Calabi product Lagrangian immersions in complex projective space and complex hyperbolic space
Starting from two Lagrangian immersions and a Legendre curve $\tilde{\gamma}(t)$ in $\mathbb{S}^3(1)$ (or in $\mathbb{H}_1^3(1)$), it is possible to construct a new Lagrangian immersion in $\mathbb{CP}^n$ (or in $\mathbb{CH}^n$), which is called a warped product Lagrangian immersion. When $\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, r_2e^{i(- \frac{r_1}{r_2}at)})$ (or $\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, r_2e^{i(\frac{r_1}{r_2}at)})$), where $r_1$, $r_2$, and $a$ are positive constants with $r_1^2+r_2^2=1$ (or $-r_1^2+r_2^2=-1$), we call the new Lagrangian immersion a Calabi product Lagrangian immersion. In this paper, we study the inverse problem: how to determine from the properties of the second fundamental form whether a given Lagrangian immersion of $\mathbb{CP}^n$ or $\mathbb{CH}^n$ is a Calabi product Lagrangian immersion. When the Calabi product is minimal, or is Hamiltonian minimal, or has parallel second fundamental form, we give some further characterizations.