Related papers: Classifying spaces for homogeneous manifolds and t…

The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show…

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is also equivalent to the description of all flat submanifolds with flat normal bundle in a pseudo-Euclidean space.…

We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of…

Here, we classify Lie groups acting isometrically on compact Lorentz manifolds, and in particular we describe the geometric structure of compact homogeneous Lorentz manifolds.

A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$,…

The complete classification of classical $r$-matrices generating quantum deformations of the (3+1)-dimensional (A)dS and Poincar\'e groups such that their Lorentz sector is a quantum subgroup is presented. It is found that there exists…

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space and anti-de Sitter 3-space is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group…

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.

We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test…

The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-)Finslerian manifolds; under some additional…

We investigate the mapping class groups of a class of non-Hausdorff topological spaces which includes finite spaces. We show that the mapping class group of a finite space is isomorphic to the homeomorphism group of its $T_0$ quotient. As a…

Several classes of irreducible orthogonal representations of compact Lie groups that are of importance in Differential Geometry have the property that the second osculating spaces of all of their nontrivial orbits coincide with the…

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits…

What does it mean for a shape to change continuously? Over the space of convex regions, there is only one "reasonable" answer. However, over a broader class of regions, such as the class of star-shaped regions, there can be many different…

A detailed study is made of the noncommutative geometry of $R^3_q$, the quantum space covariant under the quantum group $SO_q(3)$. For each of its two $SO_q(3)$-covariant differential calculi we find its metric, the corresponding frame and…

Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a…

Locally symmetric spaces like $SL(n,\mathbb Z)\backslash SL_n(\mathbb R)/SO(n)$ contain immersed compact flat manifolds of dimension equal to the real rank. We give a lower bound for the contribution of these cycles to the homology of…

Hamiltonian structures for spatially compact locally homogeneous vacuum universes are investigated, provided that the set of dynamical variables contains the \Teich parameters, parameterizing the purely global geometry. One of the key…

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$.…

The N-dimensional Cayley-Klein scheme allows the simultaneous description of $3^N$ geometries (symmetric orthogonal homogeneous spaces) by means of a set of Lie algebras depending on $N$ real parameters. We present here a quantum…