- #1
InbredDummy
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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 parallel 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
-
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, I am so lost on this material. differential geometry really isn't my strong point
a. Show that if a curve C is a line of curvature and a geodesic then C is a plane curve.
-
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 parallel 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
-
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, I am so lost on this material. differential geometry really isn't my strong point