  In this work we characterize certain immersed closed hypersurfaces of some
ambient manifolds via the second eigenvalue of the Jacobi operator. First, we
characterize the Clifford torus as the surface which maximizes the second
eigenvalue of the Jacobi operator among all closed immersed orientable surfaces
of $\mathbb S^3$ with genus bigger than zero. After, we characterize the slices
of the warped product $I\times_h\mathbb S^n$, under a suitable hypothesis on
the warping function $h:I\subset\mathbb R\to\mathbb R$, as the only
hypersurfaces which saturate a certain integral inequality involving the second
eigenvalue of the Jacobi operator. As a consequence, we obtain that if $\Sigma$
is a closed immersed hypersurface of $\mathbb R\times\mathbb S^n$, then the
second eigenvalue of the Jacobi operator of $\Sigma$ satisfies $\lambda_2\le n$
and the slices are the only hypersurfaces which satisfy $\lambda_2=n$.
