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**Geodesics**

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**Jaspers****Member**- Registered: 2019-05-24
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A geodesic on a smooth manifold *M* with an affine connection ∇ is defined as a curve *γ*(*t*) such that parallel transport along the curve preserves the tangent vector to the curve, i.e.

at each point along the curve, where

is the derivative with respect to *t*.

Using local coordinates on *M*, we can write the geodesic equation (using the summation convention) as the ordinary differential equation

where

are the coordinates of the curve *γ*(*t*) and

are the Christoffel symbols of the connection ∇.

[Source: Wikipedia]

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