58Axx Differentiable manifolds and differential forms
This subtopic introduces the core ideas in differentiable manifolds and differential forms, 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.
Specific topics
58A03 Topos-theoretic approach to differentiable manifolds
Overview
Topos-theoretic approach to differentiable manifolds. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A05 Differentiable manifolds, foundations
Overview
Differentiable manifolds, foundations. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A07 Real-analytic and Nash manifolds
Overview
Real-analytic and Nash manifolds. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A10 Differential forms in global analysis
Overview
Differential forms in global analysis. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A12 de Rham theory in global analysis
Overview
de Rham theory in global analysis. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A14 Hodge theory in global analysis
Overview
Hodge theory in global analysis. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A15 Exterior differential systems in global analysis
Overview
Exterior differential systems in global analysis. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A17 Pfaffian systems
Overview
Pfaffian systems. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A20 Jets in global analysis
Overview
Jets in global analysis. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A25 Currents in global analysis
Overview
Currents in global analysis. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A30 Vector distributions (subbundles of the tangent bundles)
Overview
Vector distributions (subbundles of the tangent bundles). This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A32 Natural bundles
Overview
Natural bundles. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A35 Stratified sets
Overview
Stratified sets. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A40 Differential spaces
Overview
Differential spaces. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks
58A50 Supermanifolds and graded manifolds
Overview
Supermanifolds and graded manifolds. This topic develops differentiable manifolds and differential forms, including local/global smooth structures and integration frameworks.
Related Wikipedia Page
Wikipedia: Differentiable manifold
Useful Links
Key Ideas
- smooth atlas and tangent structure
- differential forms and exterior calculus
- integration and cohomological tools on manifolds
Typical Uses
Used as a foundational language for modern geometry, topology, and geometric analysis.
Applications
- Differential geometry
- Geometric mechanics
- Global analysis prerequisites
References
Recommended Textbooks