On each triangle! The first term contributes f MG. Since each edge occurs twice in opposite directions, the various f a6 Kg cancel. The angles around each vertex sum to 27T, so the angle term contributes 27TV.
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The last term contributes 7TF. The Gauss map of a surface in R3. Consider such a surface as pictured in Figure 8. For the purposes of illustration, suppose Kl O. We want to consider the derivative Dn, called the Weingarten map. If we move in the x-direction from Pl toward a point P2, n turns in the x-direction an amount proportional to IKll, but positive while Kl J.
If we move instead in the y-direction from Pl toward a point P3, n turns in the negative y-direction an amount proportional to IK The second column of Dn is [ 0 -K2 J. Hence This identity holds in any orthonormal coordinates. The Gauss map of a hypersurface.
Hopf in [Ho]. The Gauss-Bonnet-Chern Theorem. Amazingly enough, a generalization of the Gauss-Bonnet Theorem 8. An extrinsic proof was obtained by C. Allendoerfer [All] and W. Fenchel [Fen] around , an intrinsic proof by S. Chern [Ch] in Because Nash's theorem was not proved until Actually Chern used the language of differential forms and moving frames. He defined G as the Pfaffian a square root of the determinant of certain curvature forms. His pioneering work on fiber bundles launched the modern era in differential geometry.
Parallel transport. A vectorfield on a curve is called parallel if its covariant derivative along the curve vanishes [see 6. In Euclidean space, a parallel vectorfield is constant-that is, the vectors are all "parallel. If 'Y is a closed curve, the result X l of parallel-transporting X around the curve will be at some angle a from the starting vector X O. By the Gauss-Bonnet formula 8. Hence the Gaussian curvature may be interpreted as the net amount a vector turns under parallel transport around a small closed curve.
More generally, in a higher-dimensional Riemannian manifold M, R ijk1 may be interpreted as the amount a vector turns in the ei,erplane under parallel transport around a small closed curve in the eberplane. For example, heading east along a circle of latitude in the northern unit hemisphere involves curving to the left think of a small circle around the north pole. For latitude near the equator, this effect is small, and a parallel vectorfield ends up pointing slightly to the right, i.
Sure enough, the enclosed area also is almost 21T, the area of the whole northern hemisphere. We have already seen the infinitesimal version of this interpretation of Riemannian curvature in formula 6. Rjkl gives the amount the j component of the original vector X contributes to the i component of the change.
A proof of Gauss-Bonnet in R3. Ambar Sengupta has shown me a simple proof of the Gauss-Bonnet formula 8. Then, of course, the Gauss-Bonnet theorem 8. The proof begins with a simple proof of the formula for a geodesic triangle on the unit sphere 8. Each pair of great circles bounds two congruent lunes L;, L; with angles ai' The lunes Li intersect in 6.
Let y t t:5 1 be the curve bounding R. Let X t be a parallel vectorfield on y, so X t is a multiple of n. Let a be the angle from X O to X l. Here area R' denotes the algebraic area of R', negative if G is negative and n reverses orientation. Similarly, JilR' Kg - 2'7T must be interpreted so that, for example, if G is negative, it switches sign too. This chapter discusses geodesics and some theorems that draw global conclusions from local curvature hypotheses.
For example, Bonnet's Theorem 9. Cheeger and Ebin provide a beautiful reference [CEl on such topics in global Riemannian geometry. Let M be a smooth Riemannian manifold. Recall that by the theory of differential equations, there is a unique geodesic through every point in every direction. Assume that M is geodesic ally complete-that is, geodesics may be continued indefinitely. The geodesic may overlap itself, as the equator winds repeatedly around the sphere.
This condition means that M has no boundary and no missing points. The exponential map. The exponential map Expp at a point p in M maps the tangent space TpM into M by sending a vector v in TpM to the point in M a distance Iv I along the geodesic from p in the direction v. See Figure 9. Then See Figure 9. For any point p in a smooth Riemannian manifold M, Expp is a smooth diffeomorphism at O. It provides very nice coordinates called normal coordinates in a neighborhood of p. Compare to Theorem 3.
Simple means at most one; convex means at least one. Moreover, that geodesic is the shortest path in all of M between the two points. The Hopf-Rinow Theorem says that as long as M is connected, there is a geodesic giving the shortest path between any two points. In particular, Expp maps TpM onto M.
The curvature of SO n. As an example, we now compute the curvature of SO n. In Chapter 5 we defined the second fundamental tensor of a submanifold M of R n by the turning rate K of unit tangents along each slice curve. Figure 9. The singular point q for Expp is called a conjugate point. Conjugate points and Jacobi fields.
Although Expp is a diffeomorphism at 0, it need not be a diffeomorphism at all points v E TpM. For example, let M be the unit sphere and let p be the north pole. Such conjugate points q are characterized by a variation "Jacobi" vectorfield I along the geodesic, vanishing at p and q, for which the second variation of length is zero. In other words, let 'YsCt result from letting a finite piece of geodesic 'Yo t flow a distance sll t I in the direction l t.
It turns out that once a geodesic passes a conjugate point, it is no longer the shortest geodesic from p. We state the following theorem as an early example of the relationship between curvature and conjugate points see [CE, Rauch's Thm. If the sectional curvature K at every point for every section is bounded above by a constant K o, then the distance from any point to a conjugate point is at least 7TI In particular, if the sectional curvature K is nonpositive, there are no conjugate points, and Expp is a local diffeomorphism at every point.
We say that Expp is a submersion or a covering map. Cut points and injectivity radius. A cut point is the last point on a geodesic from p to which the geodesic remains the shortest path from p. Geodesics on the cylinder originating at p stop being shortest paths at the cut point q.
Alternatively, the cut point q could be like the antipodal point on the cylinder of Figure 9. Inside the locus of cut points, Expp is injective, a diffeomorphism. The infimum of distances from any point to a cut point is called the injectiuity radius of the manifold. For example, the injectivity radius of a cylinder of radius a is 7Ta. Bounding the sectional curvature does not bound the injectivity radius away from O.
A cylinder with Gauss curvature 0 can have an arbitrarily small radius and injectivity radius.
Likewise, hyperbolic manifolds with negative curvature can have a small injectivity radius. A common hypothesis for global theorems is bounded geometry: sectional curvature bounded above and injectivity radius bounded below. Bonnet's Theorem. Bonnet's Theorem draws a global conclusion from a local, curvature hypothesis: Let M be a smooth connected Riemannian manifold with sectional curuature bounded below by a positiue constant Ko. We now give a proof sketch beginning with three lemmas. The first lemma relates the second variation of the length of a geodesic to sectional curvature K.
Let l' be a finite piece of geodesic with unit tangent T. Let W be an orthogonal, parallel unit variational vectorfield on 1'. Let W be the unit upward vectorfie1d.
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