Sub-Riemannian Geometry :: General Theory and Examples.
A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach.
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge :
Cambridge University Press,
2009.
|
| Series: | Encyclopedia of mathematics and its applications.
|
| Subjects: | |
| Online Access: | DE-862 DE-863 |
| Summary: | A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach. |
| Item Description: | 9.6 Volume Element on Heisenberg Manifolds. |
| Physical Description: | 1 online resource (384 pages) |
| Bibliography: | Includes bibliographical references (pages 363-366) and index. |
| ISBN: | 9781107096097 110709609X 9781107089839 1107089832 9781139195966 1139195964 1107104149 9781107104143 |
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MARC
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| 100 | 1 | |a Calin, Ovidiu. | |
| 245 | 1 | 0 | |a Sub-Riemannian Geometry : |b General Theory and Examples. |
| 260 | |a Cambridge : |b Cambridge University Press, |c 2009. | ||
| 300 | |a 1 online resource (384 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
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| 490 | 1 | |a Encyclopedia of Mathematics and its Applications ; |v v. 126 | |
| 505 | 0 | |a Cover; Half Title; Series Page; Title; Copyright; Dedication; Contents; Preface; Part I General Theory; 1 Introductory Chapter; 1.1 Differentiable Manifolds; 1.2 Submanifolds; 1.3 Distributions; 1.4 Integral Curves of a Vector Field; 1.5 Independent One-Forms; 1.6 Distributions Defined by One-Forms; 1.7 Integrability of One-Forms; 1.8 Elliptic Functions; 1.9 Exterior Differential Systems; 1.10 Formulas Involving Lie Derivative; 1.11 Pfaff Systems; 1.12 Characteristic Vector Fields; 1.13 Lagrange-Charpit Method; 1.14 Eiconal Equation on the Euclidean Space; 1.15 Hamilton-Jacobi Equation on Rn. | |
| 505 | 8 | |a 2 Basic Properties2.1 Sub-Riemannian Manifolds; 2.2 The Existence of Sub-Riemannian Metrics; 2.3 Systems of Orthonormal Vector Fields at a Point; 2.4 Bracket-Generating Distributions; 2.5 Non-Bracket-Generating Distributions; 2.6 Cyclic Bracket Structures; 2.7 Strong Bracket-Generating Condition; 2.8 Nilpotent Distributions; 2.9 The Horizontal Gradient; 2.10 The Intrinsic and Extrinsic Ideals; 2.11 The Induced Connection and Curvature Forms; 2.12 The Iterated Extrinsic Ideals; 3 Horizontal Connectivity; 3.1 Teleman's Theorem; 3.2 Carathéodory's Theorem; 3.3 Thermodynamical Interpretation. | |
| 505 | 8 | |a 3.4 A Global Nonconnectivity Example3.5 Chow's Theorem; 4 The Hamilton-Jacobi Theory; 4.1 The Hamilton-Jacobi Equation; 4.2 Length-Minimizing Horizontal Curves; 4.3 An Example: The Heisenberg Distribution; 4.4 Sub-Riemannian Eiconal Equation; 4.5 Solving the Hamilton-Jacobi Equation; 5 The Hamiltonian Formalism; 5.1 The Hamiltonian Function; 5.2 Normal Geodesics and Their Properties; 5.3 The Nonholonomic Constraint; 5.4 The Covariant Sub-Riemannian Metric; 5.5 Covariant and Contravariant Sub-Riemannian Metrics; 5.6 The Acceleration Along a Horizontal Curve. | |
| 505 | 8 | |a 5.7 Horizontal and Cartesian Components5.8 Normal Geodesics as Length-Minimizing Curves; 5.9 Eigenvectors of the Contravariant Metric; 5.10 Poisson Formalism; 5.11 Invariants of a Distribution; 6 Lagrangian Formalism; 6.1 Lagrange Multipliers; 6.2 Singular Minimizers; 6.3 Regular Implies Normal; 6.4 The Euler-Lagrange Equations; 7 Connections on Sub-Riemannian Manifolds; 7.1 The Horizontal Connection; 7.2 The Torsion of the Horizontal Connection; 7.3 Horizontal Divergence; 7.4 Connections on Sub-Riemannian Manifolds; 7.5 Parallel Transport Along Horizontal Curves. | |
| 505 | 8 | |a 7.6 The Curvature of a Connection7.7 The Induced Curvature; 7.8 The Metrical Connection; 7.9 The Flat Connection; 8 Gauss' Theory of Sub-Riemannian Manifolds; 8.1 The Second Fundamental Form; 8.2 The Adapted Connection; 8.3 The Adapted Weingarten Map; 8.4 The Variational Problem; 8.5 The Case of the Sphere S3; Part II Examples and Applications; 9 Heisenberg Manifolds; 9.1 The Quantum Origins of the Heisenberg Group; 9.2 Basic Definitions and Properties; 9.3 Determinants of Skew-Symmetric Matrices; 9.4 Heisenberg Manifolds as Contact Manifolds; 9.5 The Curvature Two-Form. | |
| 500 | |a 9.6 Volume Element on Heisenberg Manifolds. | ||
| 520 | |a A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach. | ||
| 588 | 0 | |a Print version record. | |
| 504 | |a Includes bibliographical references (pages 363-366) and index. | ||
| 650 | 0 | |a Geodesics (Mathematics) |0 http://id.loc.gov/authorities/subjects/sh85053967 | |
| 650 | 0 | |a Geometry, Riemannian. |0 http://id.loc.gov/authorities/subjects/sh85054159 | |
| 650 | 0 | |a Riemannian manifolds. |0 http://id.loc.gov/authorities/subjects/sh85114045 | |
| 650 | 0 | |a Submanifolds. |0 http://id.loc.gov/authorities/subjects/sh85129484 | |
| 650 | 6 | |a Géodésiques (Mathématiques) | |
| 650 | 6 | |a Géométrie de Riemann. | |
| 650 | 6 | |a Variétés de Riemann. | |
| 650 | 6 | |a Sous-variétés (Mathématiques) | |
| 650 | 7 | |a MATHEMATICS |x Geometry |x Analytic. |2 bisacsh | |
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| 650 | 7 | |a Geometry, Riemannian |2 fast | |
| 650 | 7 | |a Riemannian manifolds |2 fast | |
| 650 | 7 | |a Submanifolds |2 fast | |
| 700 | 1 | |a Chang, Der-Chen. | |
| 758 | |i has work: |a Sub-Riemannian geometry (Text) |1 https://id.oclc.org/worldcat/entity/E39PCFtFpQdhXx64cptvhCTJpd |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Calin, Ovidiu. |t Sub-Riemannian Geometry : General Theory and Examples. |d Cambridge : Cambridge University Press, ©2009 |z 9780521897303 |
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Record in the Search Index
| DE-BY-FWS_katkey | ZDB-4-EBA-ocn850148893 |
|---|---|
| _version_ | 1829094962897092608 |
| adam_text | |
| any_adam_object | |
| author | Calin, Ovidiu |
| author2 | Chang, Der-Chen |
| author2_role | |
| author2_variant | d c c dcc |
| author_facet | Calin, Ovidiu Chang, Der-Chen |
| author_role | |
| author_sort | Calin, Ovidiu |
| author_variant | o c oc |
| building | Verbundindex |
| bvnumber | localFWS |
| callnumber-first | Q - Science |
| callnumber-label | QA649 |
| callnumber-raw | QA649 .C27 2009 |
| callnumber-search | QA649 .C27 2009 |
| callnumber-sort | QA 3649 C27 42009 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 370 |
| collection | ZDB-4-EBA |
| contents | Cover; Half Title; Series Page; Title; Copyright; Dedication; Contents; Preface; Part I General Theory; 1 Introductory Chapter; 1.1 Differentiable Manifolds; 1.2 Submanifolds; 1.3 Distributions; 1.4 Integral Curves of a Vector Field; 1.5 Independent One-Forms; 1.6 Distributions Defined by One-Forms; 1.7 Integrability of One-Forms; 1.8 Elliptic Functions; 1.9 Exterior Differential Systems; 1.10 Formulas Involving Lie Derivative; 1.11 Pfaff Systems; 1.12 Characteristic Vector Fields; 1.13 Lagrange-Charpit Method; 1.14 Eiconal Equation on the Euclidean Space; 1.15 Hamilton-Jacobi Equation on Rn. 2 Basic Properties2.1 Sub-Riemannian Manifolds; 2.2 The Existence of Sub-Riemannian Metrics; 2.3 Systems of Orthonormal Vector Fields at a Point; 2.4 Bracket-Generating Distributions; 2.5 Non-Bracket-Generating Distributions; 2.6 Cyclic Bracket Structures; 2.7 Strong Bracket-Generating Condition; 2.8 Nilpotent Distributions; 2.9 The Horizontal Gradient; 2.10 The Intrinsic and Extrinsic Ideals; 2.11 The Induced Connection and Curvature Forms; 2.12 The Iterated Extrinsic Ideals; 3 Horizontal Connectivity; 3.1 Teleman's Theorem; 3.2 Carathéodory's Theorem; 3.3 Thermodynamical Interpretation. 3.4 A Global Nonconnectivity Example3.5 Chow's Theorem; 4 The Hamilton-Jacobi Theory; 4.1 The Hamilton-Jacobi Equation; 4.2 Length-Minimizing Horizontal Curves; 4.3 An Example: The Heisenberg Distribution; 4.4 Sub-Riemannian Eiconal Equation; 4.5 Solving the Hamilton-Jacobi Equation; 5 The Hamiltonian Formalism; 5.1 The Hamiltonian Function; 5.2 Normal Geodesics and Their Properties; 5.3 The Nonholonomic Constraint; 5.4 The Covariant Sub-Riemannian Metric; 5.5 Covariant and Contravariant Sub-Riemannian Metrics; 5.6 The Acceleration Along a Horizontal Curve. 5.7 Horizontal and Cartesian Components5.8 Normal Geodesics as Length-Minimizing Curves; 5.9 Eigenvectors of the Contravariant Metric; 5.10 Poisson Formalism; 5.11 Invariants of a Distribution; 6 Lagrangian Formalism; 6.1 Lagrange Multipliers; 6.2 Singular Minimizers; 6.3 Regular Implies Normal; 6.4 The Euler-Lagrange Equations; 7 Connections on Sub-Riemannian Manifolds; 7.1 The Horizontal Connection; 7.2 The Torsion of the Horizontal Connection; 7.3 Horizontal Divergence; 7.4 Connections on Sub-Riemannian Manifolds; 7.5 Parallel Transport Along Horizontal Curves. 7.6 The Curvature of a Connection7.7 The Induced Curvature; 7.8 The Metrical Connection; 7.9 The Flat Connection; 8 Gauss' Theory of Sub-Riemannian Manifolds; 8.1 The Second Fundamental Form; 8.2 The Adapted Connection; 8.3 The Adapted Weingarten Map; 8.4 The Variational Problem; 8.5 The Case of the Sphere S3; Part II Examples and Applications; 9 Heisenberg Manifolds; 9.1 The Quantum Origins of the Heisenberg Group; 9.2 Basic Definitions and Properties; 9.3 Determinants of Skew-Symmetric Matrices; 9.4 Heisenberg Manifolds as Contact Manifolds; 9.5 The Curvature Two-Form. |
| ctrlnum | (OCoLC)850148893 |
| dewey-full | 516.3/73 516.373 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/73 516.373 |
| dewey-search | 516.3/73 516.373 |
| dewey-sort | 3516.3 273 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| id | ZDB-4-EBA-ocn850148893 |
| illustrated | Not Illustrated |
| indexdate | 2025-04-11T08:41:28Z |
| institution | BVB |
| isbn | 9781107096097 110709609X 9781107089839 1107089832 9781139195966 1139195964 1107104149 9781107104143 |
| language | English |
| oclc_num | 850148893 |
| open_access_boolean | |
| owner | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-862 DE-BY-FWS DE-863 DE-BY-FWS |
| physical | 1 online resource (384 pages) |
| psigel | ZDB-4-EBA FWS_PDA_EBA ZDB-4-EBA |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | Encyclopedia of mathematics and its applications. |
| series2 | Encyclopedia of Mathematics and its Applications ; |
| spelling | Calin, Ovidiu. Sub-Riemannian Geometry : General Theory and Examples. Cambridge : Cambridge University Press, 2009. 1 online resource (384 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Encyclopedia of Mathematics and its Applications ; v. 126 Cover; Half Title; Series Page; Title; Copyright; Dedication; Contents; Preface; Part I General Theory; 1 Introductory Chapter; 1.1 Differentiable Manifolds; 1.2 Submanifolds; 1.3 Distributions; 1.4 Integral Curves of a Vector Field; 1.5 Independent One-Forms; 1.6 Distributions Defined by One-Forms; 1.7 Integrability of One-Forms; 1.8 Elliptic Functions; 1.9 Exterior Differential Systems; 1.10 Formulas Involving Lie Derivative; 1.11 Pfaff Systems; 1.12 Characteristic Vector Fields; 1.13 Lagrange-Charpit Method; 1.14 Eiconal Equation on the Euclidean Space; 1.15 Hamilton-Jacobi Equation on Rn. 2 Basic Properties2.1 Sub-Riemannian Manifolds; 2.2 The Existence of Sub-Riemannian Metrics; 2.3 Systems of Orthonormal Vector Fields at a Point; 2.4 Bracket-Generating Distributions; 2.5 Non-Bracket-Generating Distributions; 2.6 Cyclic Bracket Structures; 2.7 Strong Bracket-Generating Condition; 2.8 Nilpotent Distributions; 2.9 The Horizontal Gradient; 2.10 The Intrinsic and Extrinsic Ideals; 2.11 The Induced Connection and Curvature Forms; 2.12 The Iterated Extrinsic Ideals; 3 Horizontal Connectivity; 3.1 Teleman's Theorem; 3.2 Carathéodory's Theorem; 3.3 Thermodynamical Interpretation. 3.4 A Global Nonconnectivity Example3.5 Chow's Theorem; 4 The Hamilton-Jacobi Theory; 4.1 The Hamilton-Jacobi Equation; 4.2 Length-Minimizing Horizontal Curves; 4.3 An Example: The Heisenberg Distribution; 4.4 Sub-Riemannian Eiconal Equation; 4.5 Solving the Hamilton-Jacobi Equation; 5 The Hamiltonian Formalism; 5.1 The Hamiltonian Function; 5.2 Normal Geodesics and Their Properties; 5.3 The Nonholonomic Constraint; 5.4 The Covariant Sub-Riemannian Metric; 5.5 Covariant and Contravariant Sub-Riemannian Metrics; 5.6 The Acceleration Along a Horizontal Curve. 5.7 Horizontal and Cartesian Components5.8 Normal Geodesics as Length-Minimizing Curves; 5.9 Eigenvectors of the Contravariant Metric; 5.10 Poisson Formalism; 5.11 Invariants of a Distribution; 6 Lagrangian Formalism; 6.1 Lagrange Multipliers; 6.2 Singular Minimizers; 6.3 Regular Implies Normal; 6.4 The Euler-Lagrange Equations; 7 Connections on Sub-Riemannian Manifolds; 7.1 The Horizontal Connection; 7.2 The Torsion of the Horizontal Connection; 7.3 Horizontal Divergence; 7.4 Connections on Sub-Riemannian Manifolds; 7.5 Parallel Transport Along Horizontal Curves. 7.6 The Curvature of a Connection7.7 The Induced Curvature; 7.8 The Metrical Connection; 7.9 The Flat Connection; 8 Gauss' Theory of Sub-Riemannian Manifolds; 8.1 The Second Fundamental Form; 8.2 The Adapted Connection; 8.3 The Adapted Weingarten Map; 8.4 The Variational Problem; 8.5 The Case of the Sphere S3; Part II Examples and Applications; 9 Heisenberg Manifolds; 9.1 The Quantum Origins of the Heisenberg Group; 9.2 Basic Definitions and Properties; 9.3 Determinants of Skew-Symmetric Matrices; 9.4 Heisenberg Manifolds as Contact Manifolds; 9.5 The Curvature Two-Form. 9.6 Volume Element on Heisenberg Manifolds. A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach. Print version record. Includes bibliographical references (pages 363-366) and index. Geodesics (Mathematics) http://id.loc.gov/authorities/subjects/sh85053967 Geometry, Riemannian. http://id.loc.gov/authorities/subjects/sh85054159 Riemannian manifolds. http://id.loc.gov/authorities/subjects/sh85114045 Submanifolds. http://id.loc.gov/authorities/subjects/sh85129484 Géodésiques (Mathématiques) Géométrie de Riemann. Variétés de Riemann. Sous-variétés (Mathématiques) MATHEMATICS Geometry Analytic. bisacsh Geodesics (Mathematics) fast Geometry, Riemannian fast Riemannian manifolds fast Submanifolds fast Chang, Der-Chen. has work: Sub-Riemannian geometry (Text) https://id.oclc.org/worldcat/entity/E39PCFtFpQdhXx64cptvhCTJpd https://id.oclc.org/worldcat/ontology/hasWork Print version: Calin, Ovidiu. Sub-Riemannian Geometry : General Theory and Examples. Cambridge : Cambridge University Press, ©2009 9780521897303 Encyclopedia of mathematics and its applications. http://id.loc.gov/authorities/names/n42010632 |
| spellingShingle | Calin, Ovidiu Sub-Riemannian Geometry : General Theory and Examples. Encyclopedia of mathematics and its applications. Cover; Half Title; Series Page; Title; Copyright; Dedication; Contents; Preface; Part I General Theory; 1 Introductory Chapter; 1.1 Differentiable Manifolds; 1.2 Submanifolds; 1.3 Distributions; 1.4 Integral Curves of a Vector Field; 1.5 Independent One-Forms; 1.6 Distributions Defined by One-Forms; 1.7 Integrability of One-Forms; 1.8 Elliptic Functions; 1.9 Exterior Differential Systems; 1.10 Formulas Involving Lie Derivative; 1.11 Pfaff Systems; 1.12 Characteristic Vector Fields; 1.13 Lagrange-Charpit Method; 1.14 Eiconal Equation on the Euclidean Space; 1.15 Hamilton-Jacobi Equation on Rn. 2 Basic Properties2.1 Sub-Riemannian Manifolds; 2.2 The Existence of Sub-Riemannian Metrics; 2.3 Systems of Orthonormal Vector Fields at a Point; 2.4 Bracket-Generating Distributions; 2.5 Non-Bracket-Generating Distributions; 2.6 Cyclic Bracket Structures; 2.7 Strong Bracket-Generating Condition; 2.8 Nilpotent Distributions; 2.9 The Horizontal Gradient; 2.10 The Intrinsic and Extrinsic Ideals; 2.11 The Induced Connection and Curvature Forms; 2.12 The Iterated Extrinsic Ideals; 3 Horizontal Connectivity; 3.1 Teleman's Theorem; 3.2 Carathéodory's Theorem; 3.3 Thermodynamical Interpretation. 3.4 A Global Nonconnectivity Example3.5 Chow's Theorem; 4 The Hamilton-Jacobi Theory; 4.1 The Hamilton-Jacobi Equation; 4.2 Length-Minimizing Horizontal Curves; 4.3 An Example: The Heisenberg Distribution; 4.4 Sub-Riemannian Eiconal Equation; 4.5 Solving the Hamilton-Jacobi Equation; 5 The Hamiltonian Formalism; 5.1 The Hamiltonian Function; 5.2 Normal Geodesics and Their Properties; 5.3 The Nonholonomic Constraint; 5.4 The Covariant Sub-Riemannian Metric; 5.5 Covariant and Contravariant Sub-Riemannian Metrics; 5.6 The Acceleration Along a Horizontal Curve. 5.7 Horizontal and Cartesian Components5.8 Normal Geodesics as Length-Minimizing Curves; 5.9 Eigenvectors of the Contravariant Metric; 5.10 Poisson Formalism; 5.11 Invariants of a Distribution; 6 Lagrangian Formalism; 6.1 Lagrange Multipliers; 6.2 Singular Minimizers; 6.3 Regular Implies Normal; 6.4 The Euler-Lagrange Equations; 7 Connections on Sub-Riemannian Manifolds; 7.1 The Horizontal Connection; 7.2 The Torsion of the Horizontal Connection; 7.3 Horizontal Divergence; 7.4 Connections on Sub-Riemannian Manifolds; 7.5 Parallel Transport Along Horizontal Curves. 7.6 The Curvature of a Connection7.7 The Induced Curvature; 7.8 The Metrical Connection; 7.9 The Flat Connection; 8 Gauss' Theory of Sub-Riemannian Manifolds; 8.1 The Second Fundamental Form; 8.2 The Adapted Connection; 8.3 The Adapted Weingarten Map; 8.4 The Variational Problem; 8.5 The Case of the Sphere S3; Part II Examples and Applications; 9 Heisenberg Manifolds; 9.1 The Quantum Origins of the Heisenberg Group; 9.2 Basic Definitions and Properties; 9.3 Determinants of Skew-Symmetric Matrices; 9.4 Heisenberg Manifolds as Contact Manifolds; 9.5 The Curvature Two-Form. Geodesics (Mathematics) http://id.loc.gov/authorities/subjects/sh85053967 Geometry, Riemannian. http://id.loc.gov/authorities/subjects/sh85054159 Riemannian manifolds. http://id.loc.gov/authorities/subjects/sh85114045 Submanifolds. http://id.loc.gov/authorities/subjects/sh85129484 Géodésiques (Mathématiques) Géométrie de Riemann. Variétés de Riemann. Sous-variétés (Mathématiques) MATHEMATICS Geometry Analytic. bisacsh Geodesics (Mathematics) fast Geometry, Riemannian fast Riemannian manifolds fast Submanifolds fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85053967 http://id.loc.gov/authorities/subjects/sh85054159 http://id.loc.gov/authorities/subjects/sh85114045 http://id.loc.gov/authorities/subjects/sh85129484 |
| title | Sub-Riemannian Geometry : General Theory and Examples. |
| title_auth | Sub-Riemannian Geometry : General Theory and Examples. |
| title_exact_search | Sub-Riemannian Geometry : General Theory and Examples. |
| title_full | Sub-Riemannian Geometry : General Theory and Examples. |
| title_fullStr | Sub-Riemannian Geometry : General Theory and Examples. |
| title_full_unstemmed | Sub-Riemannian Geometry : General Theory and Examples. |
| title_short | Sub-Riemannian Geometry : |
| title_sort | sub riemannian geometry general theory and examples |
| title_sub | General Theory and Examples. |
| topic | Geodesics (Mathematics) http://id.loc.gov/authorities/subjects/sh85053967 Geometry, Riemannian. http://id.loc.gov/authorities/subjects/sh85054159 Riemannian manifolds. http://id.loc.gov/authorities/subjects/sh85114045 Submanifolds. http://id.loc.gov/authorities/subjects/sh85129484 Géodésiques (Mathématiques) Géométrie de Riemann. Variétés de Riemann. Sous-variétés (Mathématiques) MATHEMATICS Geometry Analytic. bisacsh Geodesics (Mathematics) fast Geometry, Riemannian fast Riemannian manifolds fast Submanifolds fast |
| topic_facet | Geodesics (Mathematics) Geometry, Riemannian. Riemannian manifolds. Submanifolds. Géodésiques (Mathématiques) Géométrie de Riemann. Variétés de Riemann. Sous-variétés (Mathématiques) MATHEMATICS Geometry Analytic. Geometry, Riemannian Riemannian manifolds Submanifolds |
| work_keys_str_mv | AT calinovidiu subriemanniangeometrygeneraltheoryandexamples AT changderchen subriemanniangeometrygeneraltheoryandexamples |