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The Poisson summation formula for fibrations of Riemannian manifolds

S. G. Scott

TL;DR

The paper develops a fibrewise generalisation of the wave-trace Poisson summation by studying a smooth family of Laplace-type operators along the fibres of a compact Riemannian fibration. It constructs vertical square roots and wave operators as vertical Fourier integral operators, derives a fibrewise singularity expansion of the vertical wave trace with base-form valued invariants, and relates the zero-frequency data to vertical heat-trace invariants. In the Bismut superconnection setting, the leading vertical wave-invariants recover the vertical Â-genus; more generally, the approach yields cohomological and differential-form data on the base that encode fibrewise spectral geometry. This provides a comprehensive framework to extend classical Poisson summation to families and to connect fibrewise spectral data with characteristic forms, enabling new insights into index theory in families and geometric analysis on fibrations.

Abstract

We prove a wave trace singularity formula for a family of generalised Laplacians defined by a Riemannian fibre bundle; for example, the superconnection curvature operator associated to the Bismut superconnection. It is explained how this generalises Poisson summation formulae for families.

The Poisson summation formula for fibrations of Riemannian manifolds

TL;DR

The paper develops a fibrewise generalisation of the wave-trace Poisson summation by studying a smooth family of Laplace-type operators along the fibres of a compact Riemannian fibration. It constructs vertical square roots and wave operators as vertical Fourier integral operators, derives a fibrewise singularity expansion of the vertical wave trace with base-form valued invariants, and relates the zero-frequency data to vertical heat-trace invariants. In the Bismut superconnection setting, the leading vertical wave-invariants recover the vertical Â-genus; more generally, the approach yields cohomological and differential-form data on the base that encode fibrewise spectral geometry. This provides a comprehensive framework to extend classical Poisson summation to families and to connect fibrewise spectral data with characteristic forms, enabling new insights into index theory in families and geometric analysis on fibrations.

Abstract

We prove a wave trace singularity formula for a family of generalised Laplacians defined by a Riemannian fibre bundle; for example, the superconnection curvature operator associated to the Bismut superconnection. It is explained how this generalises Poisson summation formulae for families.

Paper Structure

This paper contains 12 sections, 16 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. Vertical Lagrangian distributions and FIOs
  3. Vertical distributions
  4. Vertical Lagrangian distributions and FIOs
  5. Proof of Theorem \ref{['thm1']} and Theorem \ref{['thm2']}
  6. The vertical $\psi{\rm do}$ $\textsf{Q}^{1/m}\in \Psi^*(M/B)$:
  7. The vertical wave operator $e^{it\textsf{Q}^{1/m}}$:
  8. The vertical wave trace ${\rm Tr}_{\hbox{\tiny{M/B}}}(e^{it\textsf{Q}^{1/m}})$:
  9. The singularity of $\textsf{U}(t,m,m)$ at $t=0$:
  10. Cohomology classes of wave supertrace characteristic forms:
  11. Contextualising with classical Poisson summation
  12. Proof of Proposition \ref{['tmxn']}

Key Result

Theorem 1.1

There is an asymptotic singularity expansion with differential form wave invariants and ${\rm Tr}_{\hbox{\tiny M/B}}$ the vertical trace along the fibres. The coefficient of $(t- L_\kappa + i0)^{-q + j}$ is a differential form of mixed form degree for ${\mathcal{Z}}\rightarrow B$ a subfibration of the cosphere bundle $S^*_{\hbox{\tiny{M/B}}}$ along the fibres (the fibration over $B$ with fibre $

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Proposition 3.5
  • ...and 6 more