A structured visual guide to the major mathematical areas and their relationships.
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This subtopic introduces the core ideas in global differential geometry, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Connections (general theory). This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills). This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Differential geometric aspects of gerbes and differential characters. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
$G$-structures. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Foliations (differential geometric aspects). This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
General geometric structures on manifolds. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Sub-Riemannian geometry. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Global Riemannian geometry, including pinching. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Methods of global Riemannian geometry, including PDE methods. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Geodesics in global differential geometry. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Global geometric and topological methods; gauge theory. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Rigidity results in global differential geometry. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Special Riemannian manifolds (Einstein, Sasakian, etc.). This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Hyper-Kähler and quaternionic Kähler geometry, special geometry. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Spin and Spin$^c$ geometry. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Twistor methods in differential geometry. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Issues of holonomy in differential geometry. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Differential geometry of homogeneous manifolds. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Differential geometry of symmetric spaces. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Calibrations and calibrated geometries. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Global submanifolds. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Differential geometry of immersions (minimal, prescribed curvature, tight, etc.). This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Differential geometric aspects of harmonic maps. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Geometric evolution equations (e.g. Ricci flow). This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Global surface theory (convex surfaces à la A.D. Alexandrov). This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Global differential geometry of Hermitian and Kählerian manifolds. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Other complex differential geometry. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Global differential geometry of Finsler spaces and generalizations. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Integral geometry; geometric probability. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Direct methods ($G$-spaces of Busemann, etc.). This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Geometric orders, order geometry. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.
Applications of global differential geometry to the sciences. This topic is part of global differential geometry, emphasizing manifold-scale invariants, curvature-driven phenomena, and geometric analysis methods.
Wikipedia: Riemannian geometry
Used to study manifold-wide geometric behavior and relationships between topology, curvature, and PDE.