Let
(
g
,
[
p
]
)
(\mathfrak {g},[p])
be a restricted Lie algebra over an algebraically closed field
k
k
of characteristic
p
≥
3
p\!\ge \!3
. Motivated by the behavior of geometric invariants of the so-called
(
g
,
[
p
]
)
(\mathfrak {g},[p])
-modules of constant
j
j
-rank (
j
∈
{
1
,
…
,
p
−
1
}
j \in \{1,\ldots ,p\!-\!1\}
), we study the projective variety
E
(
2
,
g
)
\mathbb {E}(2,\mathfrak {g})
of two-dimensional elementary abelian subalgebras. If
p
≥
5
p\!\ge \!5
, then the topological space
E
(
2
,
g
/
C
(
g
)
)
\mathbb {E}(2,\mathfrak {g}/C(\mathfrak {g}))
, associated to the factor algebra of
g
\mathfrak {g}
by its center
C
(
g
)
C(\mathfrak {g})
, is shown to be connected. We give applications concerning categories of
(
g
,
[
p
]
)
(\mathfrak {g},[p])
-modules of constant
j
j
-rank and certain invariants, called
j
j
-degrees.