Related papers: The Mazur-Ulam theorem
In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion and demonstrate that Bismut-Zhang's equivariant Cheeger-M\"{u}ller theorem simplifies considerably when applied to locally symmetric spaces. In a…
We prove a flat strip theorem for 2-dimensional ptolemaic spaces.
We prove that the Coleman-Mazur eigencurve is proper (over the weight space) at a large class of points.
It is proved the following theorem, if $w$ is a quasiconformal harmonic mappings between two Riemann surfaces with smooth boundary and aproximate analytic metric, then $w$ is a quasi-isometry with respect to Euclidean metric.
We define a monomial space to be a subspace of $\ltwo$ that can be approximated by spaces that are spanned by monomial functions. We describe the structure of monomial spaces.
We show that there are uncountably many geodesics between any two non-isometric $n$-dimensional normed spaces. We construct two explicit geodesics that can be used to describe all the points of the other geodesics.
We investigate Problem 155 form the "Scottish Book" due to S. Mazur and L. Sternbach. In modern terminology they asked if every bijective, locally isometric map between two real Banach spaces is always a global isometry. Recently, an…
Under certain hypothesises, we prove that a map which is an isometry for the Caratheodory infinitesimal metric at a point is an analytic isomorphism onto its image.
The main result says that every surjective isometry between two ideal Banach function spaces satisfying certain conditions can be presented as a composition of a measurable transformation of a variable and multiplication by a function.
We determine when a quasi-isometry between discrete spaces is at bounded distance from a bilipschitz map. From this we prove a geometric version of the Von Neumann conjecture on amenability. We also get some examples in geometric groups…
We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including…
In 2022, Hatori gave a sufficient condition for complex Banach spaces to have the complex Mazur--Ulam property. In this paper, we introduce a class of complex Banach spaces $B$ that do not satisfy the condition but enjoy the property that…
We show that a normed linear space is isometrically isomorphic to an inner product space if and only if it is a strongly $n$-point homogeneous metric space for any (or every) $n \geqslant 3$. The counterpart for $n=2$ is the Banach-Mazur…
We prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk--Ulam--Crofton technique. We consider waists of real and complex projective spaces, flat…
We give a new short proof for the uniqueness of the universal minimal space. The proof holds for the uniqueness of the universal object in every collection of topological dynamical systems closed under taking projective limits and…
We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric…
We determine when a matrix is similar to a partial isometry, refining a result of Halmos--McLaughlin.
We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including…
In this paper we state and prove the analogous of the principal ideal theorem of algebraic number theory for the case of 3-manifolds from the point of view of arithmetic topology.
In this article we will represent some ideas and a lot of new theorems in Euclidean plane geometry.