Alphabeta MathSymplectic Manifolds, Moser Stability, and Darboux–Weinstein Theory
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✓ 41 results · all verified · 35 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
Differential Geometry
- Connections Levi Civita and Parallel Transport
- Distributions Integral Manifolds and the Frobenius Theorem
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Riemannian Metrics Length Distance and Volume
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The De Rham Complex Homotopy and Mayer Vietoris
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
- Whitney Embedding Tubular Neighbourhoods and Approximation
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Symplectic vector space
Proposition
Symplectic vector spaces have even dimension
Definition
Symplectic orthogonal complement
Proposition
Symplectic double-orthogonal and dimension identities
Definition
Isotropic, coisotropic, symplectic, and Lagrangian subspaces
Theorem
Equivalent characterizations of Lagrangian subspaces
Proposition
Symplectic reduction of a coisotropic vector subspace
Proposition
Graphs of linear maps and Lagrangian relations
Definition
Symplectic form and symplectic manifold
Theorem
Nondegeneracy is equivalent to a nonvanishing top wedge
Corollary
Symplectic manifolds have a canonical orientation and volume form
Definition
Symplectomorphisms, local symplectomorphisms, and symplectic embeddings
Proposition
Products and opposites of symplectic manifolds
Definition
Isotropic, coisotropic, symplectic, and Lagrangian submanifolds
Proposition
Lagrangian submanifolds have half dimension
Definition
Tautological one-form on a cotangent bundle
Lemma
The tautological one-form is intrinsic and smooth
Theorem
The canonical cotangent two-form is symplectic
Proposition
Cotangent lifts are symplectomorphisms
Proposition
A graph of a one-form is Lagrangian exactly when the form is closed
Definition
Compatible complex structure on a symplectic vector space
Theorem
Compatible complex structures exist on symplectic vector spaces
Lemma
Positive-definite bundle endomorphisms have smooth positive square roots
Theorem
Every symplectic manifold admits a compatible almost-complex structure
Definition
Compatible almost-Kähler metric
Remark
Compatible almost-complex structures and Kähler geometry
Lemma
Moser pullback differentiation equation
Lemma
Smooth parametric primitives for a smooth exact family on a compact manifold
Theorem
Moser stability theorem
Theorem
Compact-support Moser stability on a noncompact manifold
Lemma
Relative Poincaré primitive near a submanifold
Theorem
Relative Moser theorem
Theorem
Darboux theorem
Corollary
Symplectic manifolds have no local invariants beyond dimension
Definition
Symplectic normal bundle of a symplectic submanifold
Theorem
Symplectic neighborhood theorem
Definition
Canonical symplectic model near the zero section of T^*L
Theorem
Weinstein Lagrangian neighborhood theorem
Proposition
A Lagrangian neighborhood germ is not uniquely determined
Proposition
Characteristic distribution of a coisotropic submanifold is involutive
Theorem
Local normal form near a coisotropic submanifold
False statement
Every nondegenerate two-form is symplectic
False statement
Symplectic manifolds can have odd dimension
False statement
Every half-dimensional submanifold is Lagrangian
False statement
The canonical cotangent symplectic form is dλ under the library convention
False statement
Cohomologous symplectic forms on a noncompact manifold are always isotopic
False statement