Alphabeta MathThe Gauss Bonnet Theorem for Riemannian Surfaces
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
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- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Modules, Exact Sequences, Projective and Injective Modules
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Algebraic Topology
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Category Theory
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Combinatorics
Differential Geometry
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Linear Algebra
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Real Analysis
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- Topology of ℝ
Set Theory
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Topology
- Classification of Compact Connected Surfaces
- Compactness
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- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Oriented Riemannian surface and positive quarter-turn
Definition
Connection one-form of an oriented orthonormal frame
Theorem
Rotation law for the surface connection form
Theorem
Gaussian curvature structure equation
Definition
Signed geodesic curvature
Proposition
Signs of geodesic curvature under reversals
Proposition
Tangent-angle formula for geodesic curvature
Definition
Signed exterior angle at an ordinary corner
Proposition
Total turning with connection and corner terms
Definition
Rotation index of a regular closed plane curve
Theorem
Hopf turning-tangent theorem with ordinary corners
Corollary
Exterior-angle sum of a planar polygon
Definition
Regular oriented surface regions with corners
Lemma
Stokes formula for finite ordinary surface corners
Lemma
Finite planar graph disk cuts and Euler count
Lemma
Finite frameable decomposition of a regular disk region
Theorem
Local Gauss-Bonnet for a frameable disk region
Theorem
Gauss-Bonnet for a geodesic triangle
Corollary
Angle-sum comparison for geodesic triangles
Theorem
Local Gauss-Bonnet for an arbitrary disk region
Definition
Curvilinear face-to-face triangulation
Definition
Geodesic triangulation with prescribed boundary arcs
Lemma
Extending a compact surface metric across its boundary
Lemma
Uniform short-geodesic scale on a compact surface
Theorem
Finite curvilinear triangulation of a compact Riemannian surface
Lemma
Finite short-geodesic polygon cellulation from curvilinear triangles
Theorem
Finite geodesic triangulation of a compact Riemannian surface
Definition
Euler characteristic of a finitely triangulated compact surface
Lemma
Euler count under elementary subdivision
Lemma
Summing local Gauss-Bonnet over a supplied triangulation
Lemma
A curvilinear triangulation gives a finite regular CW complex
Lemma
The Gauss-Bonnet expression is independent of the metric
Theorem
Topological well-definedness of the surface Euler characteristic
Proposition
Euler characteristic under finite subcomplex gluing
Theorem
Gauss-Bonnet for compact oriented surface regions with boundary and corners
Corollary
Gauss-Bonnet with smooth boundary
Theorem
Global Gauss-Bonnet for closed oriented surfaces
Corollary
Metric independence of total Gaussian curvature
Corollary
Flat closed oriented surfaces have Euler characteristic zero
Corollary
Positive curvature forces positive Euler characteristic
Theorem
Gauss-Bonnet for closed nonorientable surfaces
Corollary
Gauss-Bonnet for a geodesic polygon
Remark
Scope of classical surface Gauss-Bonnet
False statement
Geodesic curvature need not equal ambient curve curvature
False statement
A surface connection form depends on its frame
False statement
Geodesic triangles need not have Euclidean angle sum
False statement
A boundary term is necessary
False statement
The raw cell-count triple is a surface invariant
False statement
Gauss-Bonnet alone does not classify surfaces
Sources
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 9
- Ved Datar, Lectures on Riemannian Geometry
- John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997)
- Ved Datar, Lectures on Riemannian Geometry, Lectures 1–2
- R. Diestel, Graph Theory, 6th edition, Chapter 4 preview
- Jürgen Jost, Compact Riemann Surfaces: An Introduction to Contemporary Mathematics
- Panagiotis Polymerakis, On the spectrum of differential operators under Riemannian coverings
- Emil Saucan, A Note on a Theorem of Munkres
- Jurgen Jost, Compact Riemann Surfaces: An Introduction to Contemporary Mathematics
- Emil Saucan, A Note on a Theorem of Munkres (2004), arXiv:math/0403055v2
- James R. Munkres, Elementary Differential Topology, revised edition (1966)
- Allen Hatcher, Algebraic Topology
- Chris Wendl, The Gauss-Bonnet Formula, Chapter 6, Section 6.3
- Gallier and Xu, A Guide to the Classification Theorem for Compact Surfaces