| 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 M˜n(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 : Mn(c) → M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn( c) into a complex-spaceform M˜n(4 c), 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. In the same paper, the authors consider some natural twisted product decompositions on real-space-forms Mn(c) and provide some explicit constructions of adapted Lagrangian isometric: immersions into complex-space-forms Mn(4c) according to the natural twisted product decomposition.;In this thesis, besides the natural twisted product decomposition on a real-space-form Mn(c), I obtain all possible twisted product decompositions on real-space-forms Mn(c). Moreover, it is necessary to define the generalized notion of special Legendre curves in the unit hypersphere S2n+11 ↪Cn+1 2 and we call them the special Legendre translation submanifolds in S2n+11 ↪Cn+1. Similarly, we can define the special translation submanifolds in an anti-de-Sitter space-time H2n+11-1 ↪Cn+11 . Using these notions, I can provide explicit expressions of adapted Lagrangian isometric immersions of twisted product decompositions of real-space-forms Mn(c) into complex-space-forms M˜n(4c) for each case: c = 0, c > 0, and c < 0. Finally, I also find an example of a special Legendre translation submanifold in a unit hypersphere S2n+11 ↪Cn+1 and another one in an anti-de Sitter space-time H2n+11-1 ↪Cn+11 . |
