  We construct Dyson Brownian motion for $\beta \in (0,\infty]$ by adapting the
extrinsic construction of Brownian motion on Riemannian manifolds to the
geometry of group orbits within the space of Hermitian matrices. When $\beta$
is infinite, the eigenvalues evolve by Coulombic repulsion and the group orbits
evolve by motion by (minus one half times) mean curvature.
