The thing about Hopf Rinow Theorem

Lemma 1: {M} is a Riemannian manifold. {\forall p\in M}, {\exists\,} neighborhood {N} of {p} and {\epsilon>0} such that {\exp_p:B_\epsilon(0)\rightarrow N} is a diffeomorphism and any two points in {N} can ve connected by a unique geodesic with length smaller than {\epsilon}.

Lemma 2: {p\in M}, {r} is small enough such that {\exp_p:B_r(0)\rightarrow N_r(p)=\exp(B_r(0))} is a differeomorphism, then for any {q\not \in N_r(p)}, {\exists \,q'\in \partial N_r(p)} such that

\displaystyle d(p,q)=r+d(q',q)

Proof: {d(\cdot,q)} is a continuous function on {\partial N_r(p)}. Since {\partial N_r(p)} is compact, then {\exists\, q'\in \partial N_r(p)} such that

\displaystyle d(q',q)=\inf\{d(\tilde{q},q)|\tilde{q}\in\partial N_r(p)\}

Suppose {\gamma_n} is a minimizing sequence of {d(p,q)}. For any {\gamma_n}, there exists {q_n\in \partial N_r(p)\cap \gamma_n}, then

\displaystyle L(\gamma_n)\geq d(p,q_n)+d(q_n,q)\geq r+d(q',q)

Letting {n\rightarrow \infty}, {d(p,q)\geq r+d(q',q)}. By the triangle inequality,

\displaystyle d(p,q)= r+d(q',q)


Thm(Hopf-Rinow) The following statements are equivalent for Riemannian manifold {M}:
(1) {M} is a complete metric space, where the metric is induced by

\displaystyle d(p,q)=\inf\{L(\gamma)|\gamma\text{ is a curve connects } p,q\}

(2)The closed and bounded sets of {M} are compact.
(3){\exists \,p\in M} such that {\exp_p} is defined on all of {T_pM}.
(4) {\forall\,p\in M}, {\exp_p} is defined on {T_pM}
Furthermore, each of the statements {(1-4)} implies
(5) Any two points {p,q\in M} can be connected by a geodesic of shortest length. Proof: Let us prove {(3)\Rightarrow (5)} firstly.
By lemma 2, there exists {r>0} and {q'} on {\partial N_r(p)} such that

\displaystyle d(p,q)=r+d(q',q)

Suppose {q'=\exp_ptv} for some {v\in T_pM}, {||v||=1}, then {\gamma(t)=\exp_ptv} is defined on {[0,\infty)} by assumption. Consider {\{t|t\in [r,d(p,q)]\}} satisfies

\displaystyle d(p,q)=t+d(\gamma(t),q),

denote such points as {I}. {I} is nonempty as {r\in I} and it is closed by the sake of continuity.
Suppose {t_0=\max_{t\in I}t}. If {t_0=d(p,q)}, we are done. Otherwise consider {q_0=\gamma(t_0)}, by lemma 2, {\exists\,\epsilon>0} small enough, {q''\in\partial N_\epsilon(q_0)} such that

\displaystyle q\not\in N_{\epsilon}(q_0),\quad d(q_0,q)=\epsilon+d(q'',q)


\displaystyle d(p,q)=t_0+\epsilon+d(q'',q)

We are going to prove {q''=\gamma(t_0+\epsilon)}. From the triangle inequality

\displaystyle d(p,q'')\geq d(p,q)-d(q'',q)=t_0+\epsilon

\displaystyle d(p,q'')\leq d(p,q_0)+d(q_0,q'')=t_0+\epsilon

then we must have {d(p,q'')=t_0+\epsilon}. Then the union of {\gamma([0,t_0])} and the geodesic from {q_0} to {q''} constitutes a shortest curve from {p} to {q''}, therefore it must be a geodesic. By lemma 1, such geodesic is unique with given initial values. So it must concider with {\gamma(t)} in the {N_\epsilon(q_0)}, which means {q''=\gamma(t_0+\epsilon)}. Then

\displaystyle d(p,q)=t_0+\epsilon+d(\gamma(t_0+\epsilon),q)

This contradicts the fact {t_0} is maximal. So {t_0} must be {d(p,q)}, and the curve {\gamma([0,t_0])} is just the minimal geodesic connects {p} and {q}.

{(4)\Rightarrow (3)} Evidently

{(3)\Rightarrow (2)} Suppose {K} is a bounded set in {M}. From {(5)} we know, {K} can be contained in {\exp_p B_r(0)} where {B_r(0)\subset T_pM} for some {r} large enough. Since {B_r(0)} is compact and {\exp_p} is a continuous mapping, then {\exp_p B_r(0)} is compact, therefore its closed subset {K} is compact.

{(2)\Rightarrow(1)} Any cauchy sequence in {M} is bounded, its closure is compact by {(2)}. Then it must have a converge subsequence and being cauchy, it has to converge itself.

{(1)\Rightarrow (4)} For any {p\in M}, we need to prove {\exp_ptv} is defined on {[0,\infty)} for any {v\in T_pM}. Consider {t_n\nearrow T<\infty} and {\exp_p tv} is defined on each {t_n}, denote {\exp_pt_nv=q_n}.

Since {d(q_n,q_m)\leq |t_n-t_m|}, {\{q_n\}} is a cauchy sequence, {\exists\, q\in M} such that {q_n\rightarrow q}. Since there exists a neighborhood of {q}, say {N_\epsilon(q)}, such that {\forall z\in N_\epsilon(q)}, any geodesic starting from {z} can be extended at least up to length {\rho_0>0}.

For sufficiently large {n}, {q_n\in N_\epsilon(p)} and {d(q_n,q)<\rho_0}. {\exp_ptv|_{[t_n,t_{n+1}]}} is a curve starting from {\gamma(t_n)}, then it can be extended at least up to length {\rho_0}. By the uniqueness of geodesic, {\exp_ptv} can be extened beyond {q}, namely {q=\exp_pTv} is well defined. So {\exp_p} is defined on whole {[0,\infty)}.


Remark:  Should thank Bin Guo’s picture

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