We extend Witt’s theorem to several kinds of simultaneous isometries of subspaces. We determine sufficient and necessary conditions for the extension of an isometry of subspaces φ : E → E′ to an isometry φV : V → V ′ that also sends a given subspace to another, or a given self-dual flag to another, or a Witt’s decomposition to another and a special self-dual flag to another. We also determine s...

In low dimensional gravity on anti-de sitter space there exists a possibility that the asymptotic isometry is raised to a Virasoro symmetry living at spatial infinity. We discuss in detail asymptotic isometry of the most general anti-de sitter dilaton gravity in two dimensions. From analysis of the boundary dynamics it turns out that the Virasoro symmetry is not preserved in all the models. Mor...

Every ε-isometry u between real normed spaces of the same finite dimension which maps the origin to the origin may by uniformly approximated to within 2ε by a linear isometry. Under a smoothness hypothesis, necessary and sufficient conditions are obtained for the same conclusion to hold for a given ε-isometry between infinite-dimensional Banach spaces.

Let M be an irreducible n-dimensional complex projective manifold, and A → M an ample line bundle on it. Then there exists an Hermitian metric h on A such that the curvature of the unique compatible covariant derivative is −2πiω, where ω is a Kähler form on M . As is well-known, for k ≫ 0 the full linear series of global holomorphic sections of A⊗k determines a projective embedding φk : M → P (...

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear spaces. This resolves famous Tingley's problem in class

The Cauchy dual subnormality problem (for short, CDSP) asks whether the of a $2$-isometry is subnormal. In this paper, we address for cyclic $2$-isometries. view some recent developments in operator theory on function spaces

We investigate the reflexivity of the isometry group and the automorphism group of some important metric linear spaces and algebras. The paper consists of the following sections: 1. Preliminaries. 2. Sequence spaces. 3. Spaces of measurable functions. 4. Hardy spaces. 5. Banach algebras of holomorphic functions. 6. Fréchet algebras of holomorphic functions. 7. Spaces of continuous functions. In...

We investigate the quasi-isometry classification of the right-angled Coxeter groups WΓ which are 1-ended and have triangle-free defining graph Γ. We begin by characterising those WΓ which split over 2-ended subgroups, and those which are cocompact Fuchsian, in terms of properties of Γ. This allows us to apply a theorem of Papasoglu [21] to distinguish several quasi-isometry classes. We then car...

A finite set W ⊂ R is said to realize the group G if the isometry group of W is isomorphic to G. The isometry dimension of a group is the minimum dimension of a realization. It is known that the isometry dimension of G is less than |G| [1]. We show that the isometry dimension of Z2 is n. The orbit number of a group is the minimum number of orbits in a realization. We show that the groups Z2 are...