Related papers: The gap phenomenon in parabolic geometries
Among (regular, normal) parabolic geometries of type $(G,P)$, there is a locally unique maximally symmetric structure and it has symmetry dimension $\dim(G)$. The symmetry gap problem concerns the determination of the next realizable…
We extend Tanaka theory to the context of supergeometry and obtain an upper bound on the supersymmetry dimension of geometric structures related to strongly regular bracket-generating distributions on supermanifolds and their structure…
For every parabolic subgroup $P$ of a Lie supergroup $G$, the homogeneous superspace $G/P$ carries a $G$-invariant supergeometry. We address the question whether $\mathfrak{g}=\text{Lie}(G)$ is the maximal supersymmetry of this…
Supersymmetry broken geometrically in extra dimensions naturally leads to a nearly degenerate spectrum for superparticles, ameliorating the bounds from the current searches at the LHC. We present a minimal such model with a single extra…
The maximal dimension of a commutative subalgebra of the Grassmann algebra is determined. It is shown that for any commutative subalgebra there exists a commutative subalgebra which is spanned by monomials and has the same dimension. It…
In this paper we establish a gap phenomenon for immersed surfaces with arbitrary codimension, topology and boundaries that satisfy one of a family of systems of fourth-order anisotropic geometric partial differential equations. Examples…
The maximal contact symmetry dimensions for scalar ODEs of order $\ge 4$ and vector ODEs of order $\ge 3$ are well known. Using a Cartan-geometric approach, we determine for these ODEs the next largest realizable (submaximal) symmetry…
We determine the submaximal dimensions of the spaces of almost Einstein scales and normal conformal Killing fields for connected conformal manifolds. The results depend on the signature and dimension $n$ of the conformally nonflat conformal…
Parabolic cut pairs in the boundaries of relatively hyperbolic group are a new and previously unexplored phenomenon. In this paper, we give a way to create examples of relatively hyperbolic groups with parabolic cut pairs on their boundary…
The form of realistic space-time supersymmetry is fixed, by Haag-Lopuszanski-Sohnius theorem, either to the familiar form of Poincare supersymmetry or, in massless case, to that of conformal supersymmetry. We question necessity for such…
We interpret the property of having an infinitesimal symmetry as a variational property in certain geometric structures. This is achieved by establishing a one-to-one correspondence between a class of cone structures with an infinitesimal…
We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup $A$ of a $\Gamma$-symmetric theory. Depending on how anomalous $\Gamma$ is, we find that the symmetry of…
It has long been appreciated that the toroidal reduction of any gravity or supergravity to two dimensions gives rise to a scalar coset theory exhibiting an infinite-dimensional global symmetry. This symmetry is an extension of the…
Recently, the first author together with Jens Marklof studied generalizations of the classical three distance theorem to higher dimensional toral rotations, giving upper bounds in all dimensions for the corresponding numbers of distances…
We generalize the concept of affine locally symmetric spaces for parabolic geometries. We discuss mainly $|1|$--graded geometries and we show some restrictions on their curvature coming from the existence of symmetries. We use the theory of…
In our previous work "Characterization of certain homorphic geodesic cycles on Hermitian locally symmetric manifolds of the noncompact type" in "Modern methods in Complex Analysis" Annals of Math. Studies 138 (1995) 85-118, we formulated a…
We generalize the classical Tanaka result on the finiteness of symmetry algebra for non-degenerate pseudo-product structures to the case when the completely-integrable distributions defining the pseudo-product structure are no longer…
The symmetry dimension of a geometric structure is the dimension of its symmetry algebra. We investigate symmetries of almost quaternionic structures of quaternionic dimension $n$. The maximal possible symmetry is realized by the…
The aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts. This technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity…
We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the…