Math Is Fun Forum

Jaspers

Member

Registered: 2019-05-24

Posts: 54

Geodesics

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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