Invariant metrical \(f\)-structures on the sphere \(S^5\) (Q2713918)

The affine geometry of the sphere \(S^5 = SU(3)/SU(2)\) is studied. A metrical \(f\)-structure on \(S^5\) is a \((1,1)\)-tensor \(f\) such that \(f^3+f=0\) and \(g(fX,Y)+g(X,fY)=0\) for all tangent vectors \(X\), \(Y\) and a metric \(g\) on \(S^5\). A metrical \(f\)-structure is called a Killing structure if and only if \(g(X, fY)\) is the Killing form, and is called an approximate Kähler \(f\)-structure if and only if \(\nabla_fX (f) fX=0\) for all tangent vectors \(X\). The author proves that every \(SU(3)\)-invariant \(f\)-structure on \(S^5\) is an approximate Kähler \(f\)-structure and is not a Killing \(f\)-structure.