Reading: Chapter 15 through p. 292 and your lecture notes.
Problem U: Show that geodesics on the Poincaré upper half-plane are exactly what we called "lines." [Hint: Use the equation kg(t) = 0 from the second displayed formula on p. 188 using a parametrization x = t, y = y(t), unless x = const, which needs to be treated separately. The equation should then transform into y y'' + (y')2 = -1 or (yy')' = -1, which integrates directly as yy' = -t + C. This is a separable first-order differential equation and easy to solve.]
Problem V: Show that SO(3), the set (group, in fact, if you know what that means) of orthogonal 3×3 matrices with determinant 1 is a smooth submanifold of dimension 3 in R9. [Hint: Perhaps, the easiest way to do that is to write out the six defining equations (ignoring det A = 1 at first, as it almost depends on the other equations, as they imply det A = ± 1) in nine variables aij, find the corresponding Jacobi matrix, and see that its six rows are linearly independent. Finally, interpret det A = 1 as defining a connected component.]
Problem W: Show that SO(2) is diffeomorphic to the unit circle S1. [Hint: Present elements of SO(2) as rotations about the origin.]
Problem X: Show that SO(3) is diffeomorphic to the real projective space RP3. Construct a diffeomorphism; do not worry about proving that it actually is a diffeomorphism. [Hint: Present elements of SO(3) as rotations about an axis passing through the origin. Associate with such a rotation with a point in the unit ball in R3. Identify the unit ball with antipodal boundary points identified with the three-sphere with antipodal points identified. Identify that with RP3.]
Problem Y: Show that the space of configurations of a rigid line interval in the plane is a smooth manifold.
Problem Z (Not to be turned in -- just for your entertainment): Prove that the parallel transport of a vector along a curve on a Riemannian manifold of zero curvature depends only on the homotopy class of the curve. In other words, suppose that α(t) is a curve and β(t) is another curve on a given Riemannian manifold with the same endpoints α(a) = β(a) and α(b) = β(b) obtained by a smooth homotopy (deformation) γ(s,t) of α: γ(0,t) = α(t), γ(1,t) = β(t), and γ(s,a) = const, γ(s,b) = const. Show that the parallel transport of a tangent vector to the manifold along α is the same as along β. [Hint: Use covariant differentiation along the parameter s.]
Last modified: (2013-05-12 19:14:23 CDT)