GEOMETRY OF ${\mathcal{P}} R$-WARPED PRODUCTS IN PARA-KÄHLER MANIFOLDS

Abstract

In this paper, we initiate the study of ${\mathcal{P}} R$-warped products in para-Kähler manifolds and prove some fundamental results on such submanifolds. In particular, we establish a general optimal inequality for $\mathcal{P} R$-warped products in para-Kähler manifolds involving only the warping function and the second fundamental form. Moreover, we completely classify $\mathcal{P} R$-warped products in the flat para-Kähler manifold with least codimension which satisfy the equality case of the inequality. Our results provide an answer to the Open Problem (3) proposed in [19, Section 5].