Alphabeta MathRiemann Curvature and Riemannian Submanifolds
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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
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- 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
- Tensor Products of Modules
- 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
- Integration of Forms and the General Stokes Theorem
- 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 Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
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
- Matrix Differentiation and First-order Spectral Perturbation
- Matrix Norms, Condition Numbers and Numerical Stability
- 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
Measure Theory
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
- Fundamental Trigonometric Identities
- Improper and Parameter-Dependent Multiple Integrals
- 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
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Divergence Theorem and Classical Stokes
- 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
Curvature of an affine connection
Lemma
Curvature is C-infinity-linear in all three vector fields
Theorem
Curvature is a type (1,3) tensor
Proposition
Curvature is skew in its first two arguments
Proposition
Coordinate formula for the curvature tensor
Definition
Curvature of a vector-bundle connection
Proposition
Vector-bundle curvature is an endomorphism-valued two-form
Theorem
Curvature two-form structure equation
Theorem
Second Bianchi identity for a bundle connection
Proposition
Flat connections have locally path-independent parallel transport on a coordinate ball
Theorem
A flat connection admits local parallel frames
Definition
Riemann curvature four-tensor
Theorem
First Bianchi identity
Theorem
Algebraic symmetries of the Riemann tensor
Theorem
Differential second Bianchi identity
Definition
Sectional curvature
Lemma
Sectional curvature is independent of the basis of the plane
Theorem
Sectional curvatures determine the Riemann tensor
Definition
Constant sectional curvature and space form
Proposition
Curvature tensor of constant sectional curvature
Definition
Ricci curvature
Lemma
Ricci curvature is symmetric and basis independent
Definition
Scalar curvature
Proposition
Scalar curvature is twice the sum of sectional curvatures of orthonormal coordinate planes
Definition
Kulkarni–Nomizu product, trace-free Ricci tensor, and Weyl curvature
Proposition
Ricci decomposition of the Riemann tensor in dimension at least three
Theorem
Contracted second Bianchi identity
Theorem
Schur's lemma for pointwise constant sectional curvature
Theorem
A Riemannian manifold is flat iff it is locally isometric to Euclidean space
Definition
Tangential and normal projections along a Riemannian submanifold
Definition
Induced connection and second fundamental form
Theorem
The induced connection is Levi–Civita
Lemma
The second fundamental form is a symmetric normal-bundle-valued two-tensor
Definition
Normal connection
Definition
Shape operator
Theorem
Weingarten equation and adjointness of the shape operator
Theorem
Gauss equation for a Riemannian submanifold
Theorem
Codazzi equation for a Riemannian submanifold
Theorem
Ricci equation for the normal connection
Definition
Totally geodesic submanifold
Theorem
Equivalent characterizations of a totally geodesic submanifold
Definition
Principal curvatures, Gaussian curvature, and mean curvature of an oriented hypersurface
Proposition
Euclidean hypersurface sectional curvature from principal curvatures
Theorem
Gauss’s Theorema Egregium
Definition
Mean curvature vector
Proposition
First variation of volume for a normal variation
Remark
Mean curvature and minimal submanifolds
False statement
Curvature is obtained by commuting two covariant derivatives without a bracket correction
False statement
Christoffel symbols vanishing at one point implies curvature vanishes there
False statement
Sectional curvature depends on an ordered basis of the plane
False statement
Ricci curvature and scalar curvature determine the full Riemann tensor in every dimension
False statement
The second fundamental form is intrinsic to the abstract Riemannian manifold
False statement
Zero mean curvature implies a submanifold is totally geodesic
Example
The round sphere has positive constant sectional curvature
Example