Planar Geometry and Gauss-Bonnet
Let \((M,g)\) be a smooth Riemannian manifold. Write \(\nabla\) for its Levi–Civita connection and \(T_pM\) for the tangent space at \(p\). The coordinate conventions are those in tangent spaces and derivations.
On a surface there is only one tangent two-plane at each point, so sectional curvature becomes a function of the point. The Gauss–Bonnet theorem shows that its integral is constrained by topology. To reach this global statement, we first express curvature using an orthonormal frame and then relate it to the turning of a boundary curve.
The Schur theorems (Theorems 7 and 8) tell us what happens when the ambient dimension of the manifold, \(n\), is greater than or equal to 3. These clearly fail in \(n=2\), because the sectional curvature (Gauss curvature!) need not be constant. But, \(n=2\) has historically been the dominant area of study—the study of planar geometry greatly preceded the modern study of differentiable manifolds. There are many simplifications, and many rigid results, in the study of surfaces in \(\mathbb R^3\) (which turns out to be generalizable to studying \(2\)-manifolds).
At each point of a Riemannian surface there is just one tangent two-plane. Thus its sectional curvature is a function \(K(p)\); the quotient defining sectional curvature requires linearly independent \(X,Y\). Gram–Schmidt applied to coordinate vector fields produces a smooth orthonormal frame, generally not a coordinate frame. If \(X=x_1E_1+x_2E_2\) and \(Y=y_1E_1+y_2E_2\), curvature symmetries give
The equalities follow from the fact that \(\{E_1,E_2\}\) is an orthonormal frame. The question is what is the sectional curvature in \(2\)-manifolds. The following theorem answers that: the sectional curvature agrees with the Gauss curvature (see Remark 15) on \(2\)-manifolds.
Theorem 1. On a Riemannian surface, \(K(p)=\kappa(X,Y)\) for every linearly independent pair in \(T_pM\). For a surface isometrically immersed in \(\mathbb R^3\), this intrinsic curvature equals the classical Gauss curvature, the determinant of its shape operator.
Remark 2 (Frames in a chart). Choose an oriented coordinate neighborhood \(U\) and pull the metric to its image in \(\mathbb R^2\). Gram–Schmidt gives
Both denominators are strictly positive, so the frame is smooth everywhere in the chart. A simple closed curve on a general surface need not bound a disk or lie in one chart. In the local Gauss–Bonnet argument we explicitly assume a compact disk with piecewise smooth regular boundary contained in such a chart. Its boundary need not be geodesic.
Remark 3 (Connection form). For an oriented orthonormal frame define
Metric compatibility, differentiated in the identities \(\langle E_i,E_j\rangle=\delta_{ij}\), gives
These signs follow from the displayed definition. Torsion-freeness therefore gives \([E_1,E_2]=-\omega(E_1)E_1-\omega(E_2)E_2\).
Lemma 4. Let \(M\) be a 2-manifold. Then, the Gauss curvature at \(p \in M\) is given by \(K(p) = E_2 (\omega(E_1)) - E_1 (\omega(E_2)) - \omega(E_1)^2 - \omega(E_2)^2 ~.\)
Proof. Put \(\omega_i=\omega(E_i)\), and let \(\theta_i\) be the dual coframe. The exterior derivative formula and torsion-freeness give
For example \(d\theta_1(E_1,E_2)=-\theta_1([E_1,E_2])=\omega_1\), which is the value of the first right side. Expanding the product rule correctly as an identity of two-forms,
With the convention \(R(X,Y)Z=\nabla_Y\nabla_XZ-\nabla_X\nabla_YZ+\nabla_{[X,Y]}Z\), substitution of the connection formulas gives \(R(X,Y)E_1=-d\omega(X,Y)E_2\). Consequently
Equivalently \(d\omega=-K\,dA\), where \(dA=\theta_1\wedge\theta_2\). This derives the needed structural equation directly. ◻
Proof of Theorem 1. Independence of the basis of the tangent plane follows from the sectional curvature formula. For the extrinsic statement choose a local unit normal \(\nu\), write \(D_XY=\nabla_XY+h(X,Y)\nu\), and use \(D_X\nu=-A X\), so \(h(X,Y)=\langle AX,Y\rangle\). The tangential part of the ambient flat-curvature identity yields, in our sign convention,
Indeed the tangential part of \(D_YD_XZ\) is \(\nabla_Y\nabla_XZ-h(X,Z)AY\), and subtracting the analogous term gives the displayed identity. Thus for an orthonormal pair,
Changing \(\nu\) changes both principal curvatures' signs and leaves their product unchanged. This proves that the intrinsic definition agrees with classical Gauss curvature. ◻
From Theorem 1, we can now replace statements about sectional curvature with Gauss curvature, whenever we work on \(2\)-manifolds. (There is some ambiguity with using surfaces and \(2\)-manifolds interchangeably; “surfaces” are generally used with reference to \(\mathbb R^3\) as the ambient space. We will largely ignore this distinction, but will be clear when some statement of the theorem is based on the study of surfaces in \(\mathbb R^3\).)
On a smooth regular boundary arc parametrized by arclength, let \(T=\gamma'\) and let \(N\) be the unit tangent vector to the surface obtained by a positive right-angle rotation of \(T\). For the positively oriented boundary of a domain, \(N\) points inward. This is a normal to the curve within the surface, not an ambient surface normal.
Definition 5. The signed curvature is defined as
Remark 6. Differentiating \(\langle \gamma',\gamma' \rangle \equiv 1\) yields \(2 \langle \nabla_{\dif/\!\dif t} \gamma'(t), \gamma'(t) \rangle = 0\), implying \(\nabla_{\dif/\!\dif t} \gamma'(t) = \kappa_N (t) N(t)\). This means if \(\gamma\) is curving toward \(\Omega\), then \(\kappa_N (t)\) is positive, and it is negative if \(\gamma\) is curving away from \(\Omega\).
The local Gauss–Bonnet formula balances three quantities: curvature in the interior, geodesic curvature along the boundary, and the turning at corners. For a polygon with geodesic edges, the boundary integral vanishes and the formula becomes a relation between area curvature and angles.
Theorem 7 (Local Gauss–Bonnet). Let \(\Omega\) be a disk whose closure lies in an oriented coordinate chart of a smooth Riemannian surface. Suppose its positively oriented boundary is a simple closed piecewise smooth regular curve, with signed exterior turning angles \(\varepsilon_i\). Then
The disk hypothesis is essential to this right side; for a general compact surface with boundary it becomes \(2\pi\chi(\Omega)\).
Proof. On each smooth arc lift the angle of the unit tangent relative to the oriented frame to a real function \(\theta\), so
Differentiate along arclength using Remark 3 to obtain \(\nabla_TT=(\theta'+\omega(T))N\); hence \(\kappa_N=\theta'+\omega(T)\). At each corner include the signed change from the incoming to outgoing tangent. The turning-number theorem for a positively oriented simple plane curve says that the sum of these jumps and the changes along the arcs is \(2\pi\). This theorem also applies to our frame: Gram–Schmidt gives a field of orientation-preserving triangular changes of basis, homotopic through such changes to the identity on the disk, so it cannot change the boundary tangent's winding number.
Integrating and summing the arcs now gives
The middle equality is Stokes' theorem for a piecewise smooth disk; the last is the structural equation proved above. For geodesic edges \(\kappa_N=0\), so curvature is exactly the angle excess over the Euclidean polygon formula. ◻
To pass from the local formula to the whole surface, take a finite smooth triangulation and subdivide until each closed triangle lies in a coordinate chart. A smooth triangulation is a collection of smoothly parametrized curved triangles meeting along whole edges or vertices. Its existence for compact smooth surfaces is a triangulation theorem, not a consequence of the definition. We use it as a topological input. For a surface without boundary, each edge belongs to exactly two triangles.
Definition 8. Let \(M\) be a triangulated 2-manifold. The Euler characteristic of \(M\) with respect to the given triangulation is
where \(V\) is the number of vertices in the triangulation, \(E\) is the number of edges, and \(F\) is the number of faces (or, the interiors of the triangles).
Theorem 9. Let \(M\) be a differentiable, smoothly triangulated, compact, boundaryless \(2\)-manifold. Then,
where \(K\) is the Gauss curvature of \(g\), and \(\dif \operatorname{Vol}_g\) is the density.
Proof. First suppose \(M\) is connected and oriented. Apply the local formula to every triangle. A common edge has opposite boundary orientations in its two triangles, so their signed geodesic-curvature integrals cancel. If \(\theta_{ij}\) are the interior corner angles, the exterior angles are \(\pi-\theta_{ij}\), and therefore
The triangles fill a neighborhood of every vertex, so the angles around each vertex sum to \(2\pi\). Thus \(\int_MK\,dA=2\pi V-\pi F\). Counting the two incidences of each edge gives \(3F=2E\), and hence
For a nonorientable connected surface, form the orientation double cover: its points are pairs \((p,o)\), where \(o\) is one of the two orientations of \(T_pM\). Local oriented frames give smooth sheets; they also give the cover a canonical orientation. The cover is connected, since a disconnected double cover would select an orientation continuously on \(M\). Pull back the metric along \(\pi:\widetilde M\to M\). This is a locally isometric metric, not an assertion of conformality between metrics on the same manifold. Local isometry preserves curvature and area densities, so integration over the two sheets gives
Subdivide the triangulation into evenly covered neighborhoods. Every vertex, edge, and face then has two lifts, whence \(\chi(\widetilde M)=2\chi(M)\). Apply the oriented formula upstairs and divide by two. Finally a compact manifold has finitely many components; add their formulas. ◻
Remark 10. For compact 2-manifolds, Gauss–Bonnet determines the total curvature from the Euler characteristic. Combined with the classification of compact surfaces, it therefore restricts the possible signs of curvature. Recall that the classification theorem tells us every compact, connected, boundaryless, orientable 2-manifold \(M\) is homeomorphic to a sphere or a connected sum of \(n\) tori, and every nonorientable one is homeomorphic to a connected sum of \(n\) copies of the real projective plane \(\mathbb R P^2\) (\(n\) is called the genus of \(M\)).
Theorem 11 (Consequences of the Gauss-Bonnet Theorem: Classification). Let \((M,g)\) be a compact, connected, boundaryless, differentiable 2-manifold with Gauss curvature \(K\). Then,
If \(M\) is homeomorphic to the sphere or the projective plane, then \(K > 0\) somewhere.
If \(M\) is homeomorphic to the torus or the Klein bottle, then \(K \equiv 0\), or \(K\) takes on both positive and negative values.
If \(M\) is any other compact surface, then \(K < 0\) somewhere.
Proof. The classification of closed connected surfaces gives \(\chi=2-2g\) in the orientable case and \(\chi=2-h\) for a connected sum of \(h\geq1\) projective planes. Thus the three cases have positive, zero, and negative Euler characteristic respectively. Gauss–Bonnet fixes the sign of \(\int K\,dA\). A continuous nonnegative function that is positive at one point has strictly positive integral, since it is bounded below on a small disk; the same argument applies to \(-K\). This proves all three alternatives. ◻
Theorem 12 (Consequences of the Gauss-Bonnet Theorem: Homotopy Group). Let \((M,g)\) be a compact, connected, boundaryless, differentiable 2-manifold with Gauss curvature \(K\).
If \(K > 0\) everywhere on \(M\), then the universal covering manifold of \(M\) is homeomorphic to \(\mathbb S^2\), and \(\pi_1 (M)\) is either trivial or isomorphic to the group \(\mathbb Z / 2\mathbb Z\).
If \(K \leq 0\) everywhere on \(M\), then the universal covering manifold of \(M\) is homeomorphic to \(\mathbb R^2\), and \(\pi_1 (M)\) is infinite.
Proof. If \(K>0\), the integral is positive, so the preceding classification leaves only \(S^2\) and \(\mathbb RP^2\). Their simply connected covers are \(S^2\), and their fundamental groups are respectively trivial and the antipodal deck group \(\mathbb Z/2\mathbb Z\).
If \(K\leq0\), lift the metric to the universal cover. Compactness makes the base complete, and geodesics lift to show that the cover is complete. The Cartan–Hadamard theorem, proved later in these notes, makes its exponential map a diffeomorphism from \(\mathbb R^2\). If the fundamental group were finite, this universal cover would be a finite cover of a compact space and hence compact, contrary to its being \(\mathbb R^2\). The nonpositive conclusion therefore uses completeness and Cartan–Hadamard in addition to Gauss–Bonnet. ◻
Reading: Carmo (1992), Riemannian geometry.