Let
X
be a Stein manifold of complex dimension
n
>
1
endowed with a Riemannian metric
g
. We show that for every integer
k
with
n
2
≤
k
≤
n
-
1
there is a nonsingular holomorphic foliation of ...dimension
k
on
X
all of whose leaves are closed and
g
-complete. The same is true if
1
≤
k
<
n
2
provided that there is a complex vector bundle epimorphism
T
X
→
X
×
C
n
-
k
. We also show that if
F
is a proper holomorphic foliation on
C
n
(
n
>
1
)
then for any Riemannian metric
g
on
C
n
there is a holomorphic automorphism
Φ
of
C
n
such that the image foliation
Φ
∗
F
is
g
-complete. The analogous result is obtained on every Stein manifold with Varolin’s density property.
