Problem in do Carmo's book Riemannian geometry at space forms.

Question

I'm reading do Carmo's book, riemannian geometry and at page 163 is this lemma.

I dont get the fact that $\varphi(q)=q.$ Can someone fill in the details?

Answer 1

Write $\gamma_v$ the geodesic checking $\gamma_v(0)=p$ and $\gamma_v'(0)=v$. Since $\varphi$ is an isometry, $\varphi\circ\gamma_v$ is a geodesic starting at $\varphi(\gamma_v(0))=\varphi(p)=p$. Since $(\varphi\circ\gamma_v)'(0)=d\varphi_p(\gamma_v'(0))=id_p(\gamma_v'(0))=v$, then $\varphi\circ\gamma_v=\gamma_v$. Finally, $$\varphi(q)=\varphi(\exp_p(v))=\varphi(\gamma_v(1))=\gamma_v(1)=\exp_p(v)=q.$$