The Gelfand theorem and its converse in Sasakian geometry (Q1102557)

The Gelfand theorem states that for a symmetric space \(M=G/H\), where G denotes the connected component of the identity of the full isometry group of M, the algebra D(G/H) of invariant differential operators is commutative. It is known that the converse of this theorem is not true in general. The authors consider the converse in the case of \(\phi\)- symmetric spaces. Using the group of automorphisms of the Sasakian structure they obtain the commutativity of D(G/H) modulo the characteristic differential operator determined by the characteristic vector field. They also derive a characterization of \(\phi\)-symmetric spaces using the property that all geodesics are orbits of one-parameter subgroups of automorphisms.