  We investigate discretizations of a geometrically nonlinear elastic Cosserat
shell with nonplanar reference configuration originally introduced by B\^irsan,
Ghiba, Martin, and Neff in 2019. The shell model includes curvature terms up to
order 5 in the shell thickness, which are crucial to reliably simulate
high-curvature deformations such as near-folds or creases. The original model
is generalized to shells that are not homeomorphic to a subset of
$\mathbb{R}^2$. For this, we replace the originally planar parameter domain by
an abstract two-dimensional manifold, and verify that the hyperelastic shell
energy and three-dimensional reconstruction are invariant under changes of the
local coordinate systems. This general approach allows to determine the elastic
response for even non-orientable surfaces like the M\"obius strip and the Klein
bottle. We discretize the model with a geometric finite element method and,
using that geometric finite elements are $H^1$-conforming, prove that the
discrete shell model has a solution. Numerical tests then show the general
performance and versatility of the model and discretization method.
