Showing papers in "Geometriae Dedicata in 1997"
TL;DR: In this article, the energy of a unit vector field on a Riemannian manifold M is defined to be the energy energy of the mapping M → T 1M, where the unit tangent bundle T1M is equipped with the restriction of the Sasaki metric.
Abstract: The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M → T1M, where the unit tangent bundle T1M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M=S5,S7,..., and an estimate on the index is obtained.
129 citations
TL;DR: In this article, it was shown that the Zariski closure of the image of a closed orientable surface of genus at least 2 and a connected finite abelian covering with covering group $G is an arithmetic group.
Abstract: Let S be a closed orientable surface of genus at least 2 and let \(\widetilde S\) to S be a connected finite abelian covering with covering group $G$. The lifts of liftable mapping classes of S determine a central extension (by G) of a subgroup of finite index of the mapping class group of S. This extension acts on H1(\(\widetilde S\)). With a few exceptions for genus 2, we determine the Zariski closure of the image of this representation, and prove that the image is an arithmetic group.
85 citations
TL;DR: In this paper, the authors studied surfaces immersed in R3 such that the mean curvature function H satisfies the equation Δ (1/H) = 0, where Δ is the Laplace operator of the induced metric.
Abstract: In this paper we study surfaces immersed in R3 such that the mean curvature function H satisfies the equation Δ (1/H) = 0, where Δ is the Laplace operator of the induced metric. We call them HIMC surfaces. All HIMC surfaces of revolution are classified in terms of the third Painleve transcendent. In the general class of HIMC surfaces we distinguish a subclass of θ-isothermic surfaces, which is a generalization of the isothermic HIMC surfaces, and classify all the θ-isothermic HIMC surfaces in terms of the solutions of the fifth and sixth Painleve transcendents.
45 citations
TL;DR: In this article, a quasi-polarized version of the Kodaira dimension of a smooth projective variety over the complex numbers (X,L) is considered, and the conjecture that g(L) = q(X) is proved to be true.
Abstract: Let (X,L) be a quasi-polarized variety, i.e. X is a smooth projective variety over the complex numbers \(\mathbb{C}\) and L is a nef and big divisor on X. Then we conjecture that g(L) = q(X), whereg(L) is the sectional genus of L and \(q(X) = \dim H^1 (\mathcal{O}_X )\). In this paper, we treat the case \(\dim X = 2\). First we prove that this conjecture is true for \(\kappa (X) \leqslant 1\), and we classify (X,L) withg(L)=q(X), where \(\kappa (X)\) is the Kodaira dimension of X. Next we study some special cases of\(\kappa (X) = 2\).
38 citations
TL;DR: For any compact set K⊂RN, the authors constructed a hyperbolic graph CK such that the conformal dimension of CK is at most the box dimension of K.
Abstract: For any compact set K⊂RN we construct a hyperbolic graph CK, such that the conformal dimension of CK is at most the box dimension of K.
37 citations
TL;DR: In this article, the concept of a relative Tchebychev hypersurface was introduced, which extends that of affine spheres in equiaffine geometry and also that of centroaffine hypersurfaces and gave partial local and global classifications for this new class.
Abstract: We introduce the concept of a relative Tchebychev hypersurface which extends that of affine spheres in equiaffine geometry and also that of centroaffine Tchebychev hypersurfaces and give partial local and global classifications for this new class. Our tools concern a new operator and interesting properties of the traceless part of the cubic form.
32 citations
TL;DR: The necessary and sufficient condition for a given set of positive real numbers to be the dihedral angles of a hyperbolic n-simplex is given in this paper. But the necessary and necessary condition is not defined in this paper.
Abstract: We give a necessary and sufficient condition for a given set of positive real numbers to be the dihedral angles of a hyperbolic n -simplex in this note. This answers a question of W. Fenchel raised in his book, Elementary Geometry in Hyperbolic Space, (De Gruyter, Berlin, 1989, p. 174) where he obtained some necessary conditions for which six numbers have to satisfy in order to be the dihedral angles of a hyperbolic tetrahedron. We also present a simple proof of the known necessary and sufficient condition for the dihedral angles of Euclidean n-simplexes.
32 citations
TL;DR: In this paper, it was shown that each convex 3-polytope contains a path on three vertices with restricted degrees which is one of the ten types of paths.
Abstract: In this paper we prove that each convex 3-polytope contains a path on three vertices with restricted degrees which is one of the ten types. This result strengthens a theorem by Kotzig that each convex 3-polytope has an edge with the degree sum of its end vertices at most 13.
32 citations
TL;DR: In this paper, it was shown that an ovoid O of Q(4,q),q odd, is the Thas' ovoid associated with a semifield flock if and only if O represents, on the Klein quadric, a symplectic spread of PG(3,q) whose associated plane is a semifiled plane.
Abstract: We prove that an ovoid O of Q(4,q),q odd, is the Thas' ovoid associated with a semifield flock if and only if O represents, on the Klein quadric, a symplectic spread of PG(3,q), whose associated plane is a semifiled plane.
31 citations
TL;DR: In this article, it was shown that the perimeter of any convex n-gons of diameter 1 is at most n2nsin (π/2n) if and only if n has an odd factor.
Abstract: We prove that the perimeter of any convex n-gons of diameter 1 is at most n2nsin (π/2n). Equality is attained here if and only if n has an odd factor. In the latter case, there are (up to congruence) only finitely many extremal n-gons. In fact, the convex n-gons of diameter 1 and perimeter n2n sin (π/2n) are in bijective correspondence with the solutions of a diophantine problem.
30 citations
TL;DR: In this article, it was shown that if d ≥ 5, then this property holds true also for the on-line packing of a sequence of d-dimensional cubes of total volume 2.
Abstract: Almost thirty years ago Meir and Moser proved that every sequence of d -dimensional cubes of total volume 2 \(\frac{1}{2}\))d can be packed in the unit cube. We show that if d≥ 5, then this property holds true also for the on-line packing.
TL;DR: The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell-Weil lattices of elliptic curves as mentioned in this paper.
Abstract: The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell--Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L
2(q) and 2B2(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (pn-1)/2 of the symplectic group Sp2n(p) (p ≡ 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pn−1). A summary of currently known globally irreducible representations is given.
TL;DR: In this article, the Crofton formula and the principal kinematic formula of integral geometry from curvature measures of convex bodies were extended to generalized curvatures of finite unions.
Abstract: We establish extensions of the Crofton formula and, under some restrictions, of the principal kinematic formula of integral geometry from curvature measures to generalized curvature measures of convex bodies. We also treat versions for finite unions of convex bodies. As a consequence, we get a new intuitive interpretation of the area measures of Aleksandrov and Fenchel–Jessen.
TL;DR: The main theorem of as discussed by the authors describes the possible embeddings of classical groups in classical groups such that Siegel transvections act as (Siegel) transvection in the linear, symplectic or unitary groups.
Abstract: We are concerned with finite-dimensional classical groups over arbitrary commutative fields. In an orthogonal group a Siegel transvection, that is, an element centralizing l⊥ for some totally singular 2-dimensional subspace l, plays the same role as a transvection in the linear, symplectic or unitary groups. The Main Theorem of this paper describes the possible embeddings of classical groups in classical groups such that (Siegel) transvections act as (Siegel) transvections.
TL;DR: In this article, the ideal boundary of the Heisenberg group H2n+1 is shown to be a sphere S 2n-1 with a natural CR-structure and corresponding Carnot-Caratheodory metric.
Abstract: We obtain equations of geodesic lines in Heisenberg groups H2n+1and prove that the ideal boundary of the Heisenberg group H2n+1is a sphere S2n-1with a natural CR-structure and corresponding Carnot-Caratheodory metric, i.e. it is a one-point compactification of the Heisenberg group H2n-1of the next dimension in a row.
TL;DR: In this article, the homogeneous nilmanifolds of dimension 3 and 4 were classified up to isometry and the corresponding isometry groups were computed for isometry of these.
Abstract: A homogeneous nilmanifold is a nilpotent Lie group endowed with a left-invariant metric. In the present work we classify up to isometry the homogeneous nilmanifolds of dimension 3 and 4 and we compute the corresponding isometry groups.
TL;DR: Hadwiger as discussed by the authors showed that a convex body K in a Euclidean d-space, d ≥ 1, can always be convered by 2 smaller homotheti copies of K.
Abstract: In 1957, H. Hadwiger conjectured that a convex body K in a Euclidean d-space, d ≥ 1, can always be convered by 2
d
smaller homotheti copies of K. We verify this conjecture when K is the polar of a cyclic d-polytope.
TL;DR: In this article, the authors classify groups whose automorphism group has at most three orbits and their holomorph is a rank 3 permutation group, i.e., it can be seen as a rank 2 permutation.
Abstract: We classify those groups whose automorphism group has at most three orbits. In other words, we classify those groups whose holomorph is a rank 3 permutation group.
TL;DR: Using modular quotients of linear groups defined over the Eisenstein ring Z[ω], the authors constructed infinite families of finite regular or chiral polytopes of types {3,3,6, 3, 6, 3} and {6,3-6, 6}.
Abstract: Using modular quotients of linear groups defined over the Eisenstein ring Z[ω], we construct infinite families of finite regular or chiral polytopes of types {3,3,6}, {3,6,3} and {6,3,6}.
TL;DR: In this paper, it was proved that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.
Abstract: Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.
TL;DR: The average curvature of a rectifiable closed curve in R2 is its total absolute curvature divided by its length as discussed by the authors, and the average curvatures of a simple closed curve ∂ D which bounds the convex set.
Abstract: The average curvature of a rectifiable closed curve in R2 is its total absolute curvature divided by its length. If a rectifiable closed curve in R2 is contained in the interior of a convex set D then its average curvature is at least as large as the average curvature of the simple closed curve ∂ D which bounds the convex set.
TL;DR: In this paper, it was shown that the quotient module of a non-degenerate alternating bilinear form defined on V × V is a spin module for Sp(2m,K) when K is algebraically closed.
Abstract: Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers Λ
k
V for 0 ≤ k.≤ 2m There is a contraction map ∂ defined on the exterior algebra Λ, which commutes with the Sp(2m, K) action and satisfies ∂2 = 0 and ∂(Λ
k
V) ≤ Λ
k−1
V We prove that ∂(Λ
k
V)= ker ∂ ∩ Λ
k−1
V except when k=m+2. In the exceptional case, ∂(Λ
m+2
V) has codimension 2m in ker ∂ ∩ Λ
m
V and we show that the quotient module ker ∂ ∩ Λ
m
V/∂ ∩ Λ
m+2
V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in Λ
m
V and multiplicity 0 in all other components of ΛV.
TL;DR: In this article, the authors studied simply connected Lie groups with left invariant Einstein metric, negative Einstein constant and non-positive sectional curvature and constructed new examples of Einstein spaces with nonpositive curvature.
Abstract: The purpose of this article is to study some simply connected Lie groups with left invariant Einstein metric, negative Einstein constant and nonpositive sectional curvature. These Lie groups are classified if their associated metric Lie algebra s is of Iwasawa type and s = ∝A⊥n1⊥n2⊥...⊥nr, where all niare Lie algebras of Heisenberg type with [[ni,nj] = {0} for i≠j. The most important ideas of the article are based on a construction method for Einstein spaces introduced by Wolter in 1991. By this method some new examples of Einstein spaces with nonpositive curvature are constructed. In another part of the article it is shown that Damek-Ricci spaces have negative sectional curvature if and only if they are symmetric spaces.
TL;DR: In this paper, the Yamabe equation was studied on a complete Riemannian manifold of dimension n ≥ 3 whose scalar curvature S(x) is positive for all x in the manifold.
Abstract: We let (M,g) be a noncompact complete Riemannian manifold of dimension n ≥ 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on −Δu+[(n−2)/(4(n−1))]Su=qu
(n+2)/(n−2) on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.
TL;DR: In this paper, a proof of the following property of compact irreducible Hermitian symmetric spaces is given: if H = G/K where G is a compact simply connected simple Lie group, T a maximal torus of G and F(T,H) is the fixed point set of T on H, then for each pair E i, E j there is a 2-dimentional sphere N ij ⊂ H such that Ei and Ej are antipodal points of Nij.
Abstract: This paper contains a proof of the following property of compact irreducible Hermitian symmetric spaces. If H=G/K where G is a compact simply connected simple Lie group, T a maximal torus of G and F(T,H)=|E1,...,Em is the fixed point set of T on H, then for each pair E i , E j there is a 2-dimentional sphere N ij ⊂ H such that Ei and Ej are antipodal points of Nij.
TL;DR: In this paper, the authors consider a canonical Gorenstein curve C of arithmetic genus g in P g-1 (K) that admits a non-singular point P, whose Weierstrass semigroup is quasi-symmetric in the sense that the last gap is equal to 2g-2.
Abstract: We consider a canonical Gorenstein curve C of arithmetic genus g in P g-1 (K), that admits a non-singular point P, whose Weierstrass semigroup is quasi-symmetric in the sense that the last gap is equal to 2g-2. By making local considerations at the point P and the second point of the curve C on its osculating hyperplane at P we construct monomial bases for the spaces of higher order regular differentials. We give an irreducibility criterion for the canonical curve in terms of the coefficients of the quadratic relations. We also realize each quasi-symmetric numerical semigroup as the Weierstrass semigroup of a reducible canonical Gorenstein curve, but we give examples of such semigroups that cannot be realized as Weierstrass semigroups of smooth curves.
TL;DR: In this paper, the theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane.
Abstract: The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plucker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J
+ in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.
TL;DR: In this article, the authors investigated holomorphic maps between compact generalized Hopf manifolds (i.e., locally conformal Kahler manifolds with parallel Lee form) and holomorphic submersions from compact g.H. manifolds.
Abstract: We investigate holomorphic maps between compact generalized Hopf manifolds (i.e., locally conformal Kahler manifolds with parallel Lee form). We show that they preserve the canonical foliations. Moreover, we study compact complex submanifolds of g.H. manifolds and holomorphic submersions from compact g.H. manifolds.
TL;DR: In this article, the maximum principle for second-order elliptic operators is used to establish a sufficient condition for a compact hypersurface in a space form to be a geodesic sphere in terms of a pinching for the s-mean curvature.
Abstract: We use the maximum principle for second-order elliptic operators to establish a sufficient condition for a compact hypersurface in a space form to be a geodesic sphere in terms of a pinching for the s-mean curvature.
TL;DR: In this article, the authors improved the Brunn-Minkowski inequality for nonconvex sets by taking into account the volume of the convex hull of one of the sets, which is asymptotically the best possible if this set is fixed and the size of the other tends to infinity.
Abstract: We improve the Brunn–Minkowski inequality for nonconvex sets. Besides the volume of the sets, our estimate depends on the volume of the convex hull of one of the sets. The estimate is asymptotically the best possible if this set is fixed and the size of the other tends to infinity.