  In this paper, we propose a new trace finite element method for the
{Laplace-Beltrami} eigenvalue problem. The method is proposed directly on a
smooth manifold which is implicitly given by a level-set function and require
high order numerical quadrature on the surface. A comprehensive analysis for
the method is provided. We show that the eigenvalues of the discrete
Laplace-Beltrami operator coincide with only part of the eigenvalues of an
embedded problem, which further corresponds to the finite eigenvalues for a
singular generalized algebraic eigenvalue problem. The finite eigenvalues can
be efficiently solved by a rank-completing perturbation algorithm in {\it
Hochstenbach et al. SIAM J. Matrix Anal. Appl., 2019}
\cite{hochstenbach2019solving}. We prove the method has optimal convergence
rate. Numerical experiments verify the theoretical analysis and show that the
geometric consistency can improve the numerical accuracy significantly.
