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

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... ace 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.