Related papers: The Mazur-Ulam theorem
We prove that every surjective isometry from the unit sphere of a rank-2 Cartan factor $C$ onto the unit sphere of a real Banach space $Y$, admits an extension to a surjective real linear isometry from $C$ onto $Y$. The conclusion also…
Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area…
Suppose that $f_1,\ldots ,f_m : S(V)\to R$ are $m$ ($\geq 1$) continuous functions defined on the unit sphere in a Euclidean vector space $V$ of dimension $m+1$ satisfying $f_i(-v)=-f_i(v)$ for all $v\in S(V)$. The classical Borsuk-Ulam…
In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.
We introduce the notion of induced Maslov cycle, which describes and unifies geometrical and topological invariants of many apparently unrelated problems, from Real Algebraic Geometry to sub-Riemannian Geometry.
Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.
Structures of commuting semigroups of isometries under certain additional assumptions like double commutativity or dual double commutativity are found.
A proof of the isometric embedding of a given two-metric in E^3 of class C^1. The method uses the theory of first order partial differential equations. The curvature of the metric plays no role in the proof.
An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, and extend these methods to spherical, single elliptical, and hyperbolic geometry.…
This is a detailed introductory survey of the cohomological dimension theory of compact metric spaces.
We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular…
More than 20 years after Fickett attempted to prove the Hyers-Ulam stability of isometries defined on bounded subsets of $\mathbb{R}^n$ in 1981, V\"{a}is\"{a}l\"{a} improved Fickett's result significantly. In this paper, we will improve…
We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we obtain some geometric applications. Among them, we emphasize the following ones: 1.…
We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces $\Bbb U$, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry $ISO(\Bbb U)$ contains…
Stated lemma contains the assertions about isomorphism of exact m-forms and exterior differentials of regular m-maps, of linearly harmonic m-forms and exterior differentials of regular harmonic m-maps, of global minimal (n-m)-surfaces and…
We establish the Hyers-Ulam stability of certain linear first-order differential equations with singularities. We then extend these results to higher-order singular linear differential equations that can be written with these first-order…
A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and…
The classical Banach--Mazur theorem asserts that every separable Banach space admits an isometric embedding into $C[0,1]$. It is also well known that every separable Banach space embeds isometrically into $\ell^\infty$. We show that such an…
The Dold$-$Thom theorem states that for a sufficiently nice topological space, M, there is an isomorphism between the homotopy groups of the infinite symmetric product of M and the homology groups of M itself. The crux of most known proofs…
We give a short proof of Ahlfors' theorem on covering surfaces.