I have learnt the Cartan Hadamard theorem,
Let $M$ be a complete Riemannian manifold with nonpositive sectional curvature. Then $\forall x\in M, \exp_x:T_xM\to M$ has no conjugate point.
Then the notes point out completeness assumption is required since for $\mathbb{R}^3\setminus \{0,0,0\}$ with induced metric, the theorem fails.
I don't quite understand the reason. Does it mean $\exp_x$ has some conjugate points for some $x$?
By definition in the notes, if $(d\exp_x)_p$ is singular, then $p$ is called a conjugate point of map $\exp_x$ and $\exp_x(p)$ is called a conjugate point of $x$ along geodesic $\exp_x(tp)$.
I have also learnt, $(d\exp_x)_{p}$ is singular iff there exists a normal Jacobi field $U(t)$ along $\gamma(t)=\exp_x(tp)$ not identically zero such that $U(0)=U(1)=0$.
I think some geometric explanation of conjugate point might be helpful. I saw some online materials say if $p$ and $q$ are conjugate along $\gamma$, one can construct a family of geodesics that start at $p$ and $\underline{almost}$ end at $q$. I don't quite understand the reason and I am not sure whether it's useful to answer my question.