Summary: The problem of identifying unknown damping coefficient of hyperbolic equations on Riemannian manifolds is studied. For an initial boundary value problem of hyperbolic equation \({\partial_t^2}u(x, t) - {\Delta_g}u(x, t) + p(x){\partial_t}u(x, t) = 0, x \in M, 0 < t < T\), where \(M\) is a differentiable manifold with boundary \(\partial M\), the initial conditions are \(u(x, 0) = {u_0}\), \({\partial_t}u(x, 0) = {u_1}\), the boundary condition is \(u(x, t) = h\) in \(\sum = \partial M \times (0, T)\), \({u_0}, {u_1}, h\) are known. Set \(u = {u_{p_1, 0}} - {u_{p_2, 0}}\), \({p_0} = {p_1}\), \(f = {p_2} - {p_1}\), \(R = {\partial_t}{u_{p_2, 0}}\), the inverse problem \({\partial_t^2}u(x, t) - {\Delta_g}u(x, t) + {p_0}(x){\partial_t}u(x, t) = f(x)R(x, t), x \in M, 0 < t < T\) can be obtained, the initial values are \(u(x, 0) = {\partial_t}u(x, 0) = 0\), the boundary condition is \(u(x, t) = 0\) in \(\sum = \partial M \times (0, T)\). First the inverse problem is extended to \((-T, T)\). Then by introducing a cut off function, the Carleman estimate on the Riemannian manifold is given. The energy estimate of unknown coefficient \({p_0}(x)\) is established, and the Lipschitz stability of inverse problem is discussed.