Geometric properties of natural operators defined by the Riemann curvature tensor /:
A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold. The full curvature tensor is in general quite difficult to deal with. This book presents results about the geometric consequences that follow if va...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Singapore ; River Edge, NJ :
World Scientific,
©2001.
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| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold. The full curvature tensor is in general quite difficult to deal with. This book presents results about the geometric consequences that follow if various natural operators defined in terms of the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and higher order generalizations) are assumed to have constant eigenvalues or constant Jordan normal form in the appropriate domains of definition. The book presents algebraic preliminaries and various Schur type problems; deals with the skew-symmetric curvature operator in the real and complex settings and provides the classification of algebraic curvature tensors whose skew-symmetric curvature has constant rank 2 and constant eigenvalues; discusses the Jacobi operator and a higher order generalization and gives a unified treatment of the Osserman conjecture and related questions; and establishes the results from algebraic topology that are necessary for controlling the eigenvalue structures. An extensive bibliography is provided. Results are described in the Riemannian, Lorentzian, and higher signature settings, and many families of examples are displayed. |
| Beschreibung: | 1 online resource (viii, 306 pages) : illustrations |
| Bibliographie: | Includes bibliographical references (pages 293-302) and index. |
| ISBN: | 9789812799692 9812799699 1281948012 9781281948014 |
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| 245 | 1 | 0 | |a Geometric properties of natural operators defined by the Riemann curvature tensor / |c Peter B. Gilkey. |
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| 504 | |a Includes bibliographical references (pages 293-302) and index. | ||
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| 520 | |a A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold. The full curvature tensor is in general quite difficult to deal with. This book presents results about the geometric consequences that follow if various natural operators defined in terms of the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and higher order generalizations) are assumed to have constant eigenvalues or constant Jordan normal form in the appropriate domains of definition. The book presents algebraic preliminaries and various Schur type problems; deals with the skew-symmetric curvature operator in the real and complex settings and provides the classification of algebraic curvature tensors whose skew-symmetric curvature has constant rank 2 and constant eigenvalues; discusses the Jacobi operator and a higher order generalization and gives a unified treatment of the Osserman conjecture and related questions; and establishes the results from algebraic topology that are necessary for controlling the eigenvalue structures. An extensive bibliography is provided. Results are described in the Riemannian, Lorentzian, and higher signature settings, and many families of examples are displayed. | ||
| 505 | 0 | |a Ch. 1. Algebraic curvature tensors. 1.1. Introduction. 1.2. Results from linear algebra. 1.3. Self-adjoint maps of a spacelike vector space. 1.4. Clifford algebras and matrices. 1.5. Natural operators. 1.6. Algebraic curvature tensors. 1.7. Einstein and k-stein algebraic curvature tensors. 1.8. Properties of the curvature tensors R[symbol]. 1.9. Invariants of the orthogonal group. 1.10. Natural operators with constant eigenvalues. 1.11. The exponential map and Jacobi vector fields. 1.12. Geometric realizations of algebraic curvature tensors. 1.13. Schur problems. 1.14. Space forms. 1.15. Complex and para-complex space forms -- ch. 2. The Skew-symmetric curvature operator. 2.1. Introduction. 2.2. Examples. 2.3. Rank 2 algebraic curvature tensors. 2.4. Geometric realizations of rank 2 tensors. 2.5. IP algebraic curvature tensors. 2.6. Examples of IP manifolds. 2.7. Classification of IP manifolds. 2.8. Four dimensional geometry. 2.9. Seven dimensional geometry. 2.10. Eight dimensional geometry. 2.11. Almost complex IP tensors. 2.12. Higher order IP tensors -- ch. 3. The Jacobi operator. 3.1. Introduction. 3.2. Examples of Osserman tensors. 3.3. Examples of higher order Osserman tensors. 3.4. Rakic duality. 3.5. The Osserman conjecture. 3.6. Space forms and (para- )complex space forms. 3.7. The higher order Jacobi operator. 3.8. The Szabo operator -- ch. 4. Controlling the eigenvalue structure. 4.1. Introduction. 4.2. Fiber bundles. 4.3. Characteristic classes and K-theory. 4.4. Symmetric vector bundles. 4.5. Odd maps of constant rank. | |
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| 650 | 0 | |a Operator theory. |0 http://id.loc.gov/authorities/subjects/sh85095029 | |
| 650 | 6 | |a Géométrie de Riemann. | |
| 650 | 6 | |a Courbure. | |
| 650 | 6 | |a Théorie des opérateurs. | |
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| author | Gilkey, Peter B. |
| author_GND | http://id.loc.gov/authorities/names/nr90011019 |
| author_facet | Gilkey, Peter B. |
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| contents | Ch. 1. Algebraic curvature tensors. 1.1. Introduction. 1.2. Results from linear algebra. 1.3. Self-adjoint maps of a spacelike vector space. 1.4. Clifford algebras and matrices. 1.5. Natural operators. 1.6. Algebraic curvature tensors. 1.7. Einstein and k-stein algebraic curvature tensors. 1.8. Properties of the curvature tensors R[symbol]. 1.9. Invariants of the orthogonal group. 1.10. Natural operators with constant eigenvalues. 1.11. The exponential map and Jacobi vector fields. 1.12. Geometric realizations of algebraic curvature tensors. 1.13. Schur problems. 1.14. Space forms. 1.15. Complex and para-complex space forms -- ch. 2. The Skew-symmetric curvature operator. 2.1. Introduction. 2.2. Examples. 2.3. Rank 2 algebraic curvature tensors. 2.4. Geometric realizations of rank 2 tensors. 2.5. IP algebraic curvature tensors. 2.6. Examples of IP manifolds. 2.7. Classification of IP manifolds. 2.8. Four dimensional geometry. 2.9. Seven dimensional geometry. 2.10. Eight dimensional geometry. 2.11. Almost complex IP tensors. 2.12. Higher order IP tensors -- ch. 3. The Jacobi operator. 3.1. Introduction. 3.2. Examples of Osserman tensors. 3.3. Examples of higher order Osserman tensors. 3.4. Rakic duality. 3.5. The Osserman conjecture. 3.6. Space forms and (para- )complex space forms. 3.7. The higher order Jacobi operator. 3.8. The Szabo operator -- ch. 4. Controlling the eigenvalue structure. 4.1. Introduction. 4.2. Fiber bundles. 4.3. Characteristic classes and K-theory. 4.4. Symmetric vector bundles. 4.5. Odd maps of constant rank. |
| ctrlnum | (OCoLC)261121655 |
| dewey-full | 516.3/73 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/73 |
| dewey-search | 516.3/73 |
| dewey-sort | 3516.3 273 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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The book presents algebraic preliminaries and various Schur type problems; deals with the skew-symmetric curvature operator in the real and complex settings and provides the classification of algebraic curvature tensors whose skew-symmetric curvature has constant rank 2 and constant eigenvalues; discusses the Jacobi operator and a higher order generalization and gives a unified treatment of the Osserman conjecture and related questions; and establishes the results from algebraic topology that are necessary for controlling the eigenvalue structures. An extensive bibliography is provided. Results are described in the Riemannian, Lorentzian, and higher signature settings, and many families of examples are displayed.</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">Ch. 1. Algebraic curvature tensors. 1.1. Introduction. 1.2. Results from linear algebra. 1.3. Self-adjoint maps of a spacelike vector space. 1.4. Clifford algebras and matrices. 1.5. 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Examples of higher order Osserman tensors. 3.4. Rakic duality. 3.5. The Osserman conjecture. 3.6. Space forms and (para- )complex space forms. 3.7. The higher order Jacobi operator. 3.8. The Szabo operator -- ch. 4. Controlling the eigenvalue structure. 4.1. Introduction. 4.2. Fiber bundles. 4.3. Characteristic classes and K-theory. 4.4. Symmetric vector bundles. 4.5. 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| id | ZDB-4-EBA-ocn261121655 |
| illustrated | Illustrated |
| indexdate | 2024-11-27T13:16:32Z |
| institution | BVB |
| isbn | 9789812799692 9812799699 1281948012 9781281948014 |
| language | English |
| oclc_num | 261121655 |
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| publisher | World Scientific, |
| record_format | marc |
| spelling | Gilkey, Peter B. https://id.oclc.org/worldcat/entity/E39PBJrchgyDDxDB4hYD3gPRKd http://id.loc.gov/authorities/names/nr90011019 Geometric properties of natural operators defined by the Riemann curvature tensor / Peter B. Gilkey. Singapore ; River Edge, NJ : World Scientific, ©2001. 1 online resource (viii, 306 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier Includes bibliographical references (pages 293-302) and index. Print version record. A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold. The full curvature tensor is in general quite difficult to deal with. This book presents results about the geometric consequences that follow if various natural operators defined in terms of the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and higher order generalizations) are assumed to have constant eigenvalues or constant Jordan normal form in the appropriate domains of definition. The book presents algebraic preliminaries and various Schur type problems; deals with the skew-symmetric curvature operator in the real and complex settings and provides the classification of algebraic curvature tensors whose skew-symmetric curvature has constant rank 2 and constant eigenvalues; discusses the Jacobi operator and a higher order generalization and gives a unified treatment of the Osserman conjecture and related questions; and establishes the results from algebraic topology that are necessary for controlling the eigenvalue structures. An extensive bibliography is provided. Results are described in the Riemannian, Lorentzian, and higher signature settings, and many families of examples are displayed. Ch. 1. Algebraic curvature tensors. 1.1. Introduction. 1.2. Results from linear algebra. 1.3. Self-adjoint maps of a spacelike vector space. 1.4. Clifford algebras and matrices. 1.5. Natural operators. 1.6. Algebraic curvature tensors. 1.7. Einstein and k-stein algebraic curvature tensors. 1.8. Properties of the curvature tensors R[symbol]. 1.9. Invariants of the orthogonal group. 1.10. Natural operators with constant eigenvalues. 1.11. The exponential map and Jacobi vector fields. 1.12. Geometric realizations of algebraic curvature tensors. 1.13. Schur problems. 1.14. Space forms. 1.15. Complex and para-complex space forms -- ch. 2. The Skew-symmetric curvature operator. 2.1. Introduction. 2.2. Examples. 2.3. Rank 2 algebraic curvature tensors. 2.4. Geometric realizations of rank 2 tensors. 2.5. IP algebraic curvature tensors. 2.6. Examples of IP manifolds. 2.7. Classification of IP manifolds. 2.8. Four dimensional geometry. 2.9. Seven dimensional geometry. 2.10. Eight dimensional geometry. 2.11. Almost complex IP tensors. 2.12. Higher order IP tensors -- ch. 3. The Jacobi operator. 3.1. Introduction. 3.2. Examples of Osserman tensors. 3.3. Examples of higher order Osserman tensors. 3.4. Rakic duality. 3.5. The Osserman conjecture. 3.6. Space forms and (para- )complex space forms. 3.7. The higher order Jacobi operator. 3.8. The Szabo operator -- ch. 4. Controlling the eigenvalue structure. 4.1. Introduction. 4.2. Fiber bundles. 4.3. Characteristic classes and K-theory. 4.4. Symmetric vector bundles. 4.5. Odd maps of constant rank. Geometry, Riemannian. http://id.loc.gov/authorities/subjects/sh85054159 Curvature. http://id.loc.gov/authorities/subjects/sh85034911 Operator theory. http://id.loc.gov/authorities/subjects/sh85095029 Géométrie de Riemann. Courbure. Théorie des opérateurs. MATHEMATICS Geometry Analytic. bisacsh Curvature fast Geometry, Riemannian fast Operator theory fast has work: Geometric properties of natural operators defined by the Riemann curvature tensor (Text) https://id.oclc.org/worldcat/entity/E39PCFCDJTWbkJpXcd7CFYFG6q https://id.oclc.org/worldcat/ontology/hasWork Print version: Gilkey, Peter B. Geometric properties of natural operators defined by the Riemann curvature tensor. Singapore ; River Edge, NJ : World Scientific, ©2001 9810247524 9789810247522 (DLC) 2002284361 (OCoLC)49195512 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=235768 Volltext |
| spellingShingle | Gilkey, Peter B. Geometric properties of natural operators defined by the Riemann curvature tensor / Ch. 1. Algebraic curvature tensors. 1.1. Introduction. 1.2. Results from linear algebra. 1.3. Self-adjoint maps of a spacelike vector space. 1.4. Clifford algebras and matrices. 1.5. Natural operators. 1.6. Algebraic curvature tensors. 1.7. Einstein and k-stein algebraic curvature tensors. 1.8. Properties of the curvature tensors R[symbol]. 1.9. Invariants of the orthogonal group. 1.10. Natural operators with constant eigenvalues. 1.11. The exponential map and Jacobi vector fields. 1.12. Geometric realizations of algebraic curvature tensors. 1.13. Schur problems. 1.14. Space forms. 1.15. Complex and para-complex space forms -- ch. 2. The Skew-symmetric curvature operator. 2.1. Introduction. 2.2. Examples. 2.3. Rank 2 algebraic curvature tensors. 2.4. Geometric realizations of rank 2 tensors. 2.5. IP algebraic curvature tensors. 2.6. Examples of IP manifolds. 2.7. Classification of IP manifolds. 2.8. Four dimensional geometry. 2.9. Seven dimensional geometry. 2.10. Eight dimensional geometry. 2.11. Almost complex IP tensors. 2.12. Higher order IP tensors -- ch. 3. The Jacobi operator. 3.1. Introduction. 3.2. Examples of Osserman tensors. 3.3. Examples of higher order Osserman tensors. 3.4. Rakic duality. 3.5. The Osserman conjecture. 3.6. Space forms and (para- )complex space forms. 3.7. The higher order Jacobi operator. 3.8. The Szabo operator -- ch. 4. Controlling the eigenvalue structure. 4.1. Introduction. 4.2. Fiber bundles. 4.3. Characteristic classes and K-theory. 4.4. Symmetric vector bundles. 4.5. Odd maps of constant rank. Geometry, Riemannian. http://id.loc.gov/authorities/subjects/sh85054159 Curvature. http://id.loc.gov/authorities/subjects/sh85034911 Operator theory. http://id.loc.gov/authorities/subjects/sh85095029 Géométrie de Riemann. Courbure. Théorie des opérateurs. MATHEMATICS Geometry Analytic. bisacsh Curvature fast Geometry, Riemannian fast Operator theory fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85054159 http://id.loc.gov/authorities/subjects/sh85034911 http://id.loc.gov/authorities/subjects/sh85095029 |
| title | Geometric properties of natural operators defined by the Riemann curvature tensor / |
| title_auth | Geometric properties of natural operators defined by the Riemann curvature tensor / |
| title_exact_search | Geometric properties of natural operators defined by the Riemann curvature tensor / |
| title_full | Geometric properties of natural operators defined by the Riemann curvature tensor / Peter B. Gilkey. |
| title_fullStr | Geometric properties of natural operators defined by the Riemann curvature tensor / Peter B. Gilkey. |
| title_full_unstemmed | Geometric properties of natural operators defined by the Riemann curvature tensor / Peter B. Gilkey. |
| title_short | Geometric properties of natural operators defined by the Riemann curvature tensor / |
| title_sort | geometric properties of natural operators defined by the riemann curvature tensor |
| topic | Geometry, Riemannian. http://id.loc.gov/authorities/subjects/sh85054159 Curvature. http://id.loc.gov/authorities/subjects/sh85034911 Operator theory. http://id.loc.gov/authorities/subjects/sh85095029 Géométrie de Riemann. Courbure. Théorie des opérateurs. MATHEMATICS Geometry Analytic. bisacsh Curvature fast Geometry, Riemannian fast Operator theory fast |
| topic_facet | Geometry, Riemannian. Curvature. Operator theory. Géométrie de Riemann. Courbure. Théorie des opérateurs. MATHEMATICS Geometry Analytic. Curvature Geometry, Riemannian Operator theory |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=235768 |
| work_keys_str_mv | AT gilkeypeterb geometricpropertiesofnaturaloperatorsdefinedbytheriemanncurvaturetensor |