Related papers: Isometry germs and related structures

We examine the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space. A new tensorial notation is developed to present the various constructs of noncommutative geometry. In particular, this…

It is known that a connected and simply-connected Lie group admits only one left-invariant Riemannian metric up to scaling and isometry if and only if it is isomorphic to the Euclidean space, the Lie group of the real hyperbolic space, or…

Due to Janet-Cartan's theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M,g), it is in general hard to…

Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z) \ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any…

By a 2-group we mean a groupoid equipped with a weakened group structure. It is called split when it is equivalent to the semidirect product of a discrete 2-group and a one-object 2-group. By a permutation 2-group we mean the 2-group…

We investigate the geometry of the family $\cal M$ of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that sufficiently small neighborhoods of generic finite spaces in the subspace of all finite…

In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and…

We prove that if open subgroups of the groups of invertible elements in two Fourier-Stieltjes algebras are isometric as metric spaces, then the underlying locally compact groups are topologically isomorphic. We describe the structure of…

Generalized symmetries of the Einstein equations are infinitesimal transformations of the spacetime metric that formally map solutions of the Einstein equations to other solutions. The infinitesimal generators of these symmetries are…

We consider homogeneous spaces of Lie groups with compact stabilizer subgroups of two types: those with integrable invariant distributions and those with geodesic orbit invariant Riemannian metrics. The latter means that for an arbitrary…

All the Lie point symmetries of the near extremal horizon geometry equation, in the case of 4-dimensional Einstein vacuum spacetime with cosmological constant, are the diffeomorphisms of the space of the null generators of the horizon. This…

Isometries of metric spaces $(X,d)$ preserve all level sets of $d$. We formulate and prove cases of a conjecture asserting if $X$ is a complete Riemannian manifold, then a function $f:X \rightarrow X$ preserving at least one level set…

To any two-dimensional rational plane in four-dimensional space one can naturally attach a point in the Grassmannian Gr(2,4) and four lattices of rank two. Here, the first two lattices originate from the plane and its orthogonal complement…

In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal…

We consider the deformation theory of two kinds of geometric objects: foliations on one hand, pre-symplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the…

Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if…

It is known by a result of Mendes and Sampaio that the Lipschitz normal embedding of a subanalytic germ is fully characterized by the Lipschitz normal embedding of its link. In this note, we show that the result still holds for definable…

We study the isometry groups of (non-compact) Lorentzian manifolds with well-behaved causal structure, aka causal spacetimes satisfying the ``no observer horizons'' condition. Our main result is that the group of time orientation-preserving…

We associate to any Riemannian symmetric space (of finite or infinite dimension) a L$^*$-algebra, under the assumption that the curvature operator has a fixed sign. L$^*$-algebras are Lie algebras with a pleasant Hilbert space structure.…

The main theorem states that any complete connected Riemannian manifold of bounded geometry can be isometrically realized as a leaf with trivial holonomy in a compact Riemannian foliated space.