51Mxx Real and complex geometry
This subtopic introduces the core ideas in real and complex 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.
Specific topics
51M04 Elementary problems in Euclidean geometries
Overview
This topic includes classical Euclidean geometry problems focused on constructions, incidences, and metric relations in elementary settings.
Related Wikipedia Page
Wikipedia: Euclidean geometry
Useful Links
Key Ideas
- classical constructions
- angle and length relations
- geometric problem solving
Typical Uses
Central for olympiad-style and foundational geometry work.
Applications
- Mathematics education
- Contest mathematics
- Foundational geometry
References
Recommended Textbooks
51M05 Euclidean geometries (general) and generalizations
Overview
General Euclidean geometry studies axiomatic, transformational, and metric formulations and their natural extensions.
Related Wikipedia Page
Wikipedia: Euclidean geometry
Useful Links
Key Ideas
- axiomatizations
- transformations
- metric structures
Typical Uses
Provides the base framework for many geometric theories and applications.
Applications
- Computer graphics
- Robotics
- Geometric modeling
References
Recommended Textbooks
51M09 Elementary problems in hyperbolic and elliptic geometries
Overview
This area focuses on classical problem solving in non-Euclidean geometries, especially triangle and circle configurations.
Related Wikipedia Page
Wikipedia: Hyperbolic geometry
Useful Links
Key Ideas
- non-Euclidean constructions
- triangle geometry
- curvature effects
Typical Uses
Useful for understanding how classical geometric statements change with curvature.
Applications
- Geometry education
- Geometric visualization
- Theoretical models
References
Recommended Textbooks
51M10 Hyperbolic and elliptic geometries (general) and generalizations
Overview
General non-Euclidean geometry studies spaces of constant nonzero curvature and their incidence, metric, and transformation properties.
Related Wikipedia Page
Wikipedia: Non-Euclidean geometry
Useful Links
Key Ideas
- constant curvature models
- geodesics
- isometry groups
Typical Uses
Provides foundational models for geometry, topology, and mathematical physics.
Applications
- Relativity
- Geometric topology
- Group actions
References
Recommended Textbooks
51M15 Geometric constructions in real and complex geometry
Overview
Geometric constructions study what can be built from prescribed operations in real and complex settings, both classical and modern.
Related Wikipedia Page
Wikipedia: Geometric construction
Useful Links
Key Ideas
- constructibility
- algebraic solvability
- real and complex constraints
Typical Uses
Useful for linking algebraic field extensions with classical geometry.
Applications
- Computer-aided design
- Education
- Algebra-geometry interactions
References
Recommended Textbooks
51M16 Inequalities and extremum problems in real and complex geometry
Overview
This area studies optimal bounds and extremal configurations in Euclidean, hyperbolic, and complex geometric contexts.
Related Wikipedia Page
Wikipedia: Geometric inequality
Useful Links
Key Ideas
- extremal geometry
- sharp inequalities
- optimization in geometric settings
Typical Uses
Important for proving structural limits in geometric design and analysis.
Applications
- Convex optimization
- Computational geometry
- Mathematical olympiads
References
Recommended Textbooks
51M20 Polyhedra and polytopes; regular figures, division of spaces
Overview
This subfield analyzes polyhedral structures, regular tessellations, and geometric decompositions of Euclidean and related spaces.
Related Wikipedia Page
Wikipedia: Polytope
Useful Links
Key Ideas
- polyhedral combinatorics
- regular figures
- space partitioning
Typical Uses
Core for understanding discrete and computational geometry of shapes.
Applications
- Optimization
- Crystallography
- Computer graphics
References
Recommended Textbooks
51M25 Length, area and volume in real and complex geometry
Overview
Measures such as length, area, and volume are studied in classical and generalized geometric contexts, including integral-geometric viewpoints.
Related Wikipedia Page
Wikipedia: Measure (mathematics)
Useful Links
Key Ideas
- geometric measure quantities
- isoperimetric themes
- integral methods
Typical Uses
Essential for quantitative geometry and comparison theorems.
Applications
- Shape analysis
- Physics models
- Geometric probability
References
Recommended Textbooks
51M30 Line geometries and their generalizations
Overview
Line geometry treats families of lines as primary geometric objects, with links to projective, algebraic, and differential techniques.
Related Wikipedia Page
Wikipedia: Line geometry
Useful Links
Key Ideas
- line complexes
- projective line systems
- geometric correspondences
Typical Uses
Useful in kinematics, projective methods, and geometric modeling.
Applications
- Robot motion
- Computer vision
- Projective geometry
References
Recommended Textbooks
51M35 Synthetic treatment of fundamental manifolds in projective geometries
Overview
This topic develops manifold-like structures in projective geometry through synthetic axioms and incidence relations.
Related Wikipedia Page
Wikipedia: Projective geometry
Useful Links
Key Ideas
- synthetic manifolds
- projective incidence
- axiomatic manifolds
Typical Uses
Useful for comparing synthetic and analytic descriptions of projective spaces.
Applications
- Foundations of geometry
- Transformation groups
- Differential projective geometry
References
Recommended Textbooks