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Universal Bundles of Metrics and Polymetrics: A Generalization

Shouvik Datta Choudhury

TL;DR

This work generalizes metric geometry by organizing all (pseudo-)Riemannian data into a universal bundle of metrics built from open cones $\mathcal{G}^{p,q}$ and their finite polymetric products. It develops a cohesive analytic and topological framework: Fréchet/Sobolev structures for metric spaces, Ebin–Palais slices for moduli, and a robust families index theory (via spin$^c$, Hodge, and signature operators) that is compatible with equivariant, orbifold/groupoid, and foliated settings. The authors extend index-theoretic results to noncompact and coarse geometries using Roe algebras and $KK$-theory, and illustrate applications to conformal/densitized variations, coupled polymetric systems, and Finsler/sub-Riemannian layers, showing that metric choices contribute only to transgression terms. The framework yields natural additive properties of indices under sums and products, naturality under pullbacks, and invariance under polymetric deformations, thereby providing a unified scaffold linking deformation theory, moduli spaces, and large-scale geometry with explicit local and global index formulas. Overall, the paper offers a comprehensive, slice-based, polymetric perspective that unifies classical and modern index theory across compact and noncompact, equivariant, and foliated contexts with concrete computational tools and transgression phenomena.

Abstract

We formalize the ``metric bundle'' viewpoint by defining, for any smooth $n$--manifold $M$, the open fiberwise cones $\mathcal{G}^{p,q}\subset S^2\Tstar M$ of nondegenerate symmetric bilinear forms with fixed signature $(p,q)$, and we package \emph{multi\-metric} (``polymetric'') geometries as sections of finite products of such cones. This framework subsumes Riemannian and pseudo-Riemannian metrics, and admits clean extensions to conformal, densitized, Finsler, and sub-Riemannian structures. It also interfaces correctly with families index theory (Atiyah--Singer), equivariant/groupoid settings, and coarse/KK-theory. On compact $M$ the Riemannian space of metrics is convex (hence contractible), giving a transparent base for moduli and deformation theory

Universal Bundles of Metrics and Polymetrics: A Generalization

TL;DR

This work generalizes metric geometry by organizing all (pseudo-)Riemannian data into a universal bundle of metrics built from open cones and their finite polymetric products. It develops a cohesive analytic and topological framework: Fréchet/Sobolev structures for metric spaces, Ebin–Palais slices for moduli, and a robust families index theory (via spin, Hodge, and signature operators) that is compatible with equivariant, orbifold/groupoid, and foliated settings. The authors extend index-theoretic results to noncompact and coarse geometries using Roe algebras and -theory, and illustrate applications to conformal/densitized variations, coupled polymetric systems, and Finsler/sub-Riemannian layers, showing that metric choices contribute only to transgression terms. The framework yields natural additive properties of indices under sums and products, naturality under pullbacks, and invariance under polymetric deformations, thereby providing a unified scaffold linking deformation theory, moduli spaces, and large-scale geometry with explicit local and global index formulas. Overall, the paper offers a comprehensive, slice-based, polymetric perspective that unifies classical and modern index theory across compact and noncompact, equivariant, and foliated contexts with concrete computational tools and transgression phenomena.

Abstract

We formalize the ``metric bundle'' viewpoint by defining, for any smooth --manifold , the open fiberwise cones of nondegenerate symmetric bilinear forms with fixed signature , and we package \emph{multi\-metric} (``polymetric'') geometries as sections of finite products of such cones. This framework subsumes Riemannian and pseudo-Riemannian metrics, and admits clean extensions to conformal, densitized, Finsler, and sub-Riemannian structures. It also interfaces correctly with families index theory (Atiyah--Singer), equivariant/groupoid settings, and coarse/KK-theory. On compact the Riemannian space of metrics is convex (hence contractible), giving a transparent base for moduli and deformation theory
Paper Structure (64 sections, 41 theorems, 57 equations)

This paper contains 64 sections, 41 theorems, 57 equations.

Table of Contents

  1. Introduction and the foundational objects
  2. Topology and analysis on spaces of metrics and polymetrics
  3. Open metric cones and stability of signature
  4. Smooth algebra of basic operations
  5. Polymetrics as products and induced function spaces
  6. Constraint subspaces and basic variational examples
  7. Smooth dependence of geodesics and exponential maps
  8. Action of the diffeomorphism group
  9. Moduli of metrics and polymetrics: slices, orbits, and deformations
  10. Orbit geometry and the $L^2$ Riemannian structure on Met(M)
  11. Ebin--Palais slice and local model of the quotient
  12. Elliptic deformation theory for curvature-constrained subsets
  13. Polymetric moduli and product slices
  14. Quotient stack viewpoint
  15. Families index for geometric operators
  16. ...and 49 more sections

Key Result

Proposition 1.3

$\mathcal{G}^{p,q}\subset S^2\mathrm{T}^* M$ is an open subbundle. For $(p,q)=(n,0)$ and $g_0,g_1\in \mathrm{Met}(M)$, the convex path $g_t:=(1-t)g_0+t g_1$ lies in $\mathrm{Met}(M)$ for all $t\in[0,1]$. Hence $\mathrm{Met}(M)$ is convex (in particular, contractible) in the Fréchet topology of $\Gam

Theorems & Definitions (113)

  • Definition 1.1: Universal bundle of metrics of fixed signature
  • Remark 1.2: Associated-bundle description
  • Proposition 1.3: Openness and convexity (Riemannian case)
  • Definition 1.4: Polymetrics: rigorous "multi-norms"
  • Remark 1.5: Levi--Civita layer and beyond
  • Lemma 2.1: Openness of the signature locus
  • proof
  • Proposition 2.2: Stability under small perturbations
  • Proposition 2.2: Stability under small perturbations
  • proof
  • ...and 103 more