Classically, there is a strong relationship between the shape of a Riemannian manifold and the spectrum of its Laplace-operator, and, more generally, with the spectrum of Laplace type operators. In particular, for the spectrum of the Laplacian, the presence of a large first nonzero eigenvalue is related to some concentration phenomena. In the first part of the talk, I will recall these classical relations. In the second part of the talk, I will introduce the Dirichlet-to-Neumann operator on a manifold with boundary. I will survey the same types of questions in this context (existence of large first nonzero eigenvalue, concentration phenomena) without going into the technical details. This second part corresponds to work in progress with Alexandre Girouard.