Let
M
M
be a real hypersurface in
P
n
(
C
)
{P^n}({\mathbf {C}})
be the complex structure and
ξ
\xi
denote a unit normal vector field on
M
M
. We show that
M
M
is (an open subset of) a homogeneous hypersurface if and only if
M
M
has constant principal curvatures and
J
ξ
J\xi
is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically,
P
m
(
C
)
{P^m}({\mathbf {C}})
(totally geodesic),
Q
n
,
P
1
(
C
)
×
P
n
(
C
)
,
S
U
(
5
)
/
S
(
U
(
2
)
×
U
(
3
)
)
{Q^n},{P^1}({\mathbf {C}}) \times {P^n}({\mathbf {C}}),SU(5)/S(U(2) \times U(3))
and
S
O
(
10
)
/
U
(
5
)
SO(10)/U(5)
are the only complex submanifolds whose principal curvatures are constant in the sense that they depend neither on the point of the submanifold nor on the normal vector.