Multiplicity of a conjugate point ($\dim \{J:J(0)=J(t_0)=0\}$)

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 $J\mapsto D_{t}J(0)$. Thus its dimension is at most $n-1$.