Math 147A HW5
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Math 147A HW5
1. Prove the Gauss equation by using the Levi-Civita Connection and the Riemann curvature tensor.
2. Prove the Gauss’s remarkable theorem by using the Levi-Civita Connection and the Riemann curvature tensor.
3. Prove the local Gauss-Bonnet theorem by using the orthogonal parametrization.
4. Calculate all Christoffel symbols of the following Riemannian metric g on M = ((x, y, z) e R3 : x > 0}
: g(x, y, z) = (gij (x, y, z))1≤i﹐j≤3 = 、( = ╱.1 x2 ( . ( |
0 x2 0 |
0、 0 ( 1. . |
5. Consider the surface of revolution defined by a smooth positive function f : R → R>0 ,
r(u, v) = (f (u) cos v, f (u) sin v, u).
Show that the curves given by u(t) = a, where a is a constant, and v(t) = t/f (a) are geodesics if and only if f\ (a) = 0.
6. From the problem 5, compute formulae for general geodesics including meridians and parallels.
2022-09-05