Peter B. Gilkey

Gravitation, Gauge Theories and Differential Geometry

Stanford Linear Accelerator Center, Stanford, California 94305, USA and The Enrico Fermi Institute and Department of Physics, The University of Chicago, Chicago, illinois, USA Peter B. GILKEY Fine Hall, Box 37. Department of Mathematics, Princeton University, Princeton, New Jersey 08544, USA and Department of Mathematics, University of Southern California, Los Angeles, California 90007, USA and Andrew J. HANSON

The asymptotics of the Laplacian on a manifold with boundary

Let P be a second-order differential operator with leading symbol given by the tensor on a compact Riemannian manifold with boundary. We compute the asymptotics of the heat equation for Dirichlet, Neumann, and mixed boundary conditions.

Asymptotic Formulae in Spectral Geometry

A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators. Asymptotic Formulae in Spectral Geometry collects these results and computations into one book. Written by a leading pioneer in the field, it focuses on the functorial and special cases methods of computing asymptotic heat trace and heat content coefficients in the heat equation. It incorporates the work of many authors into the presentation, and includes a complete bibliography that serves as a roadmap to the literature on the subject. Geometers, mathematical physicists, and analysts alike will undoubtedly find this to be the definitive book on the subject.

A Note on Osserman Lorentzian Manifolds

Let p be a point of a Lorentzian manifold M . We show that if M is spacelike Osserman at p , then M has constant sectional curvature at p ; similarly, if M is timelike Osserman at p , then M has constant sectional curvature at p . The reverse implications are immediate. The timelike case and 4-dimensional spacelike case were first studied in [ 3 ]; we use a different approach to this case.

The boundary integrand in the formula for the signature and Euler characteristic of a Riemannian manifold with boundary

Abstract We discuss the formulas for the signature and Euler characteristic of a Riemannian manifold with boundary. We obtain boundary integrals which correct for metrics which are not product near the boundary. For the Euler characteristic, this integrand is uniquely defined by several functorial properties. We identify the integrand of the Chern-Gauss-Bonnet theorem with the integrand obtained by heat equation methods. For the signature complex, there is a similar correction term; however, there is no corresponding uniqueness theorem for this case.

Geometric properties of natural operators defined by the Riemann curvature tensor

Algebraic curvature tensors the skew-symmetric curvature operator the Jacobi operator controlling the eigenvalue structure.

Heat kernel asymptotics with mixed boundary conditions

Abstract We calculate the coefficient a 5 of the heat kernel asymptotics for an operator of Laplace type with mixed boundary conditions on a general compact manifold.

Some representations of exceptional Lie algebras

In this note, we give the dimensions of some irreducible representations of exceptional Lie algebras and algebraic groups. Similar results appear in [1] for classical groups and algebras of rank at most 4. These results were produced by computer programs developed in connection with [3], where the main result required information beyond the tables in [1]. In view of the utility of the tables in [1], it seemed worthwhile to provide tables for groups of higher rank. Although our methods are similar to those of [l], they incorporate a reduction process which permits us to push the techniques a bit further.

The Geometry of Walker Manifolds

* Basic Algebraic Notions* Basic Geometrical Notions* Walker Structures* Three-Dimensional Lorentzian Walker Manifolds* Four-Dimensional Walker Manifolds* The Spectral Geometry of the Curvature Tensor* Hermitian Geometry* Special Walker Manifolds

Residues of the eta function for an operator of Dirac type

Abstract We compute the asymptotics of Tr L 2 ( Pe −tp 2 ) where P is a first order operator of Dirac type; this is equivalent to evaluating the residues of the eta function.