# Differential Geometry and geodesics

1. Apr 16, 2007

### InbredDummy

1)
a. Show that if a curve C is a line of curvature and a geodesic then C is a plane curve.
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Pf.
Let a(s) be a parameterization of C by arc length (which i'm assuming always exists). then C is a geodesic if the covariant derivative of a(s) with respect to s is 0. We also know that if C is a geodesic, then the principal normal at each point on the curve C is paralell to the normal to S at p.

C is a line of curvature, therefore the tangent line of C is a principal direction at p. We also know that a necessary sufficient for C to be a line of curvature is that N'(s) = k(s)*a'(s), where -k(s) is the principal curvature along a'(s).

So I'm guessing i have everything I need to prove that C is a plane curve??? but i don't see what it means to be a plane curve.

So I'm really asking, what conditions need to be met for C to be a plane curve? For a plane curve, the gaussian curvature is identically zero, ie the eigenvalues of the differential of the gauss map are both 0?

Need some serious conceptual guidance.

(b) Show that if a (nonrectilinear) geodesic is a plane curve, then it is a line of curvature.
Pf.

Lost again.

2. Let v and w be vector fields along a curve a: I --> S (where S is our surface).

Prove that
d/dt <v(t), w(t)> = <Dv(t)/dt, w(t)> + <v(t), Dw(t)/dt> where Dv/dt is the covariant derivative.

Pf.

I know that in general,
d/dt <v(t), w(t)> = <dv(t)/dt, w(t)> + <v(t), dw(t)/dt>

But I can't get much more than that again.

3. Consider the torus of revolution generated by rotating the circle
(x-a)^2 + z^2 = r^2, y=0, about the z-axis (a > r > 0). The parallels generated by the points (a+r,0), (a-r, 0), (a,r) are called the maximum parallel, the minimum parallel and the upper parallel, respectively Check which of these parallels is
a) A geodesic
b) an asymptotic curve
c) a line of curvature

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So for this one, I just compute the covariant derivative at the points?

4. Compute the geodesic curvature of the upper parallel of the torus.

thanks guys, im so lost on this material. differential geometry really isnt my strong point

2. Apr 17, 2007