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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...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Singapore ; River Edge, NJ :
World Scientific,
©2001.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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.