Geometrical objects associated to a substructure (Q2889935)

A substructure on a manifold \(M\) is a \((1,1)\)-tensor field \(J :R \rightarrow R\) on a distribution \(R\) on \(M\) that admits a complementary distribution \(S\), that is \(TM=R\oplus S\). A substructure \(J\) is called polynomial if there is a polynomial \(P\) such that \(P(J)=0\). In this paper, special extensions of polynomial substructures to \((1,1)\)-tensor field on \(TM\) are studied. The authors focus on the relations of these extended structures with connections or metrics adapted to the splitting \(TM=R\oplus S\). A generalization of both Hermitian and anti-Hermitian geometry is proposed.