April 14, 2013

http://coursework.tylerlogic.com/courses/math501/homework08

With F = 0, the Christoffel symbols simplify to

From here, one would use the equation

and expand/simplify appropriately to get the desired answer. However, I was not able to find the correct sequence of expansions/simplifications of the equation resulting from the combination of the above equations.

For E = G = λ(u,v) and F = 0 we can use the previous problem to find K to be

Thus since Δlog λ = _{v} + _{u} then we have that

Now let λ = (u^{2} + v^{2} + c)^{-2}. To make the following computation easier to follow, we’ll define γ = u^{2} + v^{2} + c, i.e.
λ = γ^{-2}. We have the following value of K

Let S be a surface with geodesic coordinates X so that E = 1 and F = 0.

Since E = 1 then E

as well as

From the equations of the previous part of the problem, due to most of the Christoffel Symbols being zero, we have this small equation for the Gaussian curvature.

Let g = . Then G = g

Let W(s) = a(s)X

we have u′ = -sin. Thus by equation (1) on page 239 of do Carmo, we have the following since all Christoffel
Symbols are zero except for Γ_{11}^{1}

Solving the ODEs of

and

How much does the parallel transport of a vector rotate after one loop? ???

With the definition of α above we get

Normal Curvature Let N be the normal map on the sphere N(p) = p. Than at some p in the trace of α we have

(a) Non-length-minimizing positive curvature geodesic

The sketch in Figure 1 has a geodesic between p and q that is not length minimizing. The curve that travels “the long way” along the great circle containing p and q will be a geodesic, but not length-minimizing. The geodesic traveling “the short way” will be length-minimizing though.

(b) Non-length-minimizing zero curvature geodesic

The sketch in Figure 2 has a geodesic between p and q that is not length minimizing. Again going the long way, but this time along a horizontal circle of this cylinder.

(c) Non-length-minimizing negative curvature geodesic

The sketch in Figure 3 has a geodesic between p and q that is not length minimizing. Again going the long way, but this time around the “waist” of this surface of revolution.

(e) A surface where any two points can be joined by a geodesic, but the geodesic is only defined for a finite amount of time.

The sketch in Figure 4 has a geodesic between every two points, namely the straight line between them. However, since geodesics need to have constant velocity, then only finite time geodesics could exist.