Constraints on the Seifert invariants of orientable 3-manifolds
M
M
which admit fixed-point free
S
1
S^1
-actions and embed in
R
4
\mathbb {R}^4
are given. In particular, the generalized Euler invariant of the orbit fibration is determined up to sign by the base orbifold
B
B
unless
H
1
(
M
;
Z
)
H_1(M;\mathbb {Z})
is torsion free, in which case it can take at most one nonzero value (up to sign). An
H
2
×
E
1
\mathbb {H}^2\times \mathbb {E}^1
-manifold
M
M
with base orbifold
B
=
S
2
(
α
1
,
…
,
α
r
)
B=S^2(\alpha _1,\dots ,\alpha _r)
where all cone point orders are odd embeds in
R
4
\mathbb {R}^4
if and only if its Seifert data
S
S
is skew-symmetric.