LAGRANGIAN $H$-UMBILICAL SUBMANIFOLDS OF PARA-KÄHLER MANIFOLDS

Abstract

The notion of Lagrangian $H$-umbilical submanifolds of Kähler manifolds introduced in [3, 4] is closely related with several problems in Lagrangian geometry (cf. [7]). The classification of such submanifolds was done in a series of author's papers [3, 4, 5]. On the other hand, the study of Lagrangian submanifolds of para-Kähler manifolds was initiated very recently in [10]. In this paper we study Lagrangian $H$-submanifolds of para-Kähler manifolds. As results we prove several fundemental properties of such submanifolds. Moreover, we are able to classify Lagrangian $H$-umbilical submanifolds of the para-Kähler $n$-plane $({\mathbb E}^{2n}_n,g_0,P)$ for $n \geq 3$.