Constant Normal Curvature on Curves Lying on a Sphere?

[murmillo]

Homework Statement

What curves lying on a sphere have constant geodesic curvature?

Homework Equations

k^2 = (k_g)^2 + (K_n)^2

The Attempt at a Solution

I'm trying to understand the solution given in the back of the book. It says, a curve on a sphere will have constant curvature. But, is it true that every curve on a sphere has constant normal curvature? The definition of normal curvature I'm using is "the length of the projection of the vector kn over the normal to the surface at p."

 

[Char. Limit]

Homework Statement

What curves lying on a sphere have constant geodesic curvature?

Homework Equations

k^2 = (k_g)^2 + (K_n)^2

The Attempt at a Solution

I'm trying to understand the solution given in the back of the book. It says, a curve on a sphere will have constant curvature. But, is it true that every curve on a sphere has constant normal curvature? The definition of normal curvature I'm using is "the length of the projection of the vector kn over the normal to the surface at p."

My advice would be to calculate that length. Is it a constant? If it is, then the curve must also have constant geodesic curvature.

 

[murmillo]

But wouldn't that be difficult? I think I finally have the answer. Let C be a parametrized regular curve on the unit sphere. Let p and q be two points on C. Then there is a great circle at p that shares the same tangent vector, and by a previous proposition, C and the great circle must have the same normal curvature at p. Since the normal curvature of the great circle is 1 (because the normal vector to the circle is parallel to the normal of the sphere, and the curvature of the unit circle is 1), the normal curvature of C at p is 1, and the same argument applies to q. So any curve that lies on the unit sphere has constant normal curvature. I think that's right.