I am reading an article and I had a big question and wanted to know if colleagues could help me.
If we define an operator $P$ in a Riemannian manifold $(M^n,g)$ as follows:
For each vector field $Z$ in $M$ we have $$P(X,Y)=g(\nabla_YQ(Z),X),$$ where Q is a symmetric operator and g is Riemannian metric. Thus, what could you say about the commutativity of this operator, that is, is P symmetric? anti-symmetrical? or neither? thank you so much who can help me.