Assume is a normal geodesic, is one parameter smooth variation of curve . Denote .

is called the variation field. As we all know if the variation is geodesic one, then is a Jacobi field. If , then

Let us define and is a subspace of . Index form

for .

**Thm1:** If does not have conjugate point, then for any and . This is equivalent to say index form is positive definite.

**Thm2(Minimality of Jacobi field):** Assume does not have conjugate point. Suppose and and . If is a Jacobi field on , then

Thm1 and Thm2 are equivalent.

Thm1 Thm2:

, then

Thm2 Thm1: Obviously

One way to prove Thm1 is to prove Thm2. However, we will prove Thm1 directly.

**Proof:** Suppose , . Since does not have conjugate point, then is non-degenerate in , . There exists a neighborhood of such that is an immersion. From the fact that

is any piecewise smooth curve connecting and , then when the equality holds, must be monotone reparametirzation of

After proving this, we can construct a variation of such that the variation field is , then

Suppose for some , then for any

this means , . This means must be a Jacobi field. But and has no conjugate point, must be 0.

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