Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem
This 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.
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The index theorem
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Amer an(x Anal assume Atiyah Atiyah-Singer index theorem boundary conditions characteristic classes Chern choose Clif Clifford coefficients cohomology complete the proof compute Consequently coordinates decompose Definition dimension dimensional Dirac operator Dirac type discrete spectral resolution discuss Dolbeault complex eigenspaces eigenvalues elliptic complex elliptic operators equivariant eta invariant fiber finite follows form valued Gauss-Bonnet theorem Geom geometry Gilkey heat equation Hirzebruch holomorphic homogeneous of order index theorem integral isomorphism isospectral jets K-theory kernel Laplace type Laplacian leading symbol Lemma Let Q Levi-Civita connection line bundle linear manifolds with boundary metric module structure monomial multiplication normalized notation operator of Dirac operator of Laplace orientation orthogonal orthonormal frame Patodi polynomial positive scalar curvature Proc prove pseudo-differential operators Remark Riemannian manifold scalar curvature self-adjoint shows signature complex smooth spin complex spin structure tensor theory Topology transl unitary Univ vanishes variables vector bundle zeta function