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Gromoll--Meyer's actions and the geometry of (exotic) spacetimes

Leonardo F. Cavenaghi, Lino Grama

Abstract

Since the advent of new pairwise non-diffeomorphic structures on smooth manifolds, it has been questioned whether two topologically identical manifolds could admit different geometries. Not surprisingly, physicists have wondered whether a smooth structure assumption different from some classical known models could produce different physical meanings. In this paper, we inaugurate a very computational manner to produce physical models on classical and exotic spheres that can be built equivariantly, such as the classical Gromoll--Meyer exotic spheres. As first applications, we produce Lorentzian metrics on homeomorphic but not diffeomorphic manifolds that enjoy the same physical properties, such as geodesic completeness, positive Ricci curvature, and compatible time orientation. These constructions can be pulled back to higher models, such as exotic ten spheres bounding spin manifolds, to be approached in forthcoming papers.

Gromoll--Meyer's actions and the geometry of (exotic) spacetimes

Abstract

Since the advent of new pairwise non-diffeomorphic structures on smooth manifolds, it has been questioned whether two topologically identical manifolds could admit different geometries. Not surprisingly, physicists have wondered whether a smooth structure assumption different from some classical known models could produce different physical meanings. In this paper, we inaugurate a very computational manner to produce physical models on classical and exotic spheres that can be built equivariantly, such as the classical Gromoll--Meyer exotic spheres. As first applications, we produce Lorentzian metrics on homeomorphic but not diffeomorphic manifolds that enjoy the same physical properties, such as geodesic completeness, positive Ricci curvature, and compatible time orientation. These constructions can be pulled back to higher models, such as exotic ten spheres bounding spin manifolds, to be approached in forthcoming papers.
Paper Structure (21 sections, 25 theorems, 115 equations)

This paper contains 21 sections, 25 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Fixing the notation and a minimum of Lorentzian geometry
  3. Recordatory aspects of Linear algebra of indefinite metrics
  4. The basics of semi-Riemannian Geometry
  5. Temporal orientability
  6. Computational aspects
  7. General Notation
  8. A Cheeger approach to the Minkowski space and a first compact model
  9. The standard Minkowski spacetime
  10. $\mathrm{SU}(2)$ as a spacetime and obstructions
  11. The de Sitter spacetime and the universal covering in a $\star$-diagram
  12. A semi-Riemannian metric on the standard sphere vs. on the Gromoll and Meyer exotic sphere
  13. An induction on the construction of the Gromoll--Meyer exotic sphere
  14. Lorentzian metric via $\star$-diagrams
  15. Explicit (almost) Lorentzian metrics on homotopy spheres
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $M'\leftarrow P \rightarrow M$ be a star diagram with $\mathrm{S}^1$ as its structure group. Then there is a Lorentzian metric $\textsl{g}_{-r^2}$ in the principal stratum of $M^{reg}$ of the $\star$-action $M$, and a $\textsl{g}'_{-r^2}$ Lorentzian metric in the principal stratum ${M'}^{reg}$ o

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 1
  • Definition 2
  • Proposition 2.1
  • Definition 3
  • Proposition 2.2
  • Definition 4
  • ...and 32 more