  Recall that a submanifold of a Riemannian manifold is said to be minimal if
its mean curvature is zero. It is classical that minimal submanifolds are the
critical points of the volume function. In this paper, we examine the critical
points of the total $(2k)$-th Gauss-Bonnet curvature function, called
$(2k)$-minimal submanifolds. We prove that they are characterized by the
vanishing of a higher mean curvature, namely the $(2k+1)$-Gauss-Bonnet
curvature. Furthermore, we show that several properties of usual minimal
submanifolds can be naturally generalized to $(2k)$-minimal submanifolds.
