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Gauss-Bonnet Theorem for Simple Smooth Curves | Differential Geometry

Added: 2026-08-07

Deep Dive

Fundamental concepts of curves on surfaces, specifically the definition and geometric meaning of geodesic curvature.

Lección 33 - Curvatura geodésica | Geometría Diferencial de Curvas y Superficies | UNED

Lección 33 - Curvatura geodésica | Geometría Diferencial de Curvas y Superficies | UNED

670

Geodesic curvature is a measure of how a curve on a surface deviates from being a geodesic, defined as the scalar triple product of the curve's tangent vector, the curve's normal vector, and the surface's normal vector at each point; a curve is a geodesic if and only if its geodesic curvature equals zero, meaning the curve's acceleration vector is always perpendicular to the surface's tangent plane.

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Geodesic curvature

Geodesic curvature

1.2K

Geodesic curvature (kg) is a fundamental concept in differential geometry that measures how a curve deviates from being a geodesic on a surface; it is defined by the formula kg = ε^αβλ_αβη^β, where λ^α represents the tangent vector and η^β is a unit vector co-directional with the derivative of the tangent vector, and a curve is a geodesic if and only if its geodesic curvature equals zero.

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Geodesic curvature

Geodesic curvature

12.9K

Geodesic curvature is the component of a curve's curvature vector that lies in the tangent plane of a surface, representing how much the curve deviates from being a geodesic (shortest path) on that surface. It is calculated as the scalar triple product of the surface normal, the tangent vector, and the curvature vector, or equivalently as kg = (1/h)(∂t/∂u × v - ∂t/∂v × u), where h is the magnitude of the cross product of tangent vectors, and u and v are the components of the curvature vector in the tangent plane.

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Геометрия общей теории относительности Лекция 1. Медведев В.О.

Геометрия общей теории относительности Лекция 1. Медведев В.О.

2.4K

A geodesic satisfies ∇_γ' γ' = 0, meaning its tangent is parallel transported. Geodesic curvature measures deviation from geodesics. For a curve in a surface, geodesic curvature is the tangential component of acceleration. A curve has zero geodesic curvature iff it is a geodesic. The geodesic equation in coordinates is γ''^k + Γ^k_ij γ'^i γ'^j = 0. This concept extends to higher dimensions and is fundamental to understanding geodesic flow.

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Manifolds: Gauss Bonnet Theorem and geodesic triangles, 4-30-24 part 1

Manifolds: Gauss Bonnet Theorem and geodesic triangles, 4-30-24 part 1

282

This extensive section develops the theory of geodesics as the fundamental curves on surfaces analogous to straight lines. A curve is a geodesic if its acceleration vanishes (α'' = 0), requiring unit speed parameterization. For unit speed curves, geodesic curvature κg measures deviation from geodesicity via T' = κg n where n = JT is the normal vector. Positive curvature indicates tangent vectors turn toward the normal direction. The formula κg = dφ/ds + ω¹²(β') relates curvature to angle functions and connection forms. A regular curve is a geodesic if and only if κg = 0 everywhere. Curves with zero geodesic curvature but non-unit speed are pre-geodesics, becoming geodesics under unit speed reparameterization.

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Deep Dive

An understanding of Gaussian curvature of a surface and how it is calculated using the first and second fundamental forms.

가우스 곡률 2/2 완결편, Gaussian Curvature, First Fundamental Form, Second Fundamental Form

가우스 곡률 2/2 완결편, Gaussian Curvature, First Fundamental Form, Second Fundamental Form

626

Gaussian curvature (K) is an intrinsic property of a surface that measures how the surface bends in space, defined as the ratio of the second fundamental form to the first fundamental form: K = (eg - f²)/(EG - F²), where E, F, G are coefficients of the first fundamental form (measuring surface distances) and e, f, g are coefficients of the second fundamental form (measuring surface bending). For a sphere of radius r, K = 1/r², demonstrating that Gaussian curvature is constant across the surface. The curvature can be understood physically as the rate at which the normal vector rotates per unit surface area, connecting it to the concept of normal acceleration in physics.

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Differential Geometry - Claudio Arezzo - Lecture 07

Differential Geometry - Claudio Arezzo - Lecture 07

18.8K

The first fundamental form I_P(V, W) = ⟨V, W⟩ is the restriction of the standard inner product to the tangent space, measuring intrinsic geometry. The second fundamental form II_P(V, W) = -⟨dN_P(V), W⟩ measures extrinsic curvature. Since dN_P is symmetric, it has real eigenvalues called principal curvatures k₁ and k₂. The Gaussian curvature K = k₁k₂ is the determinant and measures intrinsic curvature. The mean curvature H = (k₁ + k₂)/2 is the average. These are the only independent invariants of a symmetric endomorphism in dimension 2.

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미분기하학 0.0.0 미분기하학 개요

미분기하학 0.0.0 미분기하학 개요

14.8K

The second fundamental form measures surface curvature away from tangent planes. Using coordinate chart φ(u,v), coefficients L, M, N come from second partial derivatives dotted with unit normal N. The quadratic form II(v) = Lv¹² + 2Mv¹v² + Nv²² measures signed normal distance. The Weingarten map W(v) = -∇_v N measures normal vector change along tangent directions, being linear and self-adjoint with eigenvalues as principal curvatures. Gauss's Theorema Egregium proved Gaussian curvature K is intrinsic, computable from first fundamental form alone. Geodesics locally minimize distance, analogous to straight lines. Gaussian curvature classifies points: K > 0 (elliptic/spherical), K = 0 (parabolic/cylindrical), K < 0 (hyperbolic/saddle).

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Διαφορική Γεωμετρία Ι | Καμπυλότητα Gauss και Μέση Καμπυλότητα Επιφάνειας | Andreas Mavridis

Διαφορική Γεωμετρία Ι | Καμπυλότητα Gauss και Μέση Καμπυλότητα Επιφάνειας | Andreas Mavridis

343

Computing surface curvature requires fundamental quantities of first order (E,F,G) and second order (e,f,g). The first fundamental form coefficients measure the metric: E = X_u·X_u, F = X_u·X_v, G = X_v·X_v. The second fundamental form coefficients measure bending: e = N·X_uu, f = N·X_uv, g = N·X_vv. Gaussian curvature K = (eg - f²)/(EG - F²) is intrinsic, depending only on metric properties. Mean curvature H = (Eg + Ge - 2Ff)/(2(EG - F²)) is extrinsic. At critical points where first derivatives vanish, these formulas simplify significantly, allowing direct computation of curvature values from second derivatives of the surface function.

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Shape Analysis Lecture 6: Surface Curvature Fundamentals

Shape Analysis Lecture 6: Surface Curvature Fundamentals

The second fundamental form is a symmetric bilinear form on the tangent plane of a surface, defined as II(v, w) = -v · dn(w), where dn is the differential of the Gauss map (shape operator). This form encodes how the surface bends in space and has eigenvalues called principal curvatures (κ₁, κ₂) whose product gives Gaussian curvature (K = κ₁κ₂) and average gives mean curvature (H = (κ₁+κ₂)/2). Gaussian curvature distinguishes surface types: positive K indicates elliptic (bowl-shaped) surfaces, negative K indicates hyperbolic (saddle-shaped) surfaces, and zero K indicates parabolic (cylindrical) surfaces. Mean curvature measures how the surface interacts with its normal vector.

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Deep Dive

Proficiency with vector calculus theorems, particularly Green's Theorem and Stokes' Theorem, which are used to relate line integrals to surface integrals.

Entendendo o Teorema de Stokes e como utilizar | Cálculo Vetorial

Entendendo o Teorema de Stokes e como utilizar | Cálculo Vetorial

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Stokes' Theorem states that for an oriented, piecewise smooth surface S with a closed, simple, piecewise smooth boundary curve C with positive orientation, the line integral of a vector field F along C equals the surface integral of the curl of F over S: ∮_C F·dr = ∬_S (∇×F)·dS. This theorem generalizes Green's Theorem from 2D to 3D space, allowing conversion between line integrals and surface integrals of curl. The positive orientation of C means that when traversing C in the direction of the surface's normal vector, the surface remains on the left side.

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What is Stokes theorem? - Formula and examples

What is Stokes theorem? - Formula and examples

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Stokes' theorem is a three-dimensional extension of Green's theorem that relates the line integral of a vector field around a closed curve C to the surface integral of the curl of that vector field over any surface S bounded by C; the theorem requires that the surface S be an oriented smooth surface with a simple, closed smooth boundary curve C having positive orientation (counterclockwise when viewed from above), and it allows evaluation of either the line integral or the surface integral depending on which is more convenient for the given problem.

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Engineering Mathematics I Maha Marathon | BAS 103 | Important Questions | AKTU Exam | By Gaurav Sir

Engineering Mathematics I Maha Marathon | BAS 103 | Important Questions | AKTU Exam | By Gaurav Sir

17.9K

This section covers fundamental theorems: Green's theorem (∮(Pdx + Qdy) = ∬(∂Q/∂x - ∂P/∂y)dA, line integral to surface integral), Divergence theorem (∯(F·dS) = ∭(∇·F)dV, surface integral to volume integral), Stokes' theorem (∮(F·dr) = ∬(∇×F)·dS, line integral to surface integral of curl). These theorems connect different types of integrals.

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PYQs on Greens and Stoke Theorem | Vector Calculus | GATE 1996 to 2023 | Short Cut Tricks

PYQs on Greens and Stoke Theorem | Vector Calculus | GATE 1996 to 2023 | Short Cut Tricks

19.2K

Green's Theorem applies to two-dimensional line integrals in the R² plane, converting ∮Pdx + Qdy to ∬(∂Q/∂x - ∂P/∂y)dA, while Stokes' Theorem applies to three-dimensional line integrals in R³, converting ∮Pdx + Qdy + Rdz to ∬(curl F)·n dS; the key decision rule is to use Green's Theorem for 2D problems and Stokes' Theorem for 3D problems, with surface area calculations using ∬√(1 + (∂z/∂x)² + (∂z/∂y)²)dA.

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Vector Calculus | ONE SHOT | Engineering Mathematics | Pradeep Giri Sir

Vector Calculus | ONE SHOT | Engineering Mathematics | Pradeep Giri Sir

124.9K

This extensive section covers advanced vector calculus theorems essential for engineering mathematics. Green's theorem establishes equivalence between line integrals around closed curves and double integrals over enclosed regions: ∮(Pdx + Qdy) = ∬(∂Q/∂x - ∂P/∂y)dA. The video demonstrates application to verify Green's theorem for specific functions over parabolic regions, including finding intersection points by solving simultaneous equations and setting up appropriate integration limits. Stokes' theorem extends this to three dimensions: ∮C F·dr = ∬S (∇×F)·n̂dS, relating surface integrals to line integrals around boundary curves. Key applications include computing normal vectors for planar surfaces (n̂ = k for z=0), determining integration limits based on surface boundaries, and recognizing that opposite limits may cancel contributions. The section emphasizes systematic problem-solving approaches for verifying theorems and evaluating complex integrals in engineering contexts.

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Deep Dive

Familiarity with the concept of the covariant derivative and frame fields (such as Cartan's moving frames) on a surface.

Covariant Derivative without Christoffel Symbols, 공변 미분, 사이비 과학, pseudo science

Covariant Derivative without Christoffel Symbols, 공변 미분, 사이비 과학, pseudo science

118

For a surface with position vector r(u,v), the covariant derivative of a vector field V = V^j ∂/∂u^j along direction ∂/∂u^i is: ∇_i V^j = ∂_i V^j + Γ^j_ki V^k. This equals the full derivative minus the normal component: ∇_i V^j = (∂_i V^j ∂/∂u^j) - (V^j ∂_i ∂/∂u^j · n)n, where n is the unit normal. The covariant derivative extracts only the tangent component, which is essential for intrinsic geometry on the surface.

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Covariant derivative

Covariant derivative

120

The covariant derivative of a vector field X tangent to a surface S along a curve α is defined as the orthogonal projection of the ordinary derivative X· onto the tangent space of S, computed as Ẋ = X· - (X· · n)n, where n is the unit normal vector to the surface; this derivative satisfies linearity properties (X+Y)̇ = Ẋ + Ȳ and the product rule (fX)̇ = ḟX + fẊ, and a curve α is a geodesic if and only if its covariant acceleration α̈ = 0.

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Differential Geometry is Impossible Without These 7 Things

Differential Geometry is Impossible Without These 7 Things

105.8K

This section completes the development of differential geometry by addressing how to differentiate vectors on curved spaces where tangent spaces differ at each point. Covariant differentiation (Ricci-Curbastro and Levi-Civita) solves this by defining derivatives that respect intrinsic geometry, using Christoffel symbols to track how vectors must change to remain parallel during transport. Cartan's method of moving frames provides an elegant geometric alternative, using orthonormal frames (tangent, normal, binormal vectors) whose evolution is captured by structure equations involving curvature and torsion. This final synthesis made differential geometry fully intrinsic, visual, and intuitive—transforming abstract concepts into computable frameworks that describe everything from planetary orbits to spacetime curvature in general relativity.

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General Relativity: Lecture 14: covariant derivatives and curvature from Frankel

General Relativity: Lecture 14: covariant derivatives and curvature from Frankel

570

A covariant differentiation operator assigns to vector X and vector field V a vector ∇_X V satisfying linearity, Leibniz rule, and smoothness. In a frame basis, ∇_X V = X^j(∂_j V^i + ω^i_jk V^k)e_i, where ω^i_jk are connection coefficients. The commutator [X,Y] of vector fields is a derivation despite being second-order. A frame has zero commutators iff it is a coordinate frame, with dual forms being closed and locally exact. The curvature tensor R(X,Y)V = ∇_X ∇_Y V - ∇_Y ∇_X V - ∇_[X,Y] V measures the failure of covariant derivatives to commute. The torsion tensor T(X,Y) = ∇_X Y - ∇_Y X - [X,Y] measures the failure of the connection to be symmetric.

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Riemann geometry -- covariant derivative

Riemann geometry -- covariant derivative

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The covariant derivative is a mathematical tool used to differentiate vector fields on curved surfaces (Riemannian geometry) by accounting for the changing tangent plane at each point; unlike ordinary derivatives, it incorporates the Christoffel symbols (Γ^i_jk), which describe how the basis vectors change as one moves along the surface, allowing entities confined to the surface to measure meaningful rates of change without reference to the external embedding space.

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Setup & Statement

Basis Expansion

Curvature Formula

Integration Step

Green's Theorem

Final Simplify

Outlook

Setup & Statement

0:00

Playing Section

1
Define smooth simple closed curve and surface parameterization.
2
State Gauss-Bonnet theorem linking geodesic and Gaussian curvature.
3
Introduce orthonormal basis for tangent space calculations.

Deep Dive

The Global Gauss-Bonnet Theorem, which links the total Gaussian curvature of a compact surface to its topological Euler characteristic.

(06/02/2023) - Mestrado: Geometria Diferencial de Curvas e Superfícies - Lucas Ambrozio - Aula 14

(06/02/2023) - Mestrado: Geometria Diferencial de Curvas e Superfícies - Lucas Ambrozio - Aula 14

1.1K

The global Gauss-Bonnet theorem states that for an orientable regular surface S and a regular region Ω (a compact subset whose boundary is a finite union of traces of regular-by-parts curves), the integral of Gaussian curvature over Ω plus the integral of geodesic curvature along the boundary plus the sum of external angles at vertices equals 2π times the Euler characteristic of Ω. This theorem connects local geometric properties (curvature) with global topological properties (Euler characteristic), and can be proven by triangulating the region and applying the local Gauss-Bonnet theorem to each triangle, then using combinatorial relations between vertices, edges, and faces to derive the global result.

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Teorema de Gauss-Bonnet e o Formato de Planolândia | Palestras da Matemática

Teorema de Gauss-Bonnet e o Formato de Planolândia | Palestras da Matemática

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The Gauss-Bonnet theorem states that for any compact surface, the integral of the Gaussian curvature over the entire surface equals 2π times the Euler characteristic: ∫K dA = 2πχ. This theorem connects local geometry (curvature) to global topology (Euler characteristic). For a sphere with χ = 2, the total curvature is 4π. For a torus with χ = 0, the total curvature is 0. The presenter explains how to prove this by triangulating the surface, assuming constant curvature in each triangle, and summing the contributions.

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Topology For Beginners: Topology Meets Hyperbolic Geometry

Topology For Beginners: Topology Meets Hyperbolic Geometry

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The Gauss-Bonnet theorem states that the total Gaussian curvature of a closed surface equals its Euler characteristic. The Euler characteristic is calculated as 2 minus 2 times the genus (number of holes): χ = 2 - 2g. For a sphere (g=0), χ=2 indicating positive total curvature. For a torus (g=1), χ=0 indicating zero total curvature. For a two-hole torus (g=2), χ=-2 indicating negative total curvature. This explains why surfaces with more holes exhibit hyperbolic geometry characteristics.

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미분기하학 0.0.0 미분기하학 개요

미분기하학 0.0.0 미분기하학 개요

14.8K

The Gauss-Bonnet theorem relates the total curvature of a surface to its topology. For a compact surface without boundary, the integral of Gaussian curvature over the entire surface equals 2πχ, where χ is the Euler characteristic (a topological invariant). For a triangle on a surface, the angle defect (π minus the sum of interior angles) equals the integral of Gaussian curvature over the region enclosed by the triangle. This theorem generalizes the flat geometry result that the sum of angles in a Euclidean triangle is π.

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The Number That Refuses to Change

The Number That Refuses to Change

130

The Gauss-Bonnet theorem states that the total curvature of a surface equals 2π times its Euler characteristic (2πχ), where χ = 2 - 2g (g being the number of holes). This means the total curvature is a topological invariant: a sphere (0 holes) has total curvature 4π, a torus (1 hole) has 0, and a double torus (2 holes) has -4π. The theorem connects local geometry (curvature) with global topology (counting holes), explaining why you can't comb a hairy ball flat and why every flat map of Earth must distort reality.

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Deep Dive

Generalizing the theorem to piecewise smooth curves (curves with corners), which introduces the concept of exterior angles.

Differential Geometry - Claudio Arezzo - Lecture 16

Differential Geometry - Claudio Arezzo - Lecture 16

6.5K

For piecewise smooth curves (with finitely many vertices where the tangent vector jumps), the local Gauss-Bonnet theorem generalizes. At each vertex, there is an exterior angle Δᵢ, which is the angle by which the tangent vector must rotate to pass from one side to the other. The integral of θ' now equals 2π minus the sum of all exterior angles.

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(03/02/2022) - Mestrado: Geometria Diferencial de Curvas e Superfícies - Lucas Ambrozio - Aula 13

(03/02/2022) - Mestrado: Geometria Diferencial de Curvas e Superfícies - Lucas Ambrozio - Aula 13

793

The Gauss-Bonnet theorem states that for a piecewise smooth, closed, simple curve on a surface, the total angular variation equals 2π plus the integral of the geodesic curvature along the curve plus the sum of exterior angles at vertices, which generalizes Euclid's theorem that the sum of interior angles of a triangle equals 180 degrees.

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Geometria Diferencial - Prof. Fernando Codá Marques - Aula 10

Geometria Diferencial - Prof. Fernando Codá Marques - Aula 10

294

At each vertex of a piecewise differentiable boundary curve, the exterior angle is defined as the oriented angle between the left-hand tangent vector and the right-hand tangent vector at that vertex. The exterior angle is measured in the range (-π, π] and accounts for the turning of the curve at corners.

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mestrado2013geometriadiferencial_25042013_10

mestrado2013geometriadiferencial_25042013_10

449

A simple region is defined as the closure of an open set whose boundary consists of a finite number of piecewise differentiable curves. The boundary is assumed to be positively oriented with respect to the surface orientation. The exterior angle at a vertex is defined as the angle between the incoming tangent vector (from the left) and the outgoing tangent vector (from the right), measured in the range (-π, π]. This angle accounts for the change in direction of the curve at the vertex and is essential for the Gauss-Bonnet theorem.

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Домрина А.В. | Лекция 23 по математическому анализу II | ВМК МГУ 2021-2022

Домрина А.В. | Лекция 23 по математическому анализу II | ВМК МГУ 2021-2022

1.5K

A curve γ is piecewise smooth if it can be parameterized by t ∈ [A, B] and there exists a partition A = t₀ < t₁ < ... < tₙ = B such that each piece γ|[tᵢ₋₁, tᵢ] is a smooth curve. Examples include polygonal chains (broken lines), where the curve has corners at the vertices but each segment between vertices is smooth. For piecewise smooth curves, curvilinear integrals are defined as the sum of integrals over each smooth piece: ∫_γ f ds = Σ∫_γᵢ f ds. This allows integration over curves with corners or discontinuities in the tangent vector.

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Deep Dive

Applications to non-Euclidean geometry, such as calculating the area of geodesic triangles on spheres and hyperbolic planes.

Lecture 08

Lecture 08

455

In hyperbolic geometry, the area of a geodesic triangle in the upper half-plane model equals π minus the sum of its vertex angles (Area = π - (α + β + γ)). This result is proven by first showing that elements of PSL(2,R) preserve hyperbolic area through the Cauchy-Riemann equations, then using isometric transformations to reduce the general case to a simpler case where one vertex lies on the real axis, and finally computing the area via integration.

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The most important theorem in (differential) geometry | Euler characteristic #3

The most important theorem in (differential) geometry | Euler characteristic #3

57.9K

For a geodesic triangle T, the integral of Gaussian curvature over T equals the signed area of N(T) on the unit sphere. Since the Gauss map preserves parallel transport and holonomy, N(T) is itself a geodesic triangle on the sphere. The area of this spherical triangle equals its holonomy, which equals the angle sum minus π. Therefore, ∫∫_T κ dA = angle sum - π for any geodesic triangle.

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Дынников И.А. - Аналитическая геометрия - 16. Консультация

Дынников И.А. - Аналитическая геометрия - 16. Консультация

382

This section covers rigorous methods for surface integration and properties of geodesics on curved surfaces. Surface area can be computed through triangulation (replacing curved triangles with flat ones) or partitions of unity (using smooth functions with compact support). The area element in local coordinates is √g du dv, where g is the determinant of the first fundamental form. On the hyperbolic plane, parallel transport of a vector along a closed curve results in a rotation by an angle equal to the angular defect (2π minus the sum of interior angles). The angular defect of a geodesic polygon equals its area, given by π minus the sum of angles for a triangle. The commutator of two vector fields measures the failure of their flows to commute. A curve is both a geodesic and a line of curvature if and only if it is a straight line in the ambient space. The maximum area of a geodesic triangle on the hyperbolic plane is π, achieved by an ideal triangle with vertices at infinity.

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Non-Euclidean Geometry Explained - Hyperbolica Devlog #1

Non-Euclidean Geometry Explained - Hyperbolica Devlog #1

Non-Euclidean geometry describes spaces with constant curvature that differ fundamentally from flat Euclidean space; in spherical geometry, the sum of triangle angles exceeds 180° and parallel lines converge, while in hyperbolic geometry, the sum is less than 180° and parallel lines diverge, with both exhibiting holonomy (rotation upon returning to a starting point) and having distinct formulas for circle circumference (2πr in Euclidean, 2πsin(r) in spherical, 2πsinh(r) in hyperbolic) and triangle area (area = sum(angles) - π in spherical, area = π - sum(angles) in hyperbolic).

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Escher and Coxeter - a Mathematical Conversation - Professor Sarah Hart

Escher and Coxeter - a Mathematical Conversation - Professor Sarah Hart

12.8K

There are three fundamental geometries with different properties: (1) Euclidean geometry (flat plane) where straight lines are shortest paths and triangle angles sum to 180°; (2) Spherical geometry (surface of sphere) where geodesics are great circles and triangle angles sum to more than 180°, with the excess proportional to area; (3) Hyperbolic geometry (represented by Poincaré disc) where geodesics are arcs of circles meeting the boundary at right angles and triangle angles sum to less than 180°.

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Deep Dive

An introduction to Chern-Weil theory and the generalized Gauss-Bonnet-Chern theorem in higher-dimensional Riemannian geometry.

Mauricio Che - El teorema de Gauss-Bonnet-Chern

Mauricio Che - El teorema de Gauss-Bonnet-Chern

144

The Gauss-Bonnet-Chern theorem generalizes the classical Gauss-Bonnet theorem from surfaces to closed, orientable manifolds of even dimension. The theorem states that the integral of the Pfaffian of the curvature form (a polynomial in the curvature tensor) equals 2π^n times the Euler characteristic of the manifold, where n is half the dimension. This result connects local geometric quantities (curvature) to global topological invariants (Euler characteristic), and is a fundamental example of Chern-Weil theory, which constructs characteristic classes from curvature forms. The theorem was first proven by Chern in 1944 and has been generalized to various contexts including manifolds with singularities and Alexandrov spaces.

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Chern–Gauss–Bonnet theorem | Wikipedia audio article

Chern–Gauss–Bonnet theorem | Wikipedia audio article

106

The generalized Gauss-Bonnet theorem (also called Chern-Gauss-Bonnet theorem) states that for a compact orientable 2n-dimensional Riemannian manifold M without boundary, the Euler characteristic χ(M) equals (1/(2π)^n) times the integral of the Pfaffian of the curvature form Ω over M, expressed as ∫_M Pf(Ω) = (2π)^n χ(M), where Ω is the curvature form of the Levi-Civita connection and Pf(Ω) is a 2n-form derived from the curvature.

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NCCR SwissMAP - Introduction to Topological Field Theory

NCCR SwissMAP - Introduction to Topological Field Theory

519

Chern-Weil theory constructs a representative of the Euler class from the curvature of a connection on a vector bundle. Although the differential form depends on the choice of metric and connection, the resulting cohomology class is independent of these choices. Different choices lead to representatives differing by an exact form, which does not change the cohomology class. This allows topological information to be extracted from non-topological data, as demonstrated by the Gauss-Bonnet theorem for surfaces.

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16/11/2015 - Jean-Pierre Bourguignon - General Relativity and Geometry

16/11/2015 - Jean-Pierre Bourguignon - General Relativity and Geometry

2.4K

In the late 1930s, Shiing-Shen Chern studied generalized Lagrangians involving quadratic curvature terms. He identified three natural quadratic invariant forms on the space of curvature tensors: the norm squared of the full curvature tensor, the norm squared of the Ricci tensor, and the square of the scalar curvature. By choosing specific coefficients (1, -1/4, 1), Chern obtained an invariant that generalizes the Gauss-Bonnet theorem to higher dimensions. This work opened the door to characteristic class theory and demonstrated how ideas from Kaluza-Klein theory could solve problems in differential geometry, though Chern was unaware that similar results had appeared earlier in work by Leung-Sau.

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Gauss-Bonnet Panorama

Gauss-Bonnet Panorama

519

The Gauss-Bonnet theorem has higher-dimensional versions for even-dimensional compact manifolds. Hopf studied hypersurfaces in ambient space (codimension one surfaces), and this was later generalized. Chern provided an intrinsic proof for 2N-dimensional manifolds. The Reman curvature tensor plays a central role in these higher-dimensional generalizations. While the course covers the two-dimensional case, the higher-dimensional versions are more complex and beyond the scope of this particular course.

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Setup & Statement

Basis Expansion

Curvature Formula

Integration Step

Green's Theorem

Final Simplify

Outlook

Setup & Statement

0:00

Playing Section

1
Define smooth simple closed curve and surface parameterization.
2
State Gauss-Bonnet theorem linking geodesic and Gaussian curvature.
3
Introduce orthonormal basis for tangent space calculations.

735 views1812:30@mikethemathematicianOriginal Release: 2025-02-13

The Gauss-Bonnet Theorem states that for a simple closed smooth curve on a surface, the line integral of the geodesic curvature equals 2π minus the surface integral of the Gaussian curvature over the interior region, mathematically expressed as ∫γ kg ds = 2π - ∫∫D K dσ, where kg is the geodesic curvature, K is the Gaussian curvature, and D is the region enclosed by the curve γ.

[00:00:00]hello students in this video we'll prove the gaus banet theorem for simple closed curves let's let U of T V of T be a simple closed curve just looking c three on AB so that just means if I look at this curve and plug in the initial point and plug in the terminal point the initial point and terminal point have to be the same there's our curve we can call this curve over here gamma for example I'll just leave like that for time being looks like this there's no there's no jumps so in other words both these functions are infinitely differentiable so we can assume that u and v are infinitely differentiable and so no words there's no cusps over here I don't have a situation where there's going to be a point where the derivative doesn't exist just nice simple curve right and let R be a surface parameterization like that beautiful then gamma T which is R of U of TV f is a closed curve on the surface okay so here's your surface right like that and then the image under this parameterization of this curve gives you some curve like that right so this is your curve gamma over here okay and now what does the gaus B theorem say the gaus b theorem says in this case that if I integrate over the curve gamma the geodesic curvature that is 2 pi minus the double integral over the interior of gamma of the gausian curvature with respect to the surface area differential okay beautiful that's a wonderful theorem that relates the global curvature right on the on the interior of the curve to the gesc curvature on the boundary okay and so here's we're going to do we're going to let FG be an orthonormal basis of the tangent space to the surface at every Point okay there I set before so I've choose an orthal basis I can do this by Gram we already know properties of this orthonormal basis right then if Theta of s is the angle between F at and Gamma dot then gamma dot is going to be F I'm going to suppress the uh of s here right just cosine of theta plus Gat sin Theta because it's an orthonormal basis I can write down this expression and then this implies that the component n cross gamma dot has to be what has to be F negative f sin theta plus Gat cosine Theta okay beautiful and now of course here comes the the second der has to pop up because I know if I take this expression over here and Dot this with gamma double dot that gives me the G curvature so let's do that so what's gamma double dot going to be so here's the tricky part so gamma double dot is going to be F do cosine theta plus g dot g do sin Theta then minus F cosine Theta Theta Dot and then plus G hat cosine Theta oh I have to do the D so that's going to be a sign of course right because der of cosine is s so it's going to be an F hat sin Theta th dot plus G hat cosine Theta Theta dot okay so what's my Kappa G going to be so let's do it slowly so my Kappa G is going to be what cap G is going to be well let's do this over here so we're going to have a now what do we know so here's an elementary observation so what can I say about I know that fat. Gat is equal to zero because it's an orthal basis and I also know F any derivative of f f hat dot do fat is equal to zero and G hat dot do G hat is also equal to zero because they're unit vectors right good all right so those are things that we're going to use in our calculation over here so if I do for example if I do this term over here do this term it's zero because F hat dot f hat. fat dot is zero so these first terms is give me a what an F hat dot g hat dot g hat cosine 2 Theta so the dot over there goes on the F then what do we do next then I got to do this term so this dot the G will be zero because of unit Vector so I'm going to get a minus F hat G hat Dot sin^2 Theta good and then that takes care of these terms over here right then if I have F and then F that gives me a one by unit Vector so that's going to be a just a sin squ Theta dot plus sin s Theta Theta Dot and then f.g is equal to Zer by orth orthogonality g. f is equal to zero and then g. G is equal to 1 so this going to be a plus cosine 2 Theta Theta dot right I'm going to do one thing over here I'm going to take this derivative over here on the F Dot and pass it to the g dot so all total the GS curvature is going to give me what the GS of curvature these terms just to add up to Theta dot so I get a Theta dot over here and then we have minus fat G hat dot right and then if I do that or if I put the dot over here that's a negative cosine Square Theta negative sin Square Theta gives just a negative one over here beautiful right and so now what do we know I'm going to write down what G hat is going to be right on this curve this is going to be Theta dot minus fat and then we're going to have a gat u u dot plus G hat V V Dot like that okay so that's a little bit of a nicer expression now let me integrate this okay so if I integrate kg that's the same as integrating this over here so the integral over gamma of the gsit curvature DS is going to be well if I integrate Theta dot that's a perfect derivative integrating Theta dot around the closed curve gives me 2 pi okay now that's actually I'm sweeping something important under the rug over here what I'm sweeping over here is the fact that Theta integrates over the unit circle to that that the integral over gamma Theta dot DS is equal to 2 pi this is an important relationship it's true for any simple close curve and that's known as the H lat and we won't prove that in this video because it requires homotopy Theory so we need a little bit of homotopy theory in order to prove this theorem but you can think about what happens in the case of like a circle for example if if you measure how much the the angle is changing as you Traverse the unit circle the it changes by 2 pi right that's how much the circle changes now of course you can sort of think if I deform the circle a little bit so there's the the case of the circle is easy but if I just deform the circle to some other sort of strange curve like this well in that cas case it's basically still encapsulated I can basically take that more strange curve and continuously deform it onto the unit circle so if there's a that's what a homotopy is right so in other words if I can if two curves are homotopic then I know they're going to show the same index or the same winding number right so that requires a little more work but intuitively it should be obvious that gives us 2 pi okay good and then minus this thing over here F hat and then gu U du that's a u and then a plus f g v DV like so DS okay get R the DS now it's just a regular line integral right I just put the different I write the DS as a line integral now okay and now of course I can t of green St and you you should have expected green St because it's the interior of gamma over here right so let me use green STM so is going to be 2 piun minus the double integral over the interior of gamma and then I got to do the what I got to do the U derivative of this and then minus the V derivative of the first thing the V derivative of this thing over here f g U and that's going to be a DA okay not a d Sigma right but what do we know about this when we do when we do this product I'm going to have an F UV then I'm going to have an F then I'm have an FG V and over here I'll have an F gvu so the G the when the derivatives go on the GV or when the Der goes on the GU those terms are going to cancel and you'll just have an f u v minus fvg so these terms over here are going to work out to be what those to f u g v minus f v g u and we have from our previous LMA that that's exactly equal to what that's exactly equal to the Ln minus m^2 over the square root of EG so that this whole thing was going to simplify to what this whole thing just simplifies to 2 pi minus a double integral over the interior of gamma and and then just the gausian curvature k d Sigma and what we're really using here is we're using what fact we're using the fact that this this quantity over here is equal to l Ln - m^ 2 over the < TK of EEG - f^ s right we have that from a previous video now of course that's not K but if I put it in extra Factor EG - f^2 over EG - f^2 then this this expression over here is the what that expression is the gausian curvature because now it has the right form and what's left over adjust the area differential into the surface area differential and that proves the first version of the gaus banet theorem which says that if I integrate the G of curvature over the curve that's 2 pi minus the gausian curvature over the interior of that curve beautiful result in further videos we'll extend this to the case when you don't have a smooth curve you have cusps on your curve and that basically introduces more angles into the problem thank you very much much

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