Georgia Institute of Technology College of Sciences

Given a non-null vector field X in a Riemannian manifold, a hypersurfaceis said to have a canonical principal direction relative to $X$ if theprojection of X onto the tangent space of the hypersurface gives aprincipal direction. We give different ways for building thesehypersurfaces, as well as a number of useful characterizations. Inparticular, we relate them with transnormal functions and eikonalequations. Finally, we impose the further condition of having constantmean curvature to characterize the canonical principal direction surfacesin Euclidean space as Delaunay surfaces.

We consider ensembles of $N \times N$ Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on arbitrarily small intervals converges to the semicircle law, as $N$ tends to infinity.