Suppose is a complete manifold

Then is a covering map. Moreover if is simply connected, then is diffeomorphic to .

**Remark: ** is the north pole of , is the exponential map, then will not be a metric on and are concentric circles. *Proof:* Since , we have no conjugate points of , is nonsingular on . So will be a metric on . In fact is a complete manifold. This is implied from Hopf-Rinow thm and the fact that lines through origin of are geodesics under and they can be extended to infinity.

Since is a local isometry between and , then

is a discrete set. Choose . Denote

namely will be the covering map.

For any , there exists a geodesic connects and . For any , there exists a local lifting of , say , such that . Since is a local isometry, then it is length preserving. Since ends up at some , then must be in .

Next is injective. Otherwise suppose with . Suppose and are geodesics connecting and and respectively, then both and will connect and . and will be two different geodesics because is a local isometry. This can not happen because is inside the injective domain.

is onto. Suppose be the minimal geodesic from to . Then there exist such that , then will be a geodesic in with initial speed . Therefore it must concide with . Since preserve the length, we have . is onto.

Suppose , and are two minimal geodesics from to and . Then and are different near , then and will be two different geodesics connecting and . Contracdtion implies and are disjoint.

**Remark: **If we change the restriction to , the theorem will not be true. Counter example comes from the Calabi-Yau manifold, which is compact, simply connected and has vanishing ricci curvature.

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