Why do isometries preserve geodesics (locally minimizing paths)? Under the isometry \(\phi:(M,g,\nabla)\to (\tilde M,\tilde g,\tilde \nabla)\), we have that \(\nabla=\phi^*\tilde\nabla\). 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 \(\nabla=\phi^*\tilde\nabla\). These aspects can be shown entirely within the framework tensor fields, making the results automatically global. Once we understand why \(\nabla=\phi^*\tilde\nabla\) we can answer our question.

  • First note that \(\phi^*\tilde\nabla\) is a map \(\Gamma(TM)\times\Gamma(TM)\to \Gamma(TM)\). Then we want to check linearity in the first entry and the characteristic Leibniz rule in the second. Recall that \(\phi_*(fX)=(f\circ\phi^{-1})\ \phi_*X\). Let for notation’s sake \(\psi:=\phi^{-1}:N\to M\), \(D:=\phi^*\tilde\nabla\) and the application of a derivation \(X\) on a function \(h\) be \(X.h=Xh\). Then for \(f\in C^{\infty}(M)\)
\[\begin{align} D_{fX}Y &= \psi_*\tilde\nabla_{\phi_*(fX)}\phi_*Y = \psi_*\tilde\nabla_{f\circ\psi\ \phi_*X}\phi_*Y \\ &= \psi_*(f\circ\psi\ \tilde\nabla_{\phi_*X}\phi_*Y) = (f\circ\psi\circ\phi)\ \psi_*\tilde\nabla_{\phi_*X}\phi_*Y \\ &= fD_X Y \end{align}\]

and recalling for \(g\in C^{\infty}(N)\), \((\phi_*X).g = (X.(f\circ\phi))\circ\psi\) as functions on \(N\),

\[\begin{align} D_{X}(fY) &= \psi_*\tilde\nabla_{\phi_*X}\phi_*(fY) = \psi_*\tilde\nabla_{\phi_*X}(f\circ\psi)\phi_*Y \\ &= \psi_*\left( (\phi_*X).(f\circ\psi)\phi_*Y + f\circ\psi\ \tilde\nabla_{\phi_*X}\phi_*Y \right) \\ &= \psi_*((X.(f\circ\psi\circ\phi))\circ\psi)\phi_*Y + f\psi_*\tilde\nabla_{\phi_*X}\phi_*Y \\ &= \psi_*(\underbrace{(X.f)\circ\psi)}_{\in C^{\infty}(N)}\phi_*Y + fD_{X}Y \\ &=(X.f) Y+ fD_{X}Y \end{align}\]
  • Next we show that it is torsion free, using the fact that \([\phi_* X,\phi_* Y]=\phi_*[X,Y]\). We have from torsion-freeness of \(\tilde\nabla\) that \(\tilde\nabla_{\phi_* X}\phi_* Y - \tilde\nabla_{\phi_* Y}\phi_* X = [\phi_* X,\phi_* Y] = \phi_*[X,Y]\), whereby \(D_XY-D_YX = \psi_*\phi_*[X,Y]=[X,Y]\) since pushforwards are isomorphisms.
  • Lastly we use compatibility of \(\tilde \nabla\) with \(\tilde g\) to get it for \(D\) with \(g\). Recall \(g(X,Y)=g_{Id}(X,Y)=\tilde g_{\phi}(\phi_*X,\phi_*Y)\) since \(\phi:M\to N\) is an isometry. Let \(\Gamma(TM)\ni Z=\psi_*\xi\) for \(\xi\in\Gamma(TN)\). Then since the expressions need to be equal as functions on \(M\): \(\begin{align} g(D_XY,Z) &= g(\psi_*\tilde\nabla_{\phi_*X}\phi_*Y,\psi_*\xi) = \tilde g_\phi(\tilde\nabla_{\phi_*X}\phi_*Y,\xi) = \tilde g_\phi(\tilde\nabla_{\phi_*X}\phi_*Y,\phi_* Z) \\ &= ((\phi_*X).\tilde g_{Id}(\phi_*Y,\phi_*Z))\circ\psi - \tilde g_\phi (\phi_*Y,\tilde\nabla_{\phi_*X}\phi_*Z) \\ &= X.\tilde g_\phi(\phi_* Y,\phi_* Z) - g(Y,\psi_*\tilde\nabla_{\phi_*X}\phi_*Z) \\ &= X.g(Y,Z) - g(Y,D_XZ). \end{align}\)

Thus \(\nabla=\phi^*\tilde\nabla\). As a bonus we can nicely see from this that geodesics get mapped by an isometry \(\phi:M\to N\). If \(c:I\to M\) is a geodesic then the corresponding curve \(\gamma:=\phi\circ c:I\to N\) is as well. (This curve is the flow \(\tilde\theta_t(n)=\gamma(t)\) with \(\gamma(0)=n\) from the vector field \(\phi_*c'(t)\) with \(c(0)=\psi(n)\).) \(\begin{equation} \psi_*\tilde\nabla_{\gamma'}\gamma' = \psi_*\tilde\nabla_{\phi_*c'}\phi_*c' = D_{c'}c'=\nabla_{c'}c'=0 \end{equation}\) so \(\tilde\nabla_{\gamma'}\gamma'=0\) since \(\psi_*\) is an isomorphism. (Note \(\gamma'(t) = \frac{d}{dt}(\phi\circ c(t))=T_{c(t)}\phi c'(t)=(\phi_*c')(t)\) where \(T_{c(t)}\phi:T_{c(t)}M\to T_{\gamma(t)}N\) is the tangent map or differential.)