In this paper, we establish new characterization results concerning totally umbilical hypersurfaces of the hyperbolic space $\mathbb{H}^{n+1}$, under suitable constraints on the behavior of the Lorentzian Gauss map of complete hypersurfaces having some constant higher order mean curvature. Furthermore, working with different warped product models for $\mathbb{H}^{n+1}$ and supposing that certain natural inequalities involving two consecutive higher order mean curvature functions are satisfied, we study the rigidity and the nonexistence of complete hypersurfaces immersed in $\mathbb{H}^{n+1}$.

We reconcile two classical models of edge dislocations in solids. The first, from the early 1900s, models isolated edge dislocations as line singularities in locally Euclidean manifolds. The second, from the 1950s, models continuously distributed edge dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modelled as a Weitzenböck manifold. We prove, using a weak notion of convergence, that the second model can be obtained rigorously as a homogenization limit of the first model as the density of singular edge dislocation tends to infinity.

We suggest a way to quantize, using Berezin–Toeplitz quantization, a compact hyperkähler manifold (equipped with a natural 3-plectic form), or a compact integral Kähler manifold of complex dimension n regarded as a (2n−1)-plectic manifold. We show that quantization has reasonable semiclassical properties.

Darboux Wronskian formulas allow us to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one, cannot be represented this way. It has been a long-standing problem to discover what other exceptions exist. In our previous work we proved that among transformations of total order one there are no other exceptions. Here we prove that for transformations of total order two there are no exceptions at all. We also obtain a simple explicit invariant description of all possible Darboux transformations of total order two.

In this paper, we deal with complete hypersurfaces immersed with bounded higher order mean curvatures in steady state-type spacetimes and in hyperbolic-type spaces. By applying a generalised maximum principle for the Yau's square operator [11], we obtain uniqueness results in each of these ambient spaces.