A number of processes of

surface growth In mathematics and physics, surface growth refers to models used in the dynamical study of the growth of a surface, usually by means of a stochastic differential equation of a field. Examples Popular growth models include: * KPZ equation * D ...

in areas ranging from mechanics of growing

gravitational In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...

bodies through propagating fronts of phase transitions, epitaxial growth of nanostructures and 3D printing, growth of plants, and cell mobility require ''non-Euclidean'' description because of incompatibility of boundary conditions and different mechanisms of developing stresses at interfaces. Indeed, these mechanisms result in the curving of initially flat elements of the body and changing separation between different elements of it (especially in the soft matter). Gradual accumulation of deformations under the influx of accumulating mass results in the memory-conscious grows of the body and makes strains the subject of long-range forces. As a result of all above factors, generic non-Euclidean growth is described in terms of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

with a space- and time-dependent curvature.

References

F. Sozio, M.F. Shojaei, S. Sadik, and A. Yavari, Nonlinear mechanics of thermoelastic accretion, \emph \textbf(3), 2020, 87. F. Sozio and A. Yavari, Nonlinear mechanics of accretion, \emph \textbf(4), 2019, 1813-1863. F. Sozio and A. Yavari, Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies, \emph \textbf, 2017, pp. 12-48.

References

Further reading