We consider the situation where a Riemannian manifold
M
n
{M^n}
can be isometrically immersed into spaces
N
n
+
1
(
c
)
{N^{n + 1}}(c)
and
N
n
+
q
(
c
~
)
{N^{n + q}}(\tilde c)
with constant curvatures
c
>
c
~
c > \tilde c
,
q
⩽
n
−
3
q \leqslant n - 3
, and show that this implies the existence, at each point
p
∈
M
p \in M
, of an umbilic subspace
U
p
⊂
T
p
M
{U_p} \subset {T_p}M
, for both immersions, with
U
p
⩾
n
−
q
{U_p} \geqslant n - q
. In particular, if
M
n
{M^n}
can be isometrically immersed as a hypersurface into two spaces of distinct constant curvatures,
M
n
{M^n}
is conformally flat.