(Para-)Hermitian and (para-)Kähler Submanifolds of a para-quaternionic Kähler manifold
Abstract
On a para-quaternionic Kähler manifold
(
M
˜
4n
,Q,
g
˜
)
, which is first of all a pseudo-Riemannian manifold, a natural definition of (almost) Kähler and (almost) para-Kähler submanifold
(
M
2m
,J,g)
can be given where
J=
J
1
|
M
is a (para-)complex structure on
M
which is the restriction of a section
J
1
of the para-quaternionic bundle
Q
. In this paper, we extend to such a submanifold
M
most of the results proved by Alekseevsky and Marchiafava, 2001, where Hermitian and Kähler submanifolds of a quaternionic Kähler manifold have been studied.
Conditions for the integrability of an almost (para-)Hermitian structure on
M
are given. Assuming that the scalar curvature of
M
˜
is non zero, we show that any almost (para-)Kähler submanifold is (para-)Kähler and moreover that
M
is (para-)Kähler iff it is totally (para-)complex. Considering totally (para-)complex submanifolds of maximal dimension
2n
, we identify the second fundamental form
h
of
M
with a tensor
C=
J
2
∘h∈TM⊗
S
2
T
∗
M
where
J
2
∈Q
is a compatible para-complex structure anticommuting with
J
1
. When
M
˜
4n
is a symmetric manifold the condition for a (para-)Kähler submanifold
M
2n
to be locally symmetric is given. In the case when
M
˜
is a para-quaternionic space form, it is shown, by using Gauss and Ricci equations, that a (para-)Kähler submanifold
M
2n
is curvature invariant. Moreover it is a locally symmetric Hermitian submanifold iff the
u(n)
-valued 2-form
[C,C]
is parallel. %
[C,C]:X∧Y↦[
C
X
,
C
Y
],X,Y∈TM
%(which satisfies the first and the second Bianchi identity) is parallel. Finally a characterization of \textit{parallel} Kähler and para-Kähler submanifold of maximal dimension is given.