The Gauss–Bonnet theorem

Theorem

The Gauss–Bonnet theorem

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Statement

Let \(M\) be a compact, orientable, boundaryless smooth surface equipped with a Riemannian metric \(g\), let \(K:M\to\mathbb{R}\) be its Gaussian curvature, let \(dA\) be the area form of \(g\), and let \(\chi(M)\) be the Euler characteristic of \(M\) (a purely topological invariant, independent of \(g\)). Then \[ \iint_M K\,dA \;=\; 2\pi\,\chi(M). \] More generally, if \(R\subseteq M\) is a compact region whose boundary \(\partial R\) is a finite union of piecewise-smooth simple closed curves parametrised by arc length, with geodesic curvature \(\kappa_g\) along the smooth arcs and exterior angles \(\varepsilon_1,\dots,\varepsilon_k\) at the corners, then \[ \iint_R K\,dA \;+\; \int_{\partial R}\kappa_g\,ds \;+\; \sum_{i=1}^k \varepsilon_i \;=\; 2\pi\,\chi(R). \] The closed case above is the special case \(\partial R=\varnothing\).

Why it matters

Gauss–Bonnet is the bridge between local differential-geometric data (curvature, a second-order, metric-dependent quantity computable from a single point's neighbourhood) and global topological data (Euler characteristic, a combinatorial invariant insensitive to any smooth deformation of the metric). It says that however wildly you bend a surface — stretching curvature positive here, negative there — the total signed curvature is locked to the topology and cannot change under any deformation that preserves the manifold's homeomorphism type.

It is the prototype for the entire family of index theorems (Chern–Gauss–Bonnet in higher dimensions, Riemann–Roch, Atiyah–Singer) that equate an analytic integral to a topological count, and it is the first place a student sees that curvature is not just local shape information but a conserved topological quantity in disguise.

Hypotheses

\(M\) is a smooth Riemannian surface.Without a Riemannian metric there is no Gaussian curvature \(K\) and no area form \(dA\) — the left-hand side is not defined. (A metric always exists on any smooth manifold by partition of unity, so this is a mild hypothesis, but it must be fixed before the theorem is stated.) \(M\) is compact.On a non-compact surface \(\iint_M K\,dA\) may fail to converge, or may converge to a value with no relation to any Euler characteristic. The flat plane \(\mathbb{R}^2\) has \(K\equiv 0\), so \(\iint K\,dA=0\), but it is not compact and no version of the identity \(=2\pi\chi\) is being asserted or needed; the theorem simply has no content to state. \(M\) is orientable.On a non-orientable surface \(dA\) is not a globally defined 2-form (only a density), so \(\iint_M K\,dA\) must be reinterpreted as an integral of a density. The identity survives in that reading with \(\chi(M)\) computed for the non-orientable surface (e.g. \(\mathbb{RP}^2\) has \(\chi=1\), and a round \(\mathbb{RP}^2\) of curvature \(1\) has total curvature \(2\pi\)), but the orientable statement as written, with \(dA\) an honest top-degree form, requires orientability to make sense. \(\partial M=\varnothing\) (closed case), or the boundary-integral version is used.Drop the boundary term for a surface-with-boundary and the identity fails: a flat disc \(D\subseteq\mathbb{R}^2\) has \(K\equiv 0\) and \(\chi(D)=1\), so \(\iint_D K\,dA=0\neq 2\pi=2\pi\chi(D)\); the missing \(\int_{\partial D}\kappa_g\,ds=2\pi\) (the disc's boundary circle has geodesic curvature \(1/\text{radius}\), integrating to \(2\pi\)) is exactly what restores equality. The boundary curve, if present, is piecewise smooth with well-defined exterior angles.If \(\partial R\) is merely continuous (a fractal-type curve, or a curve with no tangent line at some points), geodesic curvature and exterior angles are not defined pointwise and the boundary integral has no meaning; some regularisation (e.g. approximating by smooth curves and taking a limit of total turning) is required before any Gauss–Bonnet-type identity can even be posed.

Proof

Structure equations. Fix a point-dependent oriented orthonormal frame \((e_1,e_2)\) on an open set \(U\subseteq M\) with dual coframe \((\omega^1,\omega^2)\); let \(\omega_{12}\) be the associated connection 1-form, determined uniquely by Cartan's first structure equation \(d\omega^1=\omega_{12}\wedge\omega^2,\ d\omega^2=-\omega_{12}\wedge\omega^1\) (torsion-free condition for the Levi-Civita connection). Cartan's second structure equation then defines Gaussian curvature via \(d\omega_{12}=-K\,\omega^1\wedge\omega^2=-K\,dA\).

Definitional input: existence and uniqueness of the Levi-Civita connection (fundamental theorem of Riemannian geometry) plus the moving-frame formalism; this is how \(K\) is characterised intrinsically, independent of any embedding. C

Turning of the tangent along a curve. For a unit-speed curve \(\gamma\) in \(U\) with tangent \(T=\gamma'\), write \(T=\cos\theta\,e_1+\sin\theta\,e_2\) for a continuously varying angle \(\theta(s)\). Differentiating and comparing with the definition of geodesic curvature (\(\kappa_g = \langle \nabla_T T, JT\rangle\) for \(J\) the \(90^\circ\) rotation) gives \[\frac{d\theta}{ds} = \kappa_g(s) - \omega_{12}(T(s)).\]

Direct computation from the definitions of \(\nabla\) (Levi-Civita covariant derivative) and \(\kappa_g\) (normal component, within the tangent plane, of the acceleration of a unit-speed curve); \(\omega_{12}(T)\) is the frame's own rate of rotation relative to a parallel field along \(\gamma\). B

Local formula for a geodesic triangle. Let \(\Delta\subseteq U\) be a triangle whose three sides are geodesic arcs (so \(\kappa_g\equiv 0\) on \(\partial\Delta\)) meeting at interior angles \(\alpha,\beta,\gamma\). Integrate the relation of Step 2 around \(\partial\Delta\), traversed once with the boundary orientation, and add the three turning angles at the vertices (each exterior angle \(\varepsilon_i=\pi-\alpha_i\)): \[\oint_{\partial\Delta}\omega_{12} \;=\; \int_{\partial\Delta}\left(-\frac{d\theta}{ds}\right)ds \;+\; 0.\] Separately, the total turning of the tangent vector around any simple closed curve bounding a topological disc is \(2\pi\) (Umlaufsatz / theorem of turning tangents, in its piecewise-smooth form with corners): \[\Delta\theta_{\text{smooth arcs}} + \sum_i \varepsilon_i = 2\pi.\]

Cites the Umlaufsatz (Hopf's theorem of turning tangents) for simple closed piecewise-smooth curves bounding a disc, a genuinely independent topological input not derivable from local curvature data alone. B

Combine Steps 2–3: since \(\kappa_g=0\) on each geodesic side, \(d\theta/ds=-\omega_{12}(T)\) there, so \(\int_{\partial\Delta}(-d\theta/ds)\,ds=\oint_{\partial\Delta}\omega_{12}\). Substituting into the Umlaufsatz identity, \[-\oint_{\partial\Delta}\omega_{12} + \sum_i\varepsilon_i = 2\pi \quad\Longrightarrow\quad \oint_{\partial\Delta}\omega_{12} = \sum_i \varepsilon_i - 2\pi = \pi-(\alpha+\beta+\gamma).\]

Algebra plus \(\sum\varepsilon_i=3\pi-(\alpha+\beta+\gamma)\); routine rearrangement of Steps 2–3. A

Apply Stokes' theorem to \(\omega_{12}\) on \(\Delta\) and use Step 1: \[\oint_{\partial\Delta}\omega_{12} = \iint_\Delta d\omega_{12} = -\iint_\Delta K\,dA.\] Combining with Step 4: \[\iint_\Delta K\,dA \;=\; (\alpha+\beta+\gamma)-\pi.\] This is the local Gauss–Bonnet formula for a geodesic triangle: angle excess equals total curvature.

Stokes' theorem for a 1-form on a region with piecewise-smooth boundary (classical Green/Stokes, applicable since \(\Delta\) is a smoothly-embedded disc). B

Geodesic triangulation. Every compact smooth Riemannian surface \(M\) admits a finite triangulation \(\mathcal{T}=\{\Delta_1,\dots,\Delta_F\}\) by geodesic triangles, small enough that each \(\Delta_j\) lies in a chart where Step 5 applies, with \(V\) vertices, \(E\) edges, \(F\) faces.

Existence of a geodesic triangulation refining any given topological triangulation of a compact smooth manifold: a standard but non-trivial existence lemma (subdivide using geodesically convex neighbourhoods, whose existence follows from the Gauss lemma / normal-coordinate estimates on a compact manifold). C

Sum Step 5 over all triangles of \(\mathcal{T}\): \[\iint_M K\,dA = \sum_{j=1}^F \iint_{\Delta_j} K\,dA = \sum_{j=1}^F\Big[(\alpha_j+\beta_j+\gamma_j)-\pi\Big] = \Big(\sum_j \text{angle sums}\Big) - \pi F.\] Each interior edge is shared by exactly two triangles and each interior vertex is surrounded by angles summing to \(2\pi\) (since \(M\) is boundaryless), so \(\sum_j(\alpha_j+\beta_j+\gamma_j) = 2\pi V\).

Bookkeeping: every angle of the triangulation is counted once in the sum, and grouping by vertex, the angles meeting at a fixed interior vertex tile a full turn (\(2\pi\)) because \(M\) has no boundary and the triangulation is geodesic (no curvature is concentrated combinatorially at vertices beyond what the surrounding angle sum records). A

Hence \[\iint_M K\,dA = 2\pi V - \pi F.\] Each triangle has 3 edges and each edge borders exactly 2 triangles, so \(3F=2E\), i.e. \(F=2E-2F\), giving \(-\pi F = 2\pi F - 2\pi E\) via \(\pi F = 2\pi E-2\pi F\)... more directly \(3F=2E \Rightarrow \pi F = \tfrac{2\pi}{3}E\); substitute to eliminate \(F\) in favour of \(E,V\): \[\iint_M K\,dA = 2\pi V - \pi F = 2\pi V-2\pi E+2\pi F = 2\pi(V-E+F).\]

Substitute \(\pi F = 2\pi E-2\pi F\) (equivalent to \(3F=2E\)) into \(2\pi V-\pi F\); pure algebra using the double-counting identity \(3F=2E\) for a triangulation. A

By definition, \(V-E+F=\chi(M)\), the Euler characteristic of the triangulated surface (a combinatorial invariant, independent of the chosen triangulation, by simplicial/CW invariance of Euler characteristic). Therefore \[\iint_M K\,dA = 2\pi\,\chi(M),\] as claimed. The boundary version follows identically, replacing full \(2\pi\) angle sums at boundary vertices by \(\pi\) plus exterior-angle corrections, and retaining the geodesic-curvature integral over \(\partial R\) in place of assuming \(\kappa_g\equiv 0\) throughout.

Topological invariance of the Euler characteristic under refinement/choice of CW or simplicial structure (a standard result of algebraic topology, e.g. via simplicial homology or the classification of surfaces). B

Result

\displaystyle \iint_M K\,dA = 2\pi\,\chi(M)

Reading. The total Gaussian curvature of a closed orientable surface, however the metric bends it, is fixed by the surface's topology alone: \(2\pi\) times its Euler characteristic. Curvature can be redistributed — made more positive somewhere, more negative elsewhere — by deforming the metric, but the total is a rigid conserved quantity attached to the homeomorphism type of \(M\), not to \(g\).

Scope. Applies to any compact orientable Riemannian 2-manifold without boundary; a boundary version with the extra terms \(\int_{\partial R}\kappa_g\,ds+\sum\varepsilon_i\) applies to compact regions/surfaces with piecewise-smooth boundary. It does not by itself extend to non-compact surfaces, and the non-orientable case requires reading \(K\,dA\) as a density integral. Higher-dimensional even-dimensional analogues exist (Chern–Gauss–Bonnet, via the Pfaffian of the curvature form) but are a strictly stronger, separate theorem, not a corollary of this one.

Corollaries & converses

Sign constraint: if \(M\) is a closed orientable surface admitting a metric with \(K\gt 0\) everywhere, then \(\chi(M)\gt 0\), forcing \(M\cong S^2\) (the only closed orientable surface with positive Euler characteristic). No metric of everywhere-positive curvature exists on the torus or any higher-genus surface.
Genus formula: for a closed orientable surface of genus \(g\), \(\chi=2-2g\), so \(\iint_M K\,dA = 4\pi(1-g)\); in particular the torus (\(g=1\)) always has total curvature exactly \(0\), whatever metric it carries.
Existence of geodesic triangles with prescribed angle excess/defect: on a surface of everywhere-positive (resp. negative) curvature, every geodesic triangle has angle sum \(\gt \pi\) (resp. \(\lt \pi\)) — an immediate reading of the local formula in Step 5, independent of the global theorem.
Converse fails in the strong sense: the identity \(\iint_M K\,dA = 2\pi\chi(M)\) holding for a specific metric does not determine \(\chi(M)\) uniquely from the integral value alone without independently knowing the integral is being taken over that particular \(M\); more importantly, knowing only the number \(\iint K\,dA\) for an unspecified compact surface does not tell you which surface it is, since the theorem is an equality of two already-determined quantities, not an implication that can be run backwards to deduce topology from an unrelated curvature computation on an unknown space.
Rigidity converse (false as stated): the theorem does not assert, nor is it true, that every value of \(\chi\) forces a unique curvature distribution — only the total integral is fixed; the pointwise function \(K\) remains almost entirely free (subject only to this one integral constraint), e.g. a genus-2 surface can be given metrics with wildly different curvature functions, all integrating to \(4\pi(1-2)=-4\pi\).

Fails without

Compactness dropped: the hyperbolic plane \(\mathbb{H}^2\) (upper half-plane, curvature \(K\equiv -1\)) is non-compact with infinite area, so \(\iint_{\mathbb{H}^2} K\,dA = -\infty\), while topologically \(\mathbb{H}^2\cong\mathbb{R}^2\) would suggest \(\chi=1\); there is no finite identity to violate or satisfy — the left side simply diverges.
Boundary term omitted: the flat unit disc \(D\subseteq\mathbb{R}^2\) has \(K\equiv 0\) so \(\iint_D K\,dA=0\), yet \(\chi(D)=1\) gives \(2\pi\chi(D)=2\pi\neq 0\); the closed-surface formula fails outright on a surface with boundary unless the boundary integral \(\int_{\partial D}\kappa_g\,ds=2\pi\) (curvature of the unit circle integrated over its length \(2\pi\)) is included.
Orientability dropped, form read naively: attempting to integrate \(K\) against a chosen local orientation on the Möbius band or \(\mathbb{RP}^2\) without passing to the density interpretation gives sign-dependent, chart-dependent "answers" that do not match \(2\pi\chi\) for either sign choice, because \(dA\) as a genuine 2-form does not exist globally on a non-orientable surface.
Non-Riemannian ("curvature" undefined): on a merely topological or piecewise-linear surface with no smooth metric structure specified, \(K\) has no meaning and there is nothing on the left-hand side to compute; the discrete analogue (Descartes' theorem on angular defect) is a genuinely different, though closely related, statement and does not follow from this theorem without its own separate proof.

Common errors

Applying the closed-surface formula \(\iint K\,dA=2\pi\chi\) directly to a region with boundary (e.g. a hemisphere, a geodesic triangle) and forgetting the \(\int\kappa_g\,ds\) and exterior-angle terms, which are not automatically zero.
Confusing \(\chi(M)\) with genus \(g\) directly, forgetting the factor \(\chi=2-2g\) for closed orientable surfaces (and using the wrong formula, e.g. \(\chi=1-g\), entirely for non-orientable surfaces where \(\chi=2-k\) with \(k\) crosscaps).
Treating "total curvature is topological" as meaning \(K\) itself is constant or forced to have one sign; only the integral is constrained, not the pointwise function.
Using the Euclidean angle-sum-equals-\(\pi\) fact as if it were a hypothesis rather than the \(K\equiv 0\) special case of Step 5's local formula.
Forgetting that the triangles in the triangulation used in the proof must be geodesic (sides are geodesics) for the boundary geodesic-curvature term to vanish in Step 3–5; using an arbitrary (non-geodesic) triangulation breaks the local identity used at each face.
Sign errors in the exterior angle \(\varepsilon_i=\pi-\alpha_i\) versus interior angle \(\alpha_i\), especially at reflex/concave corners of a boundary curve.

Discussion

The theorem is named for Gauss's 1827 Disquisitiones generales circa superficies curvas, where the local formula for a geodesic triangle (Step 5 above) first appears as a striking corollary of the Theorema Egregium — Gauss's own discovery that \(K\) is an intrinsic quantity, computable from the metric alone without reference to any ambient embedding. The full global statement, unifying this with Euler's polyhedral formula \(V-E+F=2\) via triangulation, is due to Pierre Ossian Bonnet (1848), whose contribution was precisely the boundary term and the reduction of the closed case to a triangulated sum — the synthesis credited jointly ever since.

The proof given above via Cartan's moving frames and structure equations is the modern (post-1920s) formulation; it makes transparent exactly which two ingredients are doing the work: a purely local/analytic one (Stokes' theorem applied to the connection form, which is where \(K\) enters) and a purely global/topological one (the Umlaufsatz, and separately the combinatorial identity \(V-E+F=\chi\)). Seeing curvature and topology meet at the single equality \(\oint\omega_{12}=-\iint K\,dA\) is the heart of the theorem; everything else is bookkeeping to globalise it.

Gauss–Bonnet is the genus-2 (real, 2-dimensional) case of a much larger phenomenon: the Chern–Gauss–Bonnet theorem expresses the Euler characteristic of any even-dimensional closed Riemannian manifold as the integral of a curvature polynomial (the Pfaffian of the curvature form), and index theorems more generally (Atiyah–Singer) express analytic indices of elliptic operators as topological integrals in exactly this spirit. The theorem is also the geometric engine behind the classification of constant-curvature closed surfaces: since \(\iint K\,dA=2\pi\chi\) and constant \(K=c\) gives \(cA=2\pi\chi\) with \(A\gt 0\), the sign of \(c\) is forced to match the sign of \(\chi\), which is why the sphere (\(\chi=2\)) alone carries constant positive curvature, the torus (\(\chi=0\)) alone carries flat metrics, and every genus \(g\geq 2\) surface (\(\chi\lt 0\)) alone carries constant negative (hyperbolic) curvature — the uniformisation theorem's topological shadow.

Common misconception. Students often read the theorem as saying curvature is topologically determined, when only its total integral is. A genus-2 surface can be given a metric that is positively curved on a small cap and negatively curved elsewhere, provided the total still integrates to \(-4\pi\); Gauss–Bonnet is a single scalar constraint on an infinite-dimensional space of possible curvature functions, not a rigidity theorem pinning down \(K\) pointwise. A second common misconception is treating the local formula (Step 5, for a single geodesic triangle) as if it were already the full theorem; it is the seed, but the passage to a closed surface genuinely requires triangulating and invoking Euler's formula, which is Bonnet's contribution.

Worked examples

Round sphere of radius \(R\). Take \(M=S^2_R\subseteq\mathbb{R}^3\), the sphere of radius \(R\), with induced metric. Its Gaussian curvature is constant, \(K=1/R^2\).

Standard computation for a surface of revolution / standard fact for the round sphere. A

\[\iint_{S^2_R} K\,dA = \frac{1}{R^2}\cdot\text{Area}(S^2_R) = \frac{1}{R^2}\cdot 4\pi R^2 = 4\pi.\]

\(K\) constant pulls out of the integral; area of a sphere of radius \(R\) is \(4\pi R^2\) (elementary calculus fact). A

\(S^2\) is a closed orientable genus-\(0\) surface, so \(\chi(S^2)=2-2(0)=2\), and \(2\pi\chi(S^2)=4\pi\), matching Step 2 exactly, for every \(R\).

Genus formula \(\chi=2-2g\) for closed orientable surfaces; direct application of the theorem's Result. A

\iint_{S^2_R} K\,dA = 4\pi = 2\pi\chi(S^2)\ \text{for every radius } R

Reading. Total curvature of any round sphere is \(4\pi\), independent of its radius — a first, minimal check that the theorem's right-hand side does not care how "curved" \(M\) is metrically, only what shape it is topologically.

Standard torus of revolution. Let \(T\subseteq\mathbb{R}^3\) be obtained by revolving a circle of radius \(r\) about an axis in its plane at distance \(a\gt r\) from its centre. Its Gaussian curvature (a standard computation from the first and second fundamental forms) is \[K(\theta,\varphi) = \frac{\cos\theta}{r(a+r\cos\theta)},\] where \(\theta\) parametrises the small circle and is positive on the outer half (\(-\pi/2\lt \theta\lt \pi/2\)), negative on the inner half.

Standard surface-of-revolution curvature formula from a first course in classical differential geometry (Gauss curvature of a surface of revolution via the principal curvatures). A

Directly evaluating \(\iint_T K\,dA=\int_0^{2\pi}\!\!\int_0^{2\pi} \dfrac{\cos\theta}{r(a+r\cos\theta)}\cdot r(a+r\cos\theta)\,d\theta\,d\varphi\) (using \(dA=r(a+r\cos\theta)\,d\theta\,d\varphi\)) reduces to \(\int_0^{2\pi}\!\!\int_0^{2\pi}\cos\theta\,d\theta\,d\varphi\), which vanishes by direct evaluation since \(\int_0^{2\pi}\cos\theta\,d\theta=0\).

Direct calculation: the area element exactly cancels the denominator of \(K\), a special feature of this parametrisation, then elementary integration of \(\cos\theta\) over a full period. A

Gauss–Bonnet predicts this without any of the calculation in Step 2: \(T\) is a closed orientable genus-\(1\) surface, so \(\chi(T)=2-2(1)=0\), hence \(\iint_T K\,dA = 2\pi\cdot 0 = 0\) — matching Step 2, but obtained purely from \(T\)'s topology, valid for every choice of \(a\gt r\gt 0\) and indeed for any metric on the torus whatsoever, not only this embedded one.

Direct application of the Result to genus \(g=1\); illustrates the theorem's real power — the integral is known to vanish before any explicit curvature formula is written down. B

\iint_T K\,dA = 0 = 2\pi\chi(T)\ \text{for every embedded torus of revolution, and every metric on any torus}

Reading. The positive curvature on the torus's outer rim is guaranteed, by topology alone, to be exactly cancelled by the negative curvature on its inner rim — a fact that would otherwise require the explicit calculus of Step 2 to see, for this shape, and would need to be redone entirely for any other embedding or metric were it not for the theorem.

Problems

A geodesic triangle on a sphere of radius \(1\) has three right angles (an "octant" triangle, one-eighth of the sphere). Verify the local Gauss–Bonnet formula (Step 5) for this triangle directly.
Solution
Angle sum \(=\alpha+\beta+\gamma=\pi/2+\pi/2+\pi/2=3\pi/2\), so angle excess \(=3\pi/2-\pi=\pi/2\). The triangle is exactly one octant of the unit sphere, whose total area is \(4\pi\), so its area is \(4\pi/8=\pi/2\). Since \(K\equiv 1\) on the unit sphere, \(\iint_\Delta K\,dA=\pi/2\), matching the angle excess \(\pi/2\) exactly, confirming Step 5.
Compute \(\iint_M K\,dA\) for a closed orientable genus-\(3\) surface, for any Riemannian metric on it.
Solution
\(\chi=2-2g=2-6=-4\), so by the theorem \(\iint_M K\,dA=2\pi\chi=-8\pi\), regardless of the metric chosen.
A flat cone is formed by cutting a wedge of angle \(\theta_0\) (where \(0\lt\theta_0\lt2\pi\)) out of the Euclidean plane and gluing the two straight edges together; the resulting surface is flat (\(K=0\)) away from the apex, but carries a curvature singularity there. Using the boundary version of Gauss–Bonnet on the disc \(R\) of geodesic radius \(\rho\) about the apex (excluding the apex itself, so \(K\equiv0\) on \(R\)), find the "concentrated curvature" that must be assigned to the apex for consistency with a closed cone of total angle \(2\pi-\theta_0\) capped off smoothly, i.e. find \(\lim_{\rho\to 0}\left(2\pi-\int_{\partial R}\kappa_g\,ds\right)\).
Solution
The boundary circle of radius \(\rho\) has ordinary Euclidean length \((2\pi-\theta_0)\rho\) (since the total angle around the apex is \(2\pi-\theta_0\) after removing the wedge), and being a flat circle it has geodesic curvature \(\kappa_g=1/\rho\) at every point, so \(\int_{\partial R}\kappa_g\,ds=\frac{1}{\rho}\cdot(2\pi-\theta_0)\rho=2\pi-\theta_0\), independent of \(\rho\). With \(K\equiv 0\) on \(R\) (topologically a disc, \(\chi(R)=1\), no corners so no exterior-angle terms), the boundary Gauss–Bonnet formula requires \(0+(2\pi-\theta_0)+0=2\pi\chi(R)=2\pi\), which only balances if a curvature "mass" of exactly \(\theta_0\) is concentrated at the excised apex. This recovers the discrete/singular Gauss–Bonnet principle: the apex carries concentrated curvature equal to its angle deficit \(\theta_0\).
(Descartes' theorem, discrete analogue) A cube has \(8\) vertices, at each of which three right-angle faces meet, so the total face-angle at each vertex is \(3\times\pi/2=3\pi/2\), giving an angular defect (deficit from \(2\pi\)) of \(2\pi-3\pi/2=\pi/2\) per vertex. Sum the angular defects over all vertices and compare with \(2\pi\chi(\text{cube})\).
Solution
Total defect \(=8\times\pi/2=4\pi\). The cube's boundary is topologically a sphere, \(\chi=2\), and \(2\pi\chi=4\pi\). The two match: this is the discrete (polyhedral) form of Gauss–Bonnet, with vertex angular defect playing the role of concentrated curvature, exactly analogous to Problem 3's cone apex, summing correctly to \(2\pi\chi\) of the polyhedron's underlying topological sphere — Descartes' theorem on angular defect, historically prior to and a discrete shadow of the smooth theorem.
Show, using Gauss–Bonnet, that the torus \(T^2\) admits no Riemannian metric of strictly positive Gaussian curvature everywhere (\(K\gt 0\) at every point).
Solution
Suppose for contradiction some metric on \(T^2\) had \(K(p)\gt 0\) for every \(p\in T^2\). Since \(T^2\) is compact, \(dA\) is a genuine positive area form with \(\iint_{T^2}dA=\text{Area}(T^2)\gt 0\) finite, and \(K\gt 0\) everywhere pointwise forces the integral \(\iint_{T^2}K\,dA\) to be strictly positive (a continuous strictly-positive function integrated against a finite positive measure on a compact space is strictly positive). But Gauss–Bonnet forces \(\iint_{T^2}K\,dA=2\pi\chi(T^2)=2\pi\cdot 0=0\), a contradiction (\(0\) cannot equal a strictly positive number). Hence no such metric exists on the torus; more generally the identical argument rules out everywhere-positive curvature on any closed orientable surface with \(\chi\leq 0\), i.e. every genus \(g\geq 1\) surface.