Alphabeta MathConnections Levi Civita and Parallel Transport
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
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
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- 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
Differential Geometry
- Euclidean Ordinary Differential Equations with Smooth Dependence
- 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
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Exterior Powers, Orientation and Hodge Duality
- 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
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- 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
- 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 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
Connection on a smooth vector bundle
Definition
Covariant derivative of a section in a vector field direction
Proposition
Connection laws in directional form
Lemma
A bundle connection is local and restricts to open sets
Definition
Connection one form in a local frame
Proposition
Local coordinate formula for a bundle connection
Theorem
Connection one form transformation law
Theorem
Local connection forms glue exactly when they obey the transformation law
Proposition
The difference of two connections is an endomorphism valued one form
Proposition
Adding an endomorphism valued one form to a connection gives a connection
Theorem
Every smooth vector bundle admits a connection
Definition
Pullback connection
Theorem
Pullback connection is well defined and functorial
Definition
Dual connection
Lemma
Finite tensor products of smooth vector bundles
Definition
Product connection on tensor and hom bundles
Proposition
Induced connections commute with contraction and permutation
Proposition
Induced connection on exterior powers is a degree zero derivation
Definition
Vector field and section along a smooth curve
Definition
Covariant derivative along a curve
Theorem
Covariant derivative along a curve is independent of frame and extension
Proposition
Local frame formula for covariant differentiation along a curve
Definition
Parallel section along a curve
Theorem
Existence and uniqueness of parallel sections
Definition
Parallel transport along a piecewise smooth curve
Theorem
Parallel transport is a linear isomorphism
Proposition
Parallel transport under reparametrization reversal and concatenation
Proposition
Pullback connections intertwine parallel transport
Definition
Metric compatible connection on a riemannian vector bundle
Proposition
A connection is metric compatible iff parallel transport is isometric
Definition
Affine connection on a smooth manifold
Definition
Christoffel symbols of an affine connection
Theorem
Christoffel symbol transformation law
Definition
Torsion tensor of an affine connection
Lemma
Torsion is c infinity bilinear and skew symmetric
Proposition
Torsion free is equivalent to symmetric christoffel symbols in coordinate frames
Definition
Levi civita connection
Lemma
Koszul formula is necessary for a levi civita connection
Theorem
The koszul formula defines an affine connection
Theorem
Fundamental theorem of riemannian geometry
Proposition
Christoffel formula for the levi civita connection
Proposition
Levi civita parallel transport preserves lengths angles and volume
Proposition
Levi civita connection commutes with musical isomorphisms
Proposition
Gradient hessian and divergence connection formulas
Proposition
The riemannian hessian is symmetric
Remark
Holonomy of a connection
False statement
A connection is c infinity linear in the section being differentiated
False statement
Christoffel symbols are components of a tensor
False statement
Parallel transport depends only on the endpoints of a curve
False statement
Every affine connection is the levi civita connection of a riemannian metric
False statement
Torsion free means curvature free
False statement
Every connection on a riemannian vector bundle is metric compatible
Example
The euclidean levi civita connection
Sources
- Ved Datar, Lectures on Riemannian Geometry, section 5.1
- Brian Conrad, Linear algebra operations on vector bundles, section 3, pp.7–9; local Hom charts, with a choice-free second-countability proof supplied here
- Ved Datar, Lectures on Riemannian Geometry, section 4.1
- Ved Datar, Lectures on Riemannian Geometry, Lemma 5.1.1 and Corollary 5.1.2
- Ved Datar, Lectures on Riemannian Geometry