Related papers: Fundamental theorem of hyperbolic geometry without…
Let $F$ be a proper essential immersed surface in a hyperbolic 3-manifold $M$ with boundary disjoint from a torus boundary component $T$ of $M$. Let $\alpha$ be the set of coannular slopes of $F$ on $T$. The main theorem of the paper shows…
We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism $f: M \to\mathbb{R}^n$ is bijective if and only if $H_{n-1}(M)=0$ and the pre-image of…
Let $T: A\to B$ be a (not necessarily surjective) linear isometry between two real JB$^*$-triples. Then for each $a\in A$ there exists a tripotent $u_a$ in the bidual, $B'',$ of $B$ such that \begin{enumerate}[$(a)$] \item…
We study the homology of Riemannian manifolds of finite volume that are covered by an $r$-fold product $(\mathbb{H}^2)^r = \mathbb{H}^2 \times \ldots \times \mathbb{H}^2$ of hyperbolic planes. Using a variation of a method developed by…
We prove the following: 1. Let epsilon>0 and let S_1,S_2 be two closed hyperbolic surfaces. Then there exists locally-isometric covers S'_i of S_i (for i=1,2) such that there is a (1+\epsilon) bi-Lipschitz homeomorphism between S'_1 and…
We prove that if $X$ is a real rearrangement-invariant function space on $[0,1]$, which is not isometrically isomorphic to $L_2,$ then every surjective isometry $T:X\to X$ is of the form $Tf(s)=a(s)f(\sigma(s))$ for a Borel function $a$ and…
About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed…
We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not…
In order to generalize the results of Mazur-Ulam and Vogt, we shall prove that any map T which preserves equality of distance with T(0)=0 between two F-spaces without surjective condition is linear. Then, as a special case linear isometries…
We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and…
Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary.…
We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k\subset…
The embedding theorem arises in several problems from analysis and geometry. The purpose of this paper is to provide a deeper understanding of analysis and geometry with a particular focus on embedding theorems on spaces of homogeneous type…
When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry-invariant? This question was answered for one-ended unimodular graphs in…
In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for…
The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion.…
In this talk we shall show a perfect fluid cosmological model and its properties. The model possesses an orthogonally transitive abelian two-dimensional group of isometries that corresponds to cylindrical symmetry. The matter content is a…
After having investigated the real conic sections and their isoptic curves in the hyperbolic plane $\bH^2$ we consider the problem of the isoptic curves of generalized conic sections in the extended hyperbolic plane. This topic is widely…
We use Herbrand's theorem to give a new proof that Euclid's parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a…
Let F be a commutative field of characteristic 0, G_n: F^n \times F^n -> F, G_n((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say that g:R^n->F^n preserves distance d>=0 if for each x,y \in R^n |x-y|=d implies…