It is well known that totally geodesic Lagrangian submanifolds
of a complex-space-form
M˜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˜n(4c). In order to do so,
associated with each twisted product decomposition of a real-space-form
of the form
f1I1×…
×fkIk×1Nn−k(c), we introduce
a canonical 1-form,
called the twistor form of the twisted product decomposition.
Roughly speaking, our main result
says that if the twistor 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˜n(4c). Conversely, if
L: Mn(c)→
M˜n(4c) is a non-totally geodesic
Lagrangian isometric immersion of a
real-space-form Mn(c) of
constant sectional curvature c into a complex-space-form
M˜n(4c), then
Mn(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.
An Immersion of an $n$-dimensional Real Space Form into an $n$-dimensional Complex Space Form