Assume that $C_m(p)$ is a cut locus of $p$ and $C(p) $ is a conjugate locus (Here we consider first conjugate locus). Then we have tangent cut locus $TC_m(p)\subset T_pM$ and tangent conjugate locus $TC(p)\subset T_pM$.
We want to find a Riemannian manifold $M$ s.t. there is a point $p\in M$ s.t. $$ TC_m(p)\ \bigcap\ TC(p)=\emptyset \ \ast$$
(1) In any two dimensional sphere $S^2$, $C_m(p)$ is a tree. Here vertices of $C_m(p)$ is in $C(p)$ so that $\ast$ does not hold.
(2) If $$S^2 =\bigg\{ (x,y,z)\in \mathbb{R}^3 \bigg| \frac{x^2+z^2 }{a^2} + \frac{y^2}{b^2} =1 \bigg\} ,\ b>a>0, $$
then let $p=(a,0,0)$. Then in this case $TC_m(p)\neq TC(p)$.
Question : In $S^3$, is there a metric with $\ast$ ?
Thank you in advance.