Explicit Construction of Lagrangian Isometric Immersion of a Real-space-form Mⁿ(c) into a Complex-space-form M̃ ⁿ(4c)

In [4], it is proved that there exists a 'unique' adapted Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M̃n(4c) of constant sectional curvature 4c associated with each twisted product decomposition of a real-space-form if its twistor form is twisted closed. Conversely, if L: M n(c) → M̃n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) into a complex-space-form M̃n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the immersion L is determined by the corresponding adapted Lagrangian isometric immersion of the twisted product decomposition. It is natural to ask the explicit expressions of adapted Lagrangian isometric immersions of twisted product decompositions of real-space-forms Mn(c) into complex-space-forms M̃n(4c) for each case: c = 0, c > 0 and c < 0. © 2002 Cambridge Philosophical Society.