Solving The Jacobi Equation
Let and complete this to get an orthonormal basis at . Parallel transport it to get a basis all along . This gives an orthonormal basis with . The Jacobi field can be written in co-ordinates in terms of this basis as and thus
and the Jacobi equation can be rewritten as a system
for each . This way we get a linear ordinary differential equation (ODE). Since this ODE has smooth coefficients we have that solutions exist for all and are unique, given and, for all .
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