Home | Pullback connections and why isometries preserve geodesics

Why do isometries preserve geodesics (locally minimizing paths)? Under the isometry $ϕ : (M, g, \nabla) \to (\tilde{M}, \tilde{g}, \tilde{\nabla})$ , we have that $\nabla = ϕ^{*} \tilde{\nabla}$ . In other words, the only Levi-Civita connection on $M$ 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 $M$ , which is torsion-free and compatible with the metric $g$ . Uniqueness of the connection on $M$ satisfying these properties then gives $\nabla = ϕ^{*} \tilde{\nabla}$ . These aspects can be shown entirely within the framework tensor fields, making the results automatically global. Once we understand why $\nabla = ϕ^{*} \tilde{\nabla}$ we can answer our question.

First note that $ϕ^{*} \tilde{\nabla}$ is a map $Γ (T M) \times Γ (T M) \to Γ (T M)$ . Then we want to check linearity in the first entry and the characteristic Leibniz rule in the second. Recall that $ϕ_{*} (f X) = (f \circ ϕ^{- 1}) ϕ_{*} X$ . Let for notation’s sake $ψ := ϕ^{- 1} : N \to M$ , $D := ϕ^{*} \tilde{\nabla}$ and the application of a derivation $X$ on a function $h$ be $X . h = X h$ . Then for $f \in C^{\infty} (M)$

\begin{aligned} (1) & D_{f X} Y & = ψ_{*} {\tilde{\nabla}}_{ϕ_{*} (f X)} ϕ_{*} Y = ψ_{*} {\tilde{\nabla}}_{f \circ ψ ϕ_{*} X} ϕ_{*} Y \\ (2) & = ψ_{*} (f \circ ψ {\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} Y) = (f \circ ψ \circ ϕ) ψ_{*} {\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} Y \\ (3) & = f D_{X} Y \end{aligned}

and recalling for $g \in C^{\infty} (N)$ , $(ϕ_{*} X) . g = (X . (f \circ ϕ)) \circ ψ$ as functions on $N$ ,

\begin{aligned} (4) & D_{X} (f Y) & = ψ_{*} {\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} (f Y) = ψ_{*} {\tilde{\nabla}}_{ϕ_{*} X} (f \circ ψ) ϕ_{*} Y \\ (5) & = ψ_{*} ((ϕ_{*} X) . (f \circ ψ) ϕ_{*} Y + f \circ ψ {\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} Y) \\ (6) & = ψ_{*} ((X . (f \circ ψ \circ ϕ)) \circ ψ) ϕ_{*} Y + f ψ_{*} {\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} Y \\ (7) & = ψ_{*} (\underset{\in C^{\infty} (N)}{\underset{⏟}{(X . f) \circ ψ)}} ϕ_{*} Y + f D_{X} Y \\ (8) & = (X . f) Y + f D_{X} Y \end{aligned}

Next we show that it is torsion free, using the fact that $[ϕ_{*} X, ϕ_{*} Y] = ϕ_{*} [X, Y]$ . We have from torsion-freeness of $\tilde{\nabla}$ that ${\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} Y - {\tilde{\nabla}}_{ϕ_{*} Y} ϕ_{*} X = [ϕ_{*} X, ϕ_{*} Y] = ϕ_{*} [X, Y]$ , whereby $D_{X} Y - D_{Y} X = ψ_{*} ϕ_{*} [X, Y] = [X, Y]$ since pushforwards are isomorphisms.
Lastly we use compatibility of $\tilde{\nabla}$ with $\tilde{g}$ to get it for $D$ with $g$ . Recall $g (X, Y) = g_{I d} (X, Y) = {\tilde{g}}_{ϕ} (ϕ_{*} X, ϕ_{*} Y)$ since $ϕ : M \to N$ is an isometry. Let $Γ (T M) ∋ Z = ψ_{*} ξ$ for $ξ \in Γ (T N)$ . Then since the expressions need to be equal as functions on $M$ : $\begin{aligned} (9) & g (D_{X} Y, Z) & = g (ψ_{*} {\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} Y, ψ_{*} ξ) = {\tilde{g}}_{ϕ} ({\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} Y, ξ) = {\tilde{g}}_{ϕ} ({\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} Y, ϕ_{*} Z) \\ (10) & = ((ϕ_{*} X) . {\tilde{g}}_{I d} (ϕ_{*} Y, ϕ_{*} Z)) \circ ψ - {\tilde{g}}_{ϕ} (ϕ_{*} Y, {\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} Z) \\ (11) & = X . {\tilde{g}}_{ϕ} (ϕ_{*} Y, ϕ_{*} Z) - g (Y, ψ_{*} {\tilde{\nabla}}_{ϕ_{*} X} ϕ_{*} Z) \\ (12) & = X . g (Y, Z) - g (Y, D_{X} Z) . \end{aligned}$

Thus $\nabla = ϕ^{*} \tilde{\nabla}$ . As a bonus we can nicely see from this that geodesics get mapped by an isometry $ϕ : M \to N$ . If $c : I \to M$ is a geodesic then the corresponding curve $γ := ϕ \circ c : I \to N$ is as well. (This curve is the flow ${\tilde{θ}}_{t} (n) = γ (t)$ with $γ (0) = n$ from the vector field $ϕ_{*} c^{'} (t)$ with $c (0) = ψ (n)$ .) $\begin{matrix} (13) & ψ_{*} {\tilde{\nabla}}_{γ^{'}} γ^{'} = ψ_{*} {\tilde{\nabla}}_{ϕ_{*} c^{'}} ϕ_{*} c^{'} = D_{c^{'}} c^{'} = \nabla_{c^{'}} c^{'} = 0 \end{matrix}$ so ${\tilde{\nabla}}_{γ^{'}} γ^{'} = 0$ since $ψ_{*}$ is an isomorphism. (Note $γ^{'} (t) = \frac{d}{d t} (ϕ \circ c (t)) = T_{c (t)} ϕ c^{'} (t) = (ϕ_{*} c^{'}) (t)$ where $T_{c (t)} ϕ : T_{c (t)} M \to T_{γ (t)} N$ is the tangent map or differential.)