  For any bounded regular domain $\Omega$ of a real analytic Riemannian
manifold $M$, we denote by $\lambda_{k}(\Omega)$ the $k$-th eigenvalue of the
Dirichlet Laplacian of $\Omega$. In this paper, we consider $\lambda_k$ and as
a functional upon the set of domains of fixed volume in $M$. We introduce and
investigate a natural notion of critical domain for this functional. In
particular, we obtain necessary and sufficient conditions for a domain to be
critical, locally minimizing or locally maximizing for $\lambda_k$. These
results rely on Hadamard type variational formulae that we establish in this
general setting.
