| 《自旋几何(英文版)》 preface acknowledgments introduction chapter ⅰ clifford algebras, spin groups and their representations 1. clifford algebras 2. the groups pin and spin 3. the algebras cln, and clr,s 4. the classification 5. representations 6. lie algebra structures 7. some direct applications to geometry 8. some further applications to the theory of lie groups 9. k-theory and the atiyah-bott-shapiro construction 10. kr-theory and the (1,1)-periodicity theorem chapter ⅱ spin geometry and the dirac operators 1. spin structures on vector bundles 2. spin manifolds and spin cobordism 3. clifford and spinor bundles .4. connections on spinor bundles 5. the dirac operators 6. the fundamental elliptic operators 7. clk-linear dirac operators 8. vanishing theorems and some applications chapter ⅲ index theorems 1. differential operators 2. sobolev spaces and sobolev theorems 3. pseudodifferential operators 4. elliptic operators and parametrices 5. fundamental results for elliptic operators 6. the heat kernel and the index 7. the topological invariance of the index 8. the index of a family of elliptic operators 9. the g-index 10. the clifford index 11. multiplicative sequences and the chern character 12. thom isomorphisms and the chern character defect 13. the atiyah-singer index theorem 14. fixed-point formulas for elliptic operators 15. the index theorem for families 16. families of real operators and the clk-index theorem 17. remarks on heat and supersymmetry chapter ⅳ applications in geometry and topology 1. lntegrality theorems 2. immersions of manifolds and the vector field problem 3. group actions on manifolds 4. compact manifolds of positive scalar curvature 5. positive scalar curvature and the fundamental group 6. complete manifolds of positive scalar curvature 7. the topology of the space of positive scalar curvature metrics 8. clifford multiplication and kiihler manifolds 9. pure spinors, complex structures, and twistors 10. reduced holonomy and calibrations 11. spinor cohomology and complex manifolds with vanishing first chern class 12. the positive mass conjecture in general relativity appendix a principal g-bundles appendix b classifying spaces and characteristic classes appendix c orientation classes and thom isomorphisms in k-theory appendix d spine-manifolds bibliography index notation index |
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