Meridians onf surface of revolution.

Hi Guys,

I have a homework questions that I am stuck on, it's the last in the set and I can't do it!

It reads:

Prove that the meridians are geodesics when the curve: (x(u),0,u) is revolved around the z-axis.

I have an idea of using the Geodesic Curvature formula etc but do not know where to start.

If anyone could give me a hand, please let me know!

Cheers.

Patrick.

Re: Meridians onf surface of revolution.

Consider a constant speed curve along a meridian of the revolved surface S,

where is some open interval of .

To show is a geodesic, you want to show that is orthogonal to S.

Since it has constant speed, you can say something about the relationship between and .

Consider the plane P that defines the meridian by cutting through S.

Since stays in P, so do all of its derivatives.

At any point along , P is spanned by and the normal to S at that point.

Put it all together, and you deduce something about that proves that is a geodesic.