 Complete lift of derivations -- 5 Diagonal lifts -- 5.1 Diagonal lifts -- 5.2 Applications -- 6 Horizontal lifts -- 6.1 General theory -- 6.2 Applications -- 7 Lift GD of a Riemannian G to FM -- 7.1 GD, G of type (0,2) -- 7.2 Levi-Civita connection of GD -- 7.3 Curvature of GD -- 7.4 Bundle of orthonormal frames -- 7.5 Geodesics of GD -- 7.6 Applications -- 8 Constructing G-structures on FM -- 8.1 ?-associated G-structures on FM -- 8.2 Defined by (1,1)-tensor fields -- 8.3 Application to polynomial structures on FM -- 8.4 G-structures defined by (0,2)-tensor fields -- 8.5 Applications to almost complex and Hermitian structures -- 8.6 Application to spacetime structure -- 9 Systems of connections -- 9.1 Connections on a fibred manifold -- 9.2 Principal bundle connections -- 9.3 Systems of connections -- 9.4 Universal Connections -- 9.5 Applications -- 10 The Functor Jp2 -- 10.1 The Bundle Jp2M ? M -- 10.2 The second order frame bundle -- 10.3 Second order connections -- 10.4 Geodesics of second order -- 10.5 G-structures on F2M -- 10.6 Vector fields on F2M -- 10.7 Diagonal lifts of tensor fields -- 10.8 Natural prolongations of G-structures -- 10.9 Diagonal prolongation of G-structures Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=122537 Differential Geometry of Frame Bundles [document électronique] / Luis A. Cordero ; Christopher Terence John Dodson ; SpringerLink (Online service) ; León, Manuel de . - Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer, 1989 . - X, 234 p : online resource. - (Mathematics and Its Applications; 47) .ISBN: 978-94-009-1265-6

Langues : Anglais (eng)

Tags : Mathematics Global analysis (Mathematics) Manifolds (Mathematics) Differential geometry Physics Differential Geometry Global Analysis and Analysis on Manifolds Mathematical and Computational Physics Résumé : It isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gu!ik's The Chillese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics Note de contenu : 1 The Functor Jp1 -- 1.1 The Bundle Jp1M ? M -- 1.2 Jp1G for a Lie group G -- 1.3 Jp1V for a vector space V -- 1.4 The embedding jp -- 2 Prolongation of G-structures -- 2.1 Imbedding of Jn1FM into FFM -- 2.2 Prolongation of G-structures to FM -- 2.3 Integrability -- 2.4 Applications -- 3 Vector-valued differential forms -- 3.1 General Theory -- 3.2 Applications -- 4 Prolongation of linear connections -- 4.1 Forms with values in a Lie algebra -- 4.2 Prolongation of connections -- 4.3 Complete lift of linear connections -- 4.4 Connections adapted to G-structures -- 4.5 Geodesics of ?C -- 4.6 Complete lift of derivations -- 5 Diagonal lifts -- 5.1 Diagonal lifts -- 5.2 Applications -- 6 Horizontal lifts -- 6.1 General theory -- 6.2 Applications -- 7 Lift GD of a Riemannian G to FM -- 7.1 GD, G of type (0,2) -- 7.2 Levi-Civita connection of GD -- 7.3 Curvature of GD -- 7.4 Bundle of orthonormal frames -- 7.5 Geodesics of GD -- 7.6 Applications -- 8 Constructing G-structures on FM -- 8.1 ?-associated G-structures on FM -- 8.2 Defined by (1,1)-tensor fields -- 8.3 Application to polynomial structures on FM -- 8.4 G-structures defined by (0,2)-tensor fields -- 8.5 Applications to almost complex and Hermitian structures -- 8.6 Application to spacetime structure -- 9 Systems of connections -- 9.1 Connections on a fibred manifold -- 9.2 Principal bundle connections -- 9.3 Systems of connections -- 9.4 Universal Connections -- 9.5 Applications -- 10 The Functor Jp2 -- 10.1 The Bundle Jp2M ? M -- 10.2 The second order frame bundle -- 10.3 Second order connections -- 10.4 Geodesics of second order -- 10.5 G-structures on F2M -- 10.6 Vector fields on F2M -- 10.7 Diagonal lifts of tensor fields -- 10.8 Natural prolongations of G-structures -- 10.9 Diagonal prolongation of G-structures Permalink : https://genes.bibli.fr/index.php?lvl=notice_display&id=122537 Differential Geometry of Spray and Finsler Spaces / Shen, Zhongmin / Berlin ; Heidelberg (DEU) ; New York ; Bâle (CHE) : Springer (2001)

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