Symplectic geometry on the manifold of nondegenerate \(m\)-pairs (Q1866706)

The author studies the manifold \(N^n_m\) of nondegenerate \(m\)-pairs of the real projective space \(\mathbb{R} P(n)\), i.e., the manifold whose points are pairs \((A,B)\) of an \(m\)-plane \(A\) and an \(n-m-1\) plane \(B\) not intersecting in \(\mathbb{R} P(n)\). In particular, the author constructs a hyperbolic Kählerian metric \(g\) on \(N^n_m\) which is semi-Riemannian of signature \(((m+ 1)(n-m),(m+ 1)(n-m))\) and a Kähler complex structure \(J\) on \(N^n_m\) such that \(g(JX, JY)= -g(X,Y)\) for all \(X,Y\in TM\) and the form \(\Omega(X, Y)= g(X, JY)\) is symplectic i.e., \(d\Omega= 0\). It is proved that \((N^n_m,g)\) is an Einstein manifold and that \(N^n_0\) has constant holomorphic sectional curvature. At the end, the author proves that \((N^n_m,\Omega)\) is symplectomorphic to the cotangent bundle of the Grassman manifold \(G_{m,n}\) of \(m\)-dimensional subspaces of \(\mathbb{R} P(n)\) with the standard symplectic structure of the cotangent bundle.