  In the past 50 years, the formalism of L-systems has been successfully used
and developed to model the growth of filamentous and branching biological
forms. These simulations take place in classical 2-D or 3-D Euclidean spaces.
However, various biological forms actually grow in curved, non-Euclidean,
spaces. This is for example the case of vein networks growing within curved
leaf blades, of unicellular filaments, such as pollen tubes, growing on curved
surfaces to fertilize distant ovules, of teeth patterns growing on folded
epithelia of animals, of diffusion of chemical or mechanical signals at the
surface of plant or animal tissues, etc. To model these forms growing in curved
spaces, we thus extended the formalism of L-systems to non-Euclidean spaces. In
a first step we show that this extension can be carried out by integrating
concepts of differential geometry in the notion of turtle geometry. We then
illustrate how this extension can be applied to model and program the
development of both mathematical and biological forms on curved surfaces
embedded in our Euclidean space. We provide various examples applied to plant
development. We finally show that this approach can be extended to more
abstract spaces, called abstract Riemannian spaces, that are not embedded into
any higher space, while being intrinsically curved. We suggest that this
abstract extension can be used to provide a new approach for effective modeling
of tropism phenomena and illustrate this idea on a few conceptual examples.
