Geodesic equations and Christoffel symbols

Demon117

I've been thinking about this quite a bit. So it is clear that one can determine the Christoffel symbols from the first fundamental form. Is it possible to derive the geodesics of a surface from the Christoffel symbols?

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Matterwave

Gold Member
I'm a little bit confused by the language. Is the overall metric for a manifold usually referred to as the "first fundamental form"? From my readings, I've only encountered the language "first fundamental form" in the case of embedded submanifolds for which the "first fundamental form" is the (restricted) metric on the embedded submanifold, and the second fundamental form is the extrinsic curvature.

So are you talking about geodesics on an embedded submanifold? I'm just wondering why you use the language "first fundamental form" instead of the more often seen word "metric". micromass

I'm a little bit confused by the language. Is the overall metric for a manifold usually referred to as the "first fundamental form"? From my readings, I've only encountered the language "first fundamental form" in the case of embedded submanifolds for which the "first fundamental form" is the (restricted) metric on the embedded submanifold, and the second fundamental form is the extrinsic curvature.

So are you talking about geodesics on an embedded submanifold? I'm just wondering why you use the language "first fundamental form" instead of the more often seen word "metric". Since he uses terminology like "surface" and "first fundamental form", I assume that he only works with embedded submanifolds in $\mathbb{R}^n$. A lot of introductory differential geometry books will only treat this case and don't work with general manifolds and metrics.

Demon117

Well, here is the issue. Suppose I have a helicoid parameterized by $Y(u,\theta) = (sinh(u)cos(\theta), -sinh(u)sin(\theta), \theta)$. For some point on this surface with the coordinate $(u,\theta)$, how can one easily compute the geodesic passing through that point using the first fundamental form? Or is that even possible? Call the point p.

This has bugged me for quite some time :/

nevermind.

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