From the embedding of , we have the induced metric on , is the Euclidean metric on We can write this metric explicitly under stereographic projection.

where is the north pole

and

by

will a metric on ,

so

From this we define 1-form basis , By the cartan structure equation

we can find that

Also from the metric, we can obtain the christoffel symbols

We want to verify great circles on sphere are geodesics. Under some rotation, assume the great circle is

where is the angle of with the axis, thus represents the arc length of . So after stereographic projection, we can get

Readers can verify it satisfies the equation of geodesics

**Remark:** It is easy to make mistake here. My original intension is using on and try to verify satisfies the geodesic equation. Since the image of is a line, then should be a great circle hence a geodesic.

But one can easily know fails to satisfy the geodesic equation. The subtle error I made is pointed out by Bin Guo, which lies in the parametization of a curve. The geodesics are parametrized by arc length in usual.

To illustrate this, check on , the image of is a line, which is a geodesic under the Euclidean metric. But does not satisfy the geodesic equation, in which .

For the same reason, in , is not such parameter. That is why I use to define a curve.

There is an easy way to verify that great arc is geodesic on sphere even if we don’t know what the are.

For an embedded manifold in , suppose and are two vector fields. Then

One can use this method to prove that , for any is a part of great circle.

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## Comments

good job, write more!

Thank you very much đź™‚