Abstract
We study conformal vector fields on pseudo-Riemannian manifolds which are locally gradient fields. This is closely related with a certain differential equation for the Hessian of a real function. We obtain global solutions of the oscillator and peandulum equation for the Hessian of this function on a pseudo-Riemannian manifold, generalizing previous results by M. Obata, Y. Tashiro, and Y. Kerbrat. In particular, it turns out that the pendulum equation characterizes a certain conformal type of metrics carrying a conformal vector field with infinitely many zeros.
