Mathematical proceedings of the cambridge philosophical society vol:124 pages:107-125
It is well known that totally geodesic Lagrangian submanifolds of a complex-space-form (M) over tilde(n)(4c) of constant holomorphic sectional curvature 4c are real-space-forms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form (M) over tilde(n)(4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form I-f1(1) X ... X-fk I-k X (1) Nn-k(c), we introduce a canonical 1-form, called the twister form of the twisted product decomposition. Roughly speaking, our main result says that if the twister form of such a twisted product decomposition of a simply-connected real-space-form of constant sectional curvature c is twisted closed, then it admits a 'unique' adapted Lagrangian isometric immersion into a complex-space-form (M) over tilde(n)(4c). Conversely, if L: M-n(c) --> (M) over tilde(n)(c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form M-n(c) of constant sectional curvature c into a complex-space-form (M) over tilde(n)(4c), then M-n(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.