  We illustrate the flow or wave character of the metrics and curvatures of
evolving manifolds, introducing the Riemann flow and the Riemann wave via the
bialternate product Riemannian metric. This kind of evolutions are new and very
natural to understand certain flow or wave phenomena in the nature as well as
the geometry of evolving manifolds. It possesses many interesting properties
from both mathematical and physical point of views.
  We reveal the novel features of Riemann flow and Riemann wave, as well as
their versatility, by new original results. The main results refer to: (1)
Riemann flow PDE, (3) connection between Ricci flow and Riemann flow,(4) the
meaning of the Riemann flows on constant curvature manifolds, (5) the gradient
Riemann solitons, (6) the infinitesimal deformations (linearizations) of Ricci
flow and of Riemann flow PDEs, (7) the existence of Ricci or Riemann curvature
blow-up at finite-time singularities of the flow, (8) the Riemann wave PDE and
its meaning on constant curvature manifolds, (9) the existence of Ricci or
Riemann curvature blow-up at finite-time singularities of the wave, (10) the
general form of some essential PDE systems on Riemannian manifolds.
