Given a smooth family of unparameterized curves such that through every point in every direction there passes exactly one curve, does there exist a Lagrangian with extremals being precisely this family? It is known that in dimension 2 the answer is positive. In dimension 3, it follows from the work of Douglas that the answer is, in general, negative. We generalise this result to all higher dimensions ...

For a real-analytic connected CR-hypersurface M of CR-dimension n⩾1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra s=s(M): (i) either dims=n2+4n+3 and M is spherical everywhere; (ii) or dims⩽n2+2n+2+δ2,n and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version ...

We compute the Hilbert polynomial and the Poincar´e function counting the number of fixed jet-order differential invariants of conformal metric structures modulo local diffeomorphisms, and we describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action. This resolves the local recognition problem for conformal structures.

We compute the algebra of differential invariants of unparametrized curves in the homogeneous G₂ flag varieties, namely in G₂/P. This gives a solution to the equivalence problem for such curves. We consider the cases of integral and generic curves and relate the equivalence problems for all three choices of the parabolic subgroup P.

We find generators for the algebra of rational differential invariants for general and degenerate Kundt spacetimes and relate this to other approaches to the equivalence problem for Lorentzian metrics. Special attention is given to dimensions three and four.

We consider the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. This is solved by computing differential invariants via the Lie-Tresse theorem.

Differential invariants of a (pseudo)group action can vary when restricted to invariant submanifolds (differential equations). The algebra is still governed by the Lie-Tresse theorem, but may change a lot. We describe in details the case of the motion group O(n) ⋉ Rⁿ acting on the full (unconstraint) jet-space as well as on some invariant equations.

Paraconformal or GL(2, ℝ) geometry on an <i>n</i>-dimensional manifold <i>M</i> is defined by a field of rational normal curves of degree <i>n</i> – 1 in the projectivised cotangent bundle <i>ℙT*M</i>. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of <i>GL</i>(2, ℝ) structures, ...

The topological entropy of piecewise affine maps is studied. It is shown that singularities may contribute to the entropy only if there is angular expansion and we bound the entropy via the expansion rates of the map. As a corollary we deduce that non-expanding conformal piecewise affine maps have zero topological entropy. We estimate the entropy of piecewise affine skew-products. Examples of ...

We exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy.