Special Session 20

Global Analysis and Geometry of semi-Riemannian Manifolds

Organizers: Giovanni Calvaruso (University of Salento, Italy), Anna Maria Candela (University of Bari “Aldo Moro”, Italy), Vivianadel Barco (State University of Campinas, Brazil), Paolo Piccione (University of São Paulo, Brazil)

MSC codes: 53C-XX

Description:
Semi-Riemannian geometries encompass several of the most active research topics in the framework of Differential Geometry and its applications. In particular, the study of Riemannian and Lorentzian manifolds investigates smooth manifolds endowed with metric structures, providing fundamental tools for modern geometry and mathematical physics. Current research in these areas explores the interplay between curvature, topology, analysis, and physical models.

In Riemannian Geometry, active research topics focus on curvature and its influence on topology and global analysis. Applications often arise in geometric analysis, topology, and mathematical physics. Some relevant topics in Riemannian Geometry are:

  • Curvature and comparison geometry
  • Global Riemannian geometry and topology of manifolds
  • Geometric flows (Ricci flow, mean curvature flow)
  • Spectral geometry and eigenvalue estimates
  • Rigidity, stability, and pinching phenomena
  • Einstein metrics and special geometric structures
  • Geodesics
  • Special Riemannian manifolds (Einstein, Kähler, Sasaki, etc.)
  • Group actions, Isometries, Homogeneous manifolds
  • Submanifold Theory
  • Symplectic and contact geometry
  • Lie groups and Lie algebras

Lorentzian Geometry provides the mathematical framework for General Relativity. Interactions with partial differential equations and mathematical relativity play a central role. Key research themes include:

  • Causal structures and global hyperbolicity
  • Geodesics and geodesic completeness
  • Singularities and singularity theorems
  • Lorentzian comparison geometry
  • Geometry of spacetime models in General Relativity
  • Energy conditions and curvature bounds
  • Interactions with partial differential equations

Together, these fields explore deep connections between Geometry, Analysis, and Physics, offering insights into both abstract mathematical structures and the geometry of spacetime. In both frameworks, the study of the geometry of submanifolds plays an important role. Interdisciplinary directions where Riemannian and Lorentzian geometries find deep remarkable applications are:

  • Mathematical Relativity
  • Topology and Global Analysis
  • Theoretical Physics and Cosmology