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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 ispossible 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 positiveconstants with $r_1^2+r_2^2=1$ (or $-r_1^2+r_2^2=-1$), we call the newLagrangian immersion a Calabi product Lagrangian immersion. In this paper, westudy the inverse problem: how to determine from the properties of the secondfundamental form whether a given Lagrangian immersion of $\mathbb{CP}^n$ or$\mathbb{CH}^n$ is a Calabi product Lagrangian immersion. When the Calabiproduct is minimal, or is Hamiltonian minimal, or has parallel secondfundamental form, we give some further characterizations.

Calabi Product Lagrangian Immersions in Complex Projective Space and Complex Hyperbolic Space