Alphabeta MathRiemannian Metrics Length Distance and Volume
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- 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
- Tensor Products of Modules
Differential Geometry
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Measurable Densities and Radon Volume on Manifolds
- Rank Theorems and Embedded Submanifolds
- 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
Functional Analysis
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Radon Measures and the Riesz Markov Kakutani Theorem
- Sigma Algebras and Borel Sets
- The Lebesgue Integral and the Convergence Theorems
Partial Differential Equations
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- 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
- 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
- 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 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
- Hausdorff via the Diagonal
- 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
Riemannian metric and riemannian manifold
Proposition
Coordinate criterion for a riemannian metric
Theorem
Every smooth manifold admits a riemannian metric
Definition
Pullback of a riemannian metric as a tensor
Proposition
Pullback of a riemannian metric is riemannian exactly for immersions
Definition
Riemannian isometry and local isometry
Proposition
Riemannian isometries form a group and local isometries are local diffeomorphisms
Definition
Conformal equivalence of riemannian metrics
Definition
Pointwise norm and angle from a riemannian metric
Definition
Musical isomorphisms
Theorem
The musical maps are smooth inverse bundle isomorphisms
Definition
Riemannian gradient
Proposition
The gradient is characterized by inner products
Proposition
Riemannian metrics induce metrics on dual tensor and exterior bundles
Definition
Piecewise c one curve on a manifold
Definition
Riemannian speed and length
Lemma
Riemannian length is independent of piecewise c one subdivision
Theorem
Riemannian length is invariant under orientation preserving piecewise c one reparametrization
Proposition
Length is additive under concatenation and invariant under reversal
Lemma
Local comparison of a riemannian metric with the euclidean metric
Lemma
Any two points in a connected smooth manifold can be joined by a piecewise c one curve
Definition
Riemannian distance on a connected manifold
Theorem
Riemannian distance is a metric
Theorem
The riemannian distance topology is the manifold topology
Proposition
Length dominates endpoint distance
Proposition
A smooth map with pointwise operator norm at most c is c lipschitz for riemannian distance
Corollary
Riemannian isometries preserve length and distance
Definition
Extended riemannian distance on a disconnected manifold
Definition
Distance from a point to a subset
Proposition
Distance to a nonempty subset is one lipschitz
Definition
Riemannian volume density
Lemma
The riemannian volume density is coordinate independent
Definition
Riemannian volume form on an oriented manifold
Proposition
The riemannian volume form is the unique positive unit top form
Definition
Riemannian volume of a compactly supported smooth density
Proposition
Riemannian volume is the radon measure of the riemannian density
Definition
Riemannian divergence
Proposition
Coordinate formula for riemannian divergence
Theorem
Riemannian divergence theorem
Definition
Riemannian hodge star
Theorem
Hodge star is a smooth bundle isomorphism
Proposition
Hodge star squared sign
Proposition
Riemannian inner product of compactly supported forms
Remark
The codifferential and hodge theory
False statement
The pullback of a riemannian metric by every smooth map is a riemannian metric
False statement
Every riemannian manifold has finite distance between points in different components
False statement
Riemannian distance is defined by the length of a unique shortest curve
False statement
The riemannian volume form exists on every riemannian manifold
False statement
The distance function is smooth on all of m times m
False statement
The hodge star is defined without an orientation
Example
The euclidean metric and its musical maps
Example