Triviality Results of \(\eta \)-Ricci Soliton on K-contact Manifolds

In this paper, we consider generalized \(\eta \) $\eta$ -Ricci soliton on compact and non-compact K-contact manifolds. First, it is established that if a compact K-contact metric g represents a generalized \(\eta \) $\eta$ -Ricci soliton with the potential vector field V as a Jacobi field along the Reeb vector field \(\xi \) $\xi$ , then it is \(\eta \) $\eta$ -Einstein. Next, we characterize a generalized \(\eta \) $\eta$ -Ricci soliton when the potential vector field is either a contact vector field, or a projective vector field.