- For each point x of M, and for every vector
**v**in the tangent space T_{x}M, the second derivative of the function L:T_{x}M->**R**given by - at
**v**is positive definite.

The length of γ, a differentiable curve in M is given by . Note that it's reparametrization invariant. Geodesics are curves in M whose length is extremal under functional derivatives.

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