Alphabeta MathGeodesics, the Exponential Map, Completeness, and Hopf–Rinow
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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
- 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
Complex Analysis
Differential Geometry
- Connections Levi Civita and Parallel Transport
- Euclidean Ordinary Differential Equations with Smooth Dependence
- 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
- Vector Fields Flows and Lie Derivatives
Group Theory
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
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
- 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 Complex Exponential and Euler's Formula
- 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
- Covering Spaces and Lifting
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Fundamental Group of the Circle
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Remark
Boundaryless convention for geodesic flow and Hopf–Rinow
Definition
Geodesic of an affine connection
Proposition
Geodesics have constant speed for a metric-compatible connection
Proposition
Coordinate geodesic equation
Definition
Geodesic spray
Lemma
The geodesic spray is a well-defined smooth vector field on TM
Theorem
Existence uniqueness and smooth dependence of geodesics
Proposition
Affine reparametrization of a geodesic is a geodesic
Lemma
Geodesic scaling identity
Definition
Geodesically complete Riemannian manifold
Definition
Domain and exponential map of a connection
Theorem
The exponential domain is open and the exponential map is smooth
Proposition
The exponential map scales geodesic time
Theorem
The differential of exp at zero is the identity
Lemma
Choice-free smooth inverse function theorem in Euclidean space
Theorem
Existence of normal neighborhoods
Definition
Normal neighborhood and normal coordinate chart
Proposition
Properties of normal coordinates at the center
Definition
Injectivity radius at a point and of a manifold
Proposition
Injectivity radius at each point is positive
Definition
Smooth variation and variation field of a curve
Definition
Energy of a piecewise smooth curve
Proposition
Length-energy inequality and constant-speed equality case
Theorem
First variation formula for energy
Corollary
Geodesics are exactly critical points of energy with fixed endpoints
Theorem
First variation formula for length
Theorem
Gauss lemma
Corollary
Polar form of the metric in normal coordinates
Theorem
Radial geodesics minimize length in a normal neighborhood
Corollary
Local formula for distance from the centre of a normal neighbourhood
Corollary
Sufficiently short geodesic segments are uniquely minimizing
Theorem
Existence of geodesically convex neighborhoods
Theorem
Length minimizers are constant-speed geodesics up to reparametrization
Lemma
Geodesics continue while velocity lifts remain compact
Lemma
A finite endpoint of a maximal unit-speed geodesic produces a Cauchy curve
Theorem
Metric completeness implies geodesic completeness
Lemma
Radial geodesics from one point reach every point under global exponential domain
Theorem
Hopf–Rinow theorem
Corollary
Complete connected Riemannian manifolds are proper length spaces
Corollary
Compact Riemannian manifolds are geodesically complete
Corollary
Closed embedded submanifolds of complete Riemannian manifolds are complete
Lemma
Local isometries send geodesics to geodesics
Corollary
A local isometry from a complete connected manifold has geodesically complete target image
Proposition
A Riemannian product is complete iff each factor is complete
Proposition
Incompleteness is finite-time geodesic escape
False statement
Every affinely reparametrized geodesic remains unit speed
False statement
The exponential map is always defined on all of TM
False statement
Normal coordinates make the metric Euclidean throughout the chart
False statement
Every geodesic segment is globally length minimizing
False statement
Any two points admit a minimizing geodesic
False statement
Geodesic completeness means compactness
Theorem
The universal cover of a closed hyperbolic surface is the hyperbolic plane with geometric deck action
Example
Great circles as round-sphere geodesics
Example
Hopf–Rinow on a flat cylinder
Example
Straight lines as Euclidean geodesics
Sources
- Ved Datar, Lectures on Riemannian Geometry, Lectures 15.1 and 19.2, pp.113 and 141–144
- Ved Datar, Lectures on Riemannian Geometry, Definition 15.1.1, p.113
- Ved Datar, Lectures on Riemannian Geometry, Remark 15.1.2, pp.113–114
- Ved Datar, Lectures on Riemannian Geometry, proof of Theorem 15.2.1, pp.115–117
- Ved Datar, Lectures on Riemannian Geometry, Theorem 15.2.1 and Remark 15.2.4, pp.115–117
- Ved Datar, Lectures on Riemannian Geometry, Corollary 15.2.2, pp.115–116
- Ved Datar, Lectures on Riemannian Geometry, Definition 19.2.1, p.141
- Ved Datar, Lectures on Riemannian Geometry, Definition 17.1.2, pp.127--128
- Ved Datar, Lectures on Riemannian Geometry, Proposition 17.1.4(1), p.128
- Ved Datar, Lectures on Riemannian Geometry, Proposition 17.1.4(2)--(3), p.128
- Ved Datar, Lectures on Riemannian Geometry, Proposition 17.1.4(2), p.128
- J. Lebl, Basic Analysis II, Theorem 8.5.1 and higher-regularity discussion
- Ved Datar, Lectures on Riemannian Geometry, Corollary 17.1.7, p.130
- Ved Datar, Lectures on Riemannian Geometry, Definition 17.2.1, p.130
- Ved Datar, Lectures on Riemannian Geometry, Proposition 17.2.2, pp.130--131
- Ved Datar, Lectures on Riemannian Geometry, Definition 23.3.3, pp.171--172
- Ved Datar, Lectures on Riemannian Geometry, Corollary 17.1.7 and Definition 23.3.3, pp.130 and 171--172
- Ved Datar, Lectures on Riemannian Geometry, Definitions 16.2.1--16.2.2, pp.121--122
- Ved Datar, Lectures on Riemannian Geometry, Definition 16.1.3, p.120, with the conventional factor one-half adopted here
- Ved Datar, Lectures on Riemannian Geometry, discussion after Definition 16.1.3, p.120
- Ved Datar, Lectures on Riemannian Geometry, Theorem 16.3.1 and its proof, pp.123--124
- Ved Datar, Lectures on Riemannian Geometry, Corollary 16.4.1(1) and proof Step 1, pp.124--125
- Ved Datar, Lectures on Riemannian Geometry, Theorem 16.3.1 and length calculation, pp.123--124
- Ved Datar, Lectures on Riemannian Geometry, Lemma 18.1.2 and complete proof, pp.134--135
- Ved Datar, Lectures on Riemannian Geometry, Corollary 18.1.3(1) and proof, pp.135--136
- Ved Datar, Lectures on Riemannian Geometry, Corollary 18.1.3(2)--(3) and proof, pp.135--137
- Ved Datar, Lectures on Riemannian Geometry, Proposition 19.1.2 and preceding minimality argument, pp.137 and 140
- Ved Datar, Lectures on Riemannian Geometry, Corollary 18.1.3 and proof, pp.135--137
- Ved Datar, Lectures on Riemannian Geometry, Theorem 18.0.1 and proof, pp.133--137
- Roland Steinbauer, Riemannian Geometry, Theorem 2.2.7 and proof, printed pp.49--50 (PDF pp.52--53)
- Ved Datar, Lectures on Riemannian Geometry, Theorem 18.0.1 and Corollary 18.1.3, pp.133--137
- Roland Steinbauer, Riemannian Geometry, Remark 2.3.10 and Corollary 2.3.11, printed pp.59--60
- Ved Datar, Lectures on Riemannian Geometry, proof of Proposition 20.2.2, p.151
- Ben Andrews, Geodesics and Completeness, proof of Theorem 11.5.1, printed pp.106–107
- Ben Andrews, Geodesics and Completeness, proof of Theorem 11.5.1, printed p.106
- Ved Datar, Lectures on Riemannian Geometry, Theorem 19.2.1, implication (1) implies (2), pp.141–142
- Ben Andrews, Geodesics and Completeness, Theorem 11.5.1, implication (1) implies (2), printed pp.106–107
- Ved Datar, Lectures on Riemannian Geometry, Theorem 19.2.1, implication (3) to (5), pp.142--144
- Ben Andrews, Geodesics and Completeness, Theorem 11.5.1, implication (3) to (*p), printed pp.107--108
- Ved Datar, Lectures on Riemannian Geometry, Theorem 19.2.1 and proof, pp.141--144
- Ben Andrews, Geodesics and Completeness, Theorem 11.5.1 and proof, printed pp.106--108
- Ved Datar, Lectures on Riemannian Geometry, Theorem 19.2.1 and compact-manifold consequence, pp.141--144
- Ved Datar, Lectures on Riemannian Geometry, Section 19.1, pp.139--141
- Ved Datar, Lectures on Riemannian Geometry, Chapter 20, p. 148
- Ved Datar, Lectures on Riemannian Geometry, Theorem 20.1.1 and geodesic-lifting step, pp.147--149
- Ved Datar, Lectures on Riemannian Geometry, Example 8.2.8, p.49
- Ben Andrews, Geodesics and Completeness, Theorem 11.5.1 and proof, printed pp.106--107
- Ved Datar, Lectures on Riemannian Geometry, Remark 15.1.2 and Example 15.1.3, pp. 113--114
- Ved Datar, Lectures on Riemannian Geometry, Definition 15.1.1, Example 15.1.3, Definition 17.1.2, and Theorem 19.2.1, pp. 113--114, 127--128, and 141--144
- Ved Datar, Lectures on Riemannian Geometry, Example 17.1.3, Definition 17.2.1 and Proposition 17.2.2, pp. 128, 130--131
- Ved Datar, Lectures on Riemannian Geometry, Corollary 16.4.5 and following counterexample, p. 126; Corollary 18.1.3 and proof, pp. 135--136
- Ben Andrews, Geodesics and Completeness, Theorem 11.5.1 and proof, printed pp. 106--108
- Ved Datar, Lectures on Riemannian Geometry, Corollary 24.0.2 and §§24.1, 24.3, printed pp.174–179; source PDF read 2026-09-23
- Ved Datar, Lectures on Riemannian Geometry, Proposition 15.3.1, pp. 117--118
- Ved Datar, Lectures on Riemannian Geometry, Example 15.1.3, p. 114