Jeff Cheeger

On the curvature of piecewise flat spaces

We consider analogs of the Lipschitz-Killing curvatures of smooth Riemannian manifolds for piecewise flat spaces. In the special case of scalar curvature, the definition is due to T. Regge; considerations in this spirit date back to J. Steiner. We show that if a piecewise flat space approximates a smooth space in a suitable sense, then the corresponding curvatures are close in the sense of measures.

Eta-invariants and their adiabatic limits

0. Introduction 1. Algebraic preliminaries 2. The finite dimensional case; superconnections (a) Odd dimensional base spaces (b) Even dimensional base spaces Appendix 1. Coupling to general superconnections 3. Auxiliary Grassman variables and auxiliary adiabatic limits 4. The infinite dimensional case; fibrations (a) Elementary geometry of fibrations (b) Odd dimensional base spaces (c) Even dimensional base spaces (d) The case in which the Dirac operator on the fiber is not invertible Appendix 2. Multifibrations

Families index for manifolds with boundary, superconnections, and cones. I, Families of manifolds with boundary and Dirac operators

Abstract This is Part I of a work, in which we establish a formula for the Chern character of a family of Dirac operators of Atiyah-Patodi-Singer on even-dimensional manifolds with boundary. The key tools are the superconnections of Quillen, the cone method, and the Levi-Civita superconnection. In this Part I, we construct a family of Dirac operators on manifolds with boundary and we introduce the corresponding Levi-Civita superconnections.

Families index for manifolds with boundary, superconnections and cones. II. The Chern character

Abstract This is Part II of a work in which we establish a formula for the Chern character of a family of Dirac operators of Atiyah, Patodi, and Singer. Here we establish the existence of associated superconnection heat kernels, and we prove the final index formula.