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Geodesics Outline 1. Geodesics A curve γ on a surface S is called a geodesic if 1. γ has unit speed, and 2. γ̈ is normal to S at each point. For example: • The geodesics on a sphere are great circles. • The geodesics on a cylinder are straight lines, circular cross sections, and helices that wind around the cylinder. 2. Finding Geodesics There are two major tricks for finding geodesics on a surface: 1. Any straight line on a surface is a geodesic. 2. If a surface intersects a plane at right angles, then the curve of intersection is a geodesic. In the second case, the geodesic is called a normal section of the surface. Many surfaces of interest have normal sections: • Let S be a generalized cylinder C × R, where C is a plane curve. Then each horizontal cross section C × {z0 } is a normal section of S. • Let S be a surface of revolution r = f (z). Then each meridian curve (i.e. curve of constant θ) is a normal section of S. • Let S be a surface of revolution r = f (z), and let z0 be a z-value such that f 0 (z0 ) = 0. Then the circular cross-section of S at this z-value is a normal section of S. One can prove that a planar curve on a surface is a geodesic if and only if it is either a straight line or a normal section. 3. Properties of Geodesics Given a point on a surface, there is essentially one geodesic through that point in any given direction: Theorem. Let S be a surface, let p ∈ S, and let t be a unit tangent vector to S at p. Then there exists a unique geodesic γ passing through p whose tangent vector at p is t. It follows from the uniqueness in the above theorem that two different geodesics can never be tangent at a point. Another important property of geodesics is that they are shortest-length paths: Theorem. Let γ : [a, b] → S be a geodesic. Then there exists an > 0 so that γ is the shortest path on S from γ(a) to γ(a + ). Moreover, any shortest-length path must be a geodesic: Theorem. Let S be a surface, let p, q ∈ S. Then any path from p to q on S with minimum length must be a geodesic. Note that there may not exist a minimum-length path from p to q. Hence, this theorem does not guarantee that there exists a geodesic connecting any two points on a surface. Because geodesics are locally minimum-length, they must be preserved by local isometries: Theorem. If f : S1 → S2 is a local isometry and γ is a geodesic on S1 , then f ◦γ is a geodesic on S2 . 4. Geodesic Triangles A region R on a surface is said to be simply-connected if it is homeomorphic to an open disk. Such a region cannot have any “holes” or other topological features. A geodesic triangle on a surface is a simply-connected region bounded by three geodesic segments. Such a region has three interior angles, whose sum is determined by the following theorem: Theorem. Let T be a geodesic triangle on a surface S, and let α, β, and γ be the interior angles of T . Then ZZ K dA α+β+γ−π = T where K is the Gaussian curvature of S. This is a simple case of the Gauss-Bonnet Theorem. The quantity α + β + γ − π is called the angle excess of the triangle. As a special case, we have: Theorem. Let T be a geodesic triangle on the unit sphere, with interior angles α, β, and γ. Then the area of T is equal to α + β + γ − π. That is, the angle excess of any spherical triangle is equal to the area. A similar theorem holds for the hyperbolic plane: the angle excess of any hyperbolic triangle is the negative of the area. Practice Problems 1. Let C be the cylinder x2 + y 2 = 1. (a) Find the length of the shortest path on C from the point (1, 0, 0) to the point (0, 1, 1). (Hint: Use a local isometry to unroll the cylinder.) (b) Find parametric equations for the shortest geodesic segment between these two points. 2. Let γ be the geodesic on the unit sphere satisfying γ(0) = 1 (1, 2, 2) 3 and γ̇(0) = 1 (−2, 2, −1). 3 Find a formula for γ(t). 3. In each of the following parts, let S1 and S2 be the given surfaces, and let C = S1 ∩ S2 . Determine whether C is a geodesic curve on (i) the surface S1 , and (ii) the surface S2 . (a) S1 is the unit sphere, and S2 is the plane z = x. (b) S1 is the paraboloid z = r2 , and S2 is the plane x = y. (c) S1 is the paraboloid z = r2 , and S2 is the cone z = r with the origin removed. (d) S1 is the catenoid r = cosh z, and S2 is the cylinder x2 + y 2 = 1. (e) S1 is the helicoid z = θ, and S2 is the cylinder x2 + y 2 = 1. (f) S1 is the parabolic cylinder z = x2 , and S2 is the plane x = 1.