Leseprobe
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Spin Geometry (PMS-38), Volume 38
Marie-Louise Michelsohn, H. Blaine Lawson
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.
Kundenbewertungen
Manifold, Riemannian geometry, Variable (mathematics), Fundamental group, Tensor product, Dirac operator, Fiber bundle, Geometry, Topology, Sign (mathematics), Intersection form (4-manifold), Support (mathematics), Elliptic operator, Clifford algebra, Calabi–Yau manifold, Equivariant K-theory, Projective space, Sphere theorem, Bundle metric, Vector field, Riemann curvature tensor, Hilbert scheme, Quadratic form, Spin representation, Atiyah–Singer index theorem, Isomorphism class, Hyperbolic manifold, Ricci curvature, Binary octahedral group, Cohomology, Covering space, Spin structure, Spinor, K-theory, Vector space, Gauge theory, N-sphere, Tensor calculus, Mathematical induction, Kähler manifold, Metric connection, Quaternion, Ring homomorphism, Scalar curvature, Scalar multiplication, Principal bundle, Pseudo-Riemannian manifold, Operator (physics), Unit disk, Automorphism, Disk (mathematics), Minkowski space, Cartesian product, Spinor field, Cobordism, Theorem, Cohomology ring, Dimension (vector space), Vector bundle, C*-algebra, Orthonormal basis, Quaternionic projective space, Module (mathematics), Holonomy, Pseudo-differential operator, Symplectic geometry, Tangent bundle, Riemannian manifold, Tangent space, Spinor bundle