= Multiplicity of a conjugate point
{title2=$\dim\{J:J(0)=J(t_0)=0\}$}
Along a nonconstant affinely parametrized <geodesic>, the multiplicity of a <conjugate point> at positive time $t_0$ is the dimension of the space of <Jacobi fields> vanishing at both endpoints. The tangential component of such a <Jacobi field> is affine and vanishes at both endpoints, hence is zero. Uniqueness for the <Jacobi field> equation embeds this space into the $(n-1)$-dimensional normal space by $J\mapsto D_tJ(0)$. Thus its dimension is at most $n-1$.
Back to article page
