**Lemma 1:** is a Riemannian manifold. , neighborhood of and such that is a diffeomorphism and any two points in can ve connected by a unique geodesic with length smaller than .

**Lemma 2:** , is small enough such that is a differeomorphism, then for any , such that

*Proof:* is a continuous function on . Since is compact, then such that

Suppose is a minimizing sequence of . For any , there exists , then

Letting , . By the triangle inequality,

**Thm(Hopf-Rinow)** The following statements are equivalent for Riemannian manifold :

(1) is a complete metric space, where the metric is induced by

(2)The closed and bounded sets of are compact.

(3) such that is defined on all of .

(4) , is defined on

Furthermore, each of the statements implies

(5) Any two points can be connected by a geodesic of shortest length. *Proof:* Let us prove firstly.

By lemma 2, there exists and on such that

Suppose for some , , then is defined on by assumption. Consider satisfies

denote such points as . is nonempty as and it is closed by the sake of continuity.

Suppose . If , we are done. Otherwise consider , by lemma 2, small enough, such that

then

We are going to prove . From the triangle inequality

then we must have . Then the union of and the geodesic from to constitutes a shortest curve from to , therefore it must be a geodesic. By lemma 1, such geodesic is unique with given initial values. So it must concider with in the , which means . Then

This contradicts the fact is maximal. So must be , and the curve is just the minimal geodesic connects and .

Evidently

Suppose is a bounded set in . From we know, can be contained in where for some large enough. Since is compact and is a continuous mapping, then is compact, therefore its closed subset is compact.

Any cauchy sequence in is bounded, its closure is compact by . Then it must have a converge subsequence and being cauchy, it has to converge itself.

For any , we need to prove is defined on for any . Consider and is defined on each , denote .

Since , is a cauchy sequence, such that . Since there exists a neighborhood of , say , such that , any geodesic starting from can be extended at least up to length .

For sufficiently large , and . is a curve starting from , then it can be extended at least up to length . By the uniqueness of geodesic, can be extened beyond , namely is well defined. So is defined on whole .

**Remark: ** Should thank Bin Guo’s picture