Why do isometries preserve geodesics (locally minimizing paths)?
Under the isometry , we have that . In other words, the only Levi-Civita connection on is the pullback connection.
How to prove this? The approach is to prove that the right hand side satisfies the properties of a connection on , which is torsion-free and compatible with the metric . Uniqueness of the connection on satisfying these properties then gives .
These aspects can be shown entirely within the framework tensor fields, making the results automatically global.
Once we understand why we can answer our question.
- First note that is a map . Then we want to check linearity in the first entry and the characteristic Leibniz rule in the second.
Recall that . Let for notationβs sake , and the application of a derivation on a function be . Then for
and recalling for , as functions on ,
- Next we show that it is torsion free, using the fact that . We have from torsion-freeness of that , whereby since pushforwards are isomorphisms.
- Lastly we use compatibility of with to get it for with . Recall since is an isometry. Let for . Then since the expressions need to be equal as functions on :
Thus . As a bonus we can nicely see from this that geodesics get mapped by an isometry . If is a geodesic then the corresponding curve is as well. (This curve is the flow with from the vector field with .)
so since is an isomorphism. (Note where is the tangent map or differential.)