$F(3,1)$-STRUCTURE ON THE LAGRANGIAN SPACE AND INVARIANT SUBSPACES

Jovanka Niki\'{c}

1991 Math. Subject Classification: 53B40, 53C60.

Key words: $f$-structures, invariant subspace.

Abstract: If almost product structure $P$ on the tangent space
$T(E) = T_{V}(E)\oplus T_{H}(E)$ of Lagrangian $2n$ dimensional
manifold $E$ is defined, and if $f_{v}(3,1)$-structure on
$T_{V}(E)$ is defined, then $f_{h}(3,1)$-structure on $T_{H}(E)$
are defined in the natural way. We can define $F(3,1)$-structure
on $T(E)$. The condition is given for the
reduction of the structural group of such manifolds.

Two cases of invariant submanifolds are considered. In the first case
we get an induced almost complex structure, and in the second case we
obtain an induced $f(3,1)$-structure on the invariant submanifold.