The eigenvalues of the Laplace-Beltrami operator on a closed Riemannian manifold are very natural geometric invariants. Although in many problems the Riemannian structure is kept fixed, the eigenvalues can be seen as functionals in the space of metrics. This is the suitable setting for the calculus of variations. In this vein, El Soufi and Ilias have characterised the metrics which are critical for the first eigenvalue among all metrics of fixed volume and among all metrics of fixed volume in a conformal class. In the talk, I will prove a similar characterisation for some critical metrics which are induced by embeddings into a fixed Riemannian manifold.