From the Gauss lemma to the minimality of geodesics

**Lemma:(Gauss)** Suppose ,

Here we make natural equivalence of and . This lemma means presvers the inner product.

For any there exists a neighborhood of such that the exponential map is diffeomorphism from to .

**Thm:** , for such , suppose is any geodesic ball contained in . Then the shortest length connecting a point on and is the geodesic ray from .

**Proof:** Suppose connects and , where is a curve in . Suppoe is the unit normal along , then with is perpendicular to . Then by Gauss lemma

We use the fact , because which implies .

**Remark:** I own this proof to Prof. Xiaochun Rong.

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