Math 501: Differential Geometry
April 14, 2013
1 do Carmo pg 237 problem 1
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.
2 do Carmo pg 237 problem 2
For E = G = λ(u,v) and F = 0 we can use the previous problem to find K to
Thus since Δlog λ = v + u then we have that
Now let λ = (u2 + v2 + c)-2. To make the following computation easier to follow, we’ll define γ = u2 + v2 + c, i.e.
λ = γ-2. We have the following value of K With our definition of γ, γu = 4u, γv = 4v, and γuu = γvv = 4. Continuing the equation above we obtain
Let S be a surface with geodesic coordinates X so that E = 1 and
F = 0.
(a) Christoffel Symbols
Since E = 1 then Eu = Ev = 0 and thus with F = 0 most of the Christoffel
symbols are zero. Making use of the equations in (2) on pg 232 of do Carmo, we have
as well as
(b) Gaussian Curvature
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.
(c) g for
Let g = . Then G = g2, Gu = 2ggu and Guu = 2gu2 + 2gguu, so
substituting these equations into the result of the previous part of the problem gets us
Let W(s) = a(s)Xθ + b(s)Xϕ. With
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 Γ111
Solving the ODEs of we find that
How much does the parallel transport of a vector rotate after one loop?
With the definition of α above we get
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.
Figure 1: Sketch of positive curvature surface with a geodesic between p and q which is not length-minimizing.
(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.
Figure 2: Sketch of zero curvature surface with a geodesic between p and q which is not length-minimizing.
(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.
Figure 3: Sketch of negative curvature surface with a geodesic between p and q which is not length-minimizing.
(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
Figure 4: Sketch of surface with only finite time geodesics.