Volume 14, Issue 3

We prove the equivalence of several sets of axioms for non-discrete affine buildings, by providing different types of metric, exchange and atlas conditions. We apply our result to show that the definition of a Euclidean building depends only on the topological equivalence class of the metric on the model space. The sharpness of the axioms dealing with metric conditions is illustrated in an appendix. There it is shown that a space X defined over a model space with metric d is possibly a building only if the induced distance function on X satisfies the triangle inequality.

In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n andm, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the Cauchy- Riemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention is mostly restricted to the case of low dimensions where it is possible to identify new and interesting properties and to single out the special role played by bicomplex analysis.

We deal with different properties of a smooth and strictly convex body that depend on the behavior of the planar sections of the body parallel to and close to a given tangent plane. The first topic is boundary points where any given convex domain in the tangent plane can be approximated by a sequence of suitably rescaled planar sections (so-called p-universal points). In the second topic, the given convex body is the unit ball of a Minkowski space, and oscillation properties of bisectors and trisectors in that space are considered. In either case it turns out that geometrically irregular behavior is typical, when considered from the viewpoint of Baire category.

Let M be a C 2 -smooth strictly convex closed surface in ℝ 3 and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface containing M or a plane, then M is a Euclidean sphere. Moreover, we shall see that the situation in the Euclidean plane is very different.

We study some properties of harmonic maps for Finsler manifolds and present two theorems: (1) every non-degenerate harmonic map ϕ from a compact Riemannian manifold with nonnegative Ricci curvature to a Berwald manifold with non-positive flag curvature is totally geodesic; this improves a result of He and Shen [3]. (2) Let M be a compact Riemannian manifold with non-negative Ricci curvature and let M‾ be a complete Finsler manifold with non-positive flag curvature. Then every non-degenerate smooth map ϕ : M → M with tr b ▽̃dϕ = 0 is totally geodesic; this generalizes a result of Eells and Sampson [2] from the Riemannian case to the Finsler case.

We discuss a polyhedral embedding of the classical Fricke-Klein regular map of genus 5 in ordinary space E 3 . This polyhedron was originally discovered by Grünbaum in 1999, but was recently rediscovered by Brehm andWills. We establish isomorphism of the Grünbaum polyhedron with the Fricke-Klein map, and confirm its combinatorial regularity. The Grünbaum polyhedron is among the few currently known geometrically vertex-transitive polyhedra of genus g ≥ 2, and is conjectured to be the only vertex-transitive polyhedron in this genus range that is also combinatorially regular. We also contribute a new vertex-transitive polyhedron, of genus 11, to this list, as the 7th known example. In addition we show that there are only finitely many vertex-transitive polyhedra in the entire genus range g ≥ 2.

A 2-step nilpotent Lie algebra n is said to be of type (p, q) if dim n = p + q and dim[n, n] = p. By considering a class of 2-step nilpotent Lie algebras naturally attached to graphs, we prove that there exist indecomposable, 2-step nilpotent Lie groups of type (p, q) which do not admit a nilsoliton metric for every pair (p, q) such that 21 ≤ q and q − 1 ≤ p ≤½ q 2 − 5/2q + 9. This improves a result due to Jablonski [6].

We present a complete description of all rotational linear Weingarten hypersurfaces in the Euclidean sphere S n+1 . These hypersurfaces are characterized by a linear relation aH 1 +bH 2 = c, where H 1 and H 2 stand for the first two symmetric functions of the principal curvature and a, b and c are real constants.

In this paper, we study closed spacelike hypersurfaces in anti-de Sitter space. Using the Bonnet-Myers theorem, we obtain a rigidity theorem for spacelike hypersurfaces without the constancy condition on the mean curvature or the scalar curvature.

The group SSympeo(M, ω) of strong symplectic homeomorphisms or group of ssympeomorphisms of a closed connected symplectic manifold (M, ω) was defined and studied in [2], [3]. In these papers the author uses the L (1,∞) -metric on the group Iso(M, ω) of all symplectic isotopies. In this paper we study the set SSympeo(M, ω) ∞ of ssympeomorphisms in the L ∞ - metric. We prove the equality between SSympeo(M, ω) and SSympeo(M, ω) ∞ . This generalizes Müller’s result [6] asserting that Hameo(M, ω) = Hameo(M, ω) ∞ .

The classical Barvinok bound for the sum of the Betti numbers of the intersection X of three quadrics in ℝP n says that there exists a natural number a such that b(X) ≤ n 3a . We improve this bound proving the inequality b(X) ≤ n(n+1). Moreover we show that this bound is asymptotically sharp as n goes to infinity.

Let M be a compact smooth submanifold of ℝ n that is symmetric with respect to an affine subspace L. In this paper, we prove that M has a symmetric Nash model N, which is unique up to Nash isomorphism preserving symmetries. Such a model N can be chosen to be a union of nonsingular connected components of a real algebraic set if M is a (not necessarily compact) Nash submanifold of ℝ n . If M does not intersect the mirror L, then we are able to construct a symmetric algebraic model of M too. More precisely, in this case, we show that the set of birationally nonisomorphic symmetric algebraic models of M has the power of continuum.

All triangulated d-manifolds satisfy the inequality for d ≥ 3. A triangulated d-manifold is called tight neighborly if it attains equality in this bound. For each d ≥ 3, a (2d+3)-vertex tight neighborly triangulation of the S d-1 -bundle over S 1 with β 1 = 1 was constructed by Kühnel in 1986. In this paper, it is shown that there does not exist a tight neighborly triangulated manifold with β 1 = 2. In other words, there is no tight neighborly triangulation of (S d-1 x S 1 ) #2 or (S d-1 x̲ S 1 ) #2 for d ≥ 3. A short proof of the uniqueness of Kühnel’s complexes for d ≥ 4 under the assumption β 1 ≠ 0 is also presented.