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In Do Carmo's book on Riemannian Geometry the Cut point is defined as If $(M,g)$ is a complete Riemannian manifold and $\gamma:[0,\infty)\to M$ be a normalized geodesic with $\gamma(0)=p$ then if $t>0$ is sufficiently small, $d(\gamma(0),\gamma(t))=t$ i.e $\gamma([0,t])$ is a minimizing geodesic. If $\gamma([0,t_1])$ is not minimizing then same true for $t>t_1$. Then $\gamma(t_0)$ is cut point of $p$ along $\gamma$. And cut locus is union of the cut points of $p$ along all of the geodesics that start from $p$.

My question is We are working with complete Riemannian manifold so by Hopf Rinow for any point $q\in M$ there is a length minimizing geodesic from $p$ to $q$ so that means $\gamma([0,\infty))$ is defined,so how is the cut locus is being defined? Or it is talking about all geodesics with initial point at $p$ which includes the length minimizing one also. Or there is something I am missing?

User11111
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  • Have you tried to see what happens on a standard circle? – Jason DeVito - on hiatus Aug 28 '22 at 23:24
  • I think it would consist of the antipodal point upto which geodesic is minimizing . But what I am not understanding is isn't Hopf Rinow is saying that for complete Riemannian manifold there always exists a length minimizing geodesics between two points? – User11111 Aug 29 '22 at 02:15
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    Right. If you follow a geodesic half way around a circle, it stops being minimizing. If you continue along such a geodesic, Hopf-Rinow just tells you that there is another geodesic connecting the end points. – Jason DeVito - on hiatus Aug 29 '22 at 02:35

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