Invariance Theory: The Heat Equation and the Atiyah-Singer Index TheoremThis book treats the Atiyah-Singer index theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The author uses invariance theory to identify the integrand of the index theorem for classical elliptic complexes with the invariants of the heat equation. |
Contents
Pseudodifferential operators | 1 |
Characteristic classes | 121 |
5 | 175 |
6 | 181 |
Boundary conditions | 190 |
9 | 206 |
The index theorem | 215 |
Spectral geometry | 327 |
Bibliographic information | 419 |
Notation | 509 |
Other editions - View all
Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem Peter B. Gilkey No preview available - 1984 |
Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem Peter B. Gilkey No preview available - 1995 |
Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem Peter B. Gilkey No preview available - 1995 |
Common terms and phrases
A₁ algebra Amer an(x Atiyah Atiyah-Singer index theorem boundary conditions characteristic classes Chern choose Clif coefficients cohomology compact complete the proof compute Consequently coordinates decompose define Definition det(I differential operator dimension dimensional Dirac operator Dirac type Dolbeault complex dvol eigenvalues elliptic complex elliptic operators equivariant eta invariant fiber fixed point follows formula Geom geometry Gilkey heat equation holomorphic homogeneous of order index theorem integral isomorphism isospectral jets K-theory Laplace type Laplacian leading symbol Lefschetz Lemma Levi-Civita connection linear M₁ manifolds with boundary Math metric module structure monomial normalized operator of Dirac orientation orthogonal orthonormal frame P₁ polynomial positive scalar curvature Proc prove pseudo-differential operators Remark Rham Riemannian manifold scalar curvature self-adjoint shows signature complex smooth spin complex spin structure tangent tensor theory Topology transl U₁ unitary Univ V₁ vanishes variables Vect vector bundle