We consider the initial value problem (IVP) associated to a quadratic Schr\"odinger system \begin{ equation*} \begin{ cases} i \partial_{ t} v \pm \Delta_{ g} v - v = \epsilon_{ 1} u \bar{ v}, & t \in \mathbb{ R},\; x \in M, \\[2ex] i \sigma \partial_{ t} u \pm \Delta_{ g} u - \alpha u = \frac{ \epsilon_{ 2}}{ 2} v^{ 2}, & \sigma > 0, \;\alpha \in \mathbb{ R},\; \epsilon_{ i} \in \mathbb{ C}\, (i = 1, 2),\\[2ex] (v(0), u(0)) = (v_0, u_0), \end{ cases} \end{ equation*} posed on a $d$-dimensional sphere $ \mathbb{ S}^{ d}$ or a compact Zoll manifold $M$. Considering $\sigma=\frac{ \theta}{ \beta}$ with $\theta, \beta\in \{ n^2:n\in\mathbb{ Z}\}$ we derive a bilinear Strichartz type estimate and use it to prove the local well-posedness results for given data $(v_0, u_0)\in H^s(M)\times H^s(M)$ whenever $s>\frac{ 1}{ 4}$ in the case $M = \mathbb{ S}^{ 2}$ or a Zoll manifold, and $s > \frac{ d - 2}{ 2}$ in the case $M = \mathbb{ S}^{ d}$ ($d \geq 3$) induced with the canonical metric. Moreover, in dimensions $2$ and $3$, we use a Gagliardo-Nirenberg type inequality to prove that the local solution can be extended globally in time whenever $s \geq 1$.
Categories: math.AP