  During the early-stages of embryo development, morphogenesis--- the emergence
of shape and form in living organisms--- is almost exclusively associated with
monolayers of tightly bound epithelial cells. To understand how such tissues
change their shape, we construct a fully covariant active-hydrodynamic theory.
At the cellular scale, stresses arise from apical contractility, mechanical
response and the constraint of constant cell volume. Tissue-scale deformations
emerge due to the balance between such cell-autonomous stresses and the
displacement and shear of a low Reynolds number embedding fluid. Tissues with
arbitrary curvature or shape can be described, providing a general framework
for epithelial monolayer morphology. Analysis of the stability of flat
monolayers reveals two generic shape instabilities: passive
constrained-buckling, and actively-driven tissue deformation. The active
instability can be further categorised into two types, corresponding to cell
shape changes that are either "squamous to columnar" or "regular-prism to
truncated-pyramid". The deformations resulting from the latter qualitatively
reproduce in vivo observations of the onset of both mesoderm and posterior
midgut invaginations, which take place during gastrulation in the fruit fly
Drosophila melanogaster.
