Geometry The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. Proving and Generalizing Desargues’ Two-Triangle Theorem in 3-Dimensional Projective Space Thu, 18 Dec 2014 00:10:06 +0000 With the use of only the incidence axioms we prove and generalize Desargues’ two-triangle Theorem in three-dimensional projective space considering an arbitrary number of points on each one of the two distinct planes allowing corresponding points on the two planes to coincide and three points on any of the planes to be collinear. We provide three generalizations and we define the notions of a generalized line and a triangle-connected plane set of points. Dimitrios Kodokostas Copyright © 2014 Dimitrios Kodokostas. All rights reserved. An Intrinsic Characterization of Bonnet Surfaces Based on a Closed Differential Ideal Sun, 31 Aug 2014 14:39:13 +0000 The structure equations for a two-dimensional manifold are introduced and two results based on the Codazzi equations pertinent to the study of isometric surfaces are obtained from them. Important theorems pertaining to isometric surfaces are stated and a theorem due to Bonnet is obtained. A transformation for the connection forms is developed. It is proved that the angle of deformation must be harmonic, and that the differentials of many of the important variables generate a closed differential ideal. This implies that a coordinate system exists in which many of the variables satisfy particular ordinary differential equations, and these results can be used to characterize Bonnet surfaces. Paul Bracken Copyright © 2014 Paul Bracken. All rights reserved. A Local Classification of Some Special -Metrics of Constant Flag Curvature Sun, 17 Aug 2014 08:27:29 +0000 We classify some special Finsler metrics of constant flag curvature on a manifold of dimension . Hongmei Zhu Copyright © 2014 Hongmei Zhu. All rights reserved. A Review on Unique Existence Theorems in Lightlike Geometry Mon, 07 Jul 2014 07:48:58 +0000 This is a review paper of up-to-date research done on the existence of unique null curves, screen distributions, Levi-Civita connection, symmetric Ricci tensor, and scalar curvature for a large variety of lightlike submanifolds of semi-Riemannian (in particular, Lorentzian) manifolds, supported by examples and an extensive bibliography. We also propose some open problems. K. L. Duggal Copyright © 2014 K. L. Duggal. All rights reserved. Paracomplex Paracontact Pseudo-Riemannian Submersions Wed, 07 May 2014 09:51:48 +0000 We introduce the notion of paracomplex paracontact pseudo-Riemannian submersions from almost para-Hermitian manifolds onto almost paracontact metric manifolds. We discuss the transference of structures on total manifolds and base manifolds and provide some examples. We also obtain the integrability condition of horizontal distribution and investigate curvature properties under such submersions. S. S. Shukla and Uma Shankar Verma Copyright © 2014 S. S. Shukla and Uma Shankar Verma. All rights reserved. Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere Thu, 20 Mar 2014 11:24:41 +0000 In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem. Ridha Yacoub Copyright © 2014 Ridha Yacoub. All rights reserved. Geometrical and P.D.E. Methods in the Treatment of the Theory of Shells: Comparing Euclidean and Affine Approaches Sun, 23 Feb 2014 11:34:39 +0000 The use of differential equations methods in the approach, treatment, and solution of problems in diverse areas of geometry, particularly in affine differential geometry is well known and prolific, where they have proven to be quite fruitful when it comes to the obtainment of definite results. It is perhaps lesser known that the same kind of those very same methods has been and is currently being used to treat developments in some specific areas of applied sciences, such as the theory of shells where, similarly, they can be proven to be quite effective as well. In this paper we precisely show that such is the case in two particular, related instances: the historic approach of the classical, Euclidean part of the theory pursued by Fritz John, in the past century, and the more recent expositions that we ourselves have dedicated to the affine counterpart of the theory. Salvador Gigena, Daniel Abud, and Moisés Binia Copyright © 2014 Salvador Gigena et al. All rights reserved. Vanishing Theorems on Compact Hyper-kähler Manifolds Sun, 16 Feb 2014 07:36:12 +0000 We prove that if is a -positive holomorphic line bundle on a compact hyper-kähler manifold , then for with a nonnegative integer. In a special case, and , we recover a vanishing theorem of Verbitsky’s with a little stronger assumption. Qilin Yang Copyright © 2014 Qilin Yang. All rights reserved. Moser Vector Fields and Geometry of the Mabuchi Moduli Space of Kähler Metrics Thu, 02 Jan 2014 13:50:02 +0000 There is a natural Moser type transformation along any curve in the moduli spaces of Kähler metrics. In this paper we apply this transformation to give an explicit construction of the parallel transformation along a curve in the Mabuchi moduli space of Kähler metrics. This is crucial in the proof of the equivalence between the existence of the Kähler metrics with constant scalar curvature and the geodesic stability for the type II compact almost homogeneous manifolds of cohomogeneity one mentioned in (Guan 2013). We also explain a new description of the geodesics and prove a curvature property of the moduli space, called curvature symmetric, which makes it similar to some special symmetric spaces with nonpositive curvatures, although the spaces are usually not complete. Finally, we generalize our geodesic stability conjectures in (Guan 2003) and give several results on the Lie algebra structures related to the parallel transformations. In the last section, we generalize the Futaki obstruction of the Kähler-Einstein metrics to the parallel vector fields of the invariant Mabuchi moduli space. We call the related stability the parallel stability. This includes the toric and cohomogeneity one cases as well as the spherical manifolds. Daniel Guan Copyright © 2014 Daniel Guan. All rights reserved. Note on a Class of Subsets of AG(3, q) with Intersection Numbers 1, q and n with respect to the Planes Sun, 22 Dec 2013 14:53:23 +0000 We give a new and correct proof of a result of O. Ferri and S. Ferri (1995) on -caps of in this paper; moreover we prove that sets of of type with respect to the planes of have size at most with equality if and only if is a cap. Vito Napolitano Copyright © 2013 Vito Napolitano. All rights reserved. Metric Ricci Curvature for Manifolds Wed, 20 Nov 2013 09:00:26 +0000 We introduce a metric notion of Ricci curvature for manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers theorem, for surfaces as well as for a large class of higher dimensional manifolds. David Xianfeng Gu and Emil Saucan Copyright © 2013 David Xianfeng Gu and Emil Saucan. All rights reserved. Hankel Determinant for -Valent Alpha-Convex Functions Tue, 08 Oct 2013 13:21:32 +0000 The objective of the present paper is to obtain the sharp upper bound of for p-valent α-convex functions of the form in the unit disc . Gagandeep Singh and B. S. Mehrok Copyright © 2013 Gagandeep Singh and B. S. Mehrok. All rights reserved. Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds Mon, 30 Sep 2013 11:46:17 +0000 We study a family of closed connected orientable 3-manifolds obtained by Dehn surgeries with rational coefficients along the oriented components of certain links. This family contains all the manifolds obtained by surgery along the (hyperbolic) 2-bridge knots. We find geometric presentations for the fundamental group of such manifolds and represent them as branched covering spaces. As a consequence, we prove that the surgery manifolds, arising from the hyperbolic 2-bridge knots, have Heegaard genus 2 and are 2-fold coverings of the 3-sphere branched over well-specified links. Alberto Cavicchioli, Fulvia Spaggiari, and Agnese Ilaria Telloni Copyright © 2013 Alberto Cavicchioli et al. All rights reserved. The Long-Time Behavior of the Ricci Tensor under the Ricci Flow Thu, 19 Sep 2013 08:43:17 +0000 We show that, given an immortal solution to the Ricci flow on a closed manifold with uniformly bounded curvature and diameter, the Ricci tensor goes to zero as . We also show that if there exists an immortal solution on a closed 3-dimensional manifold such that the product of the curvature and the square of the diameter is uniformly bounded, then this solution must be of type III. Christian Hilaire Copyright © 2013 Christian Hilaire. All rights reserved. Conformal Geometry of Hypersurfaces in Lorentz Space Forms Mon, 16 Sep 2013 17:33:01 +0000 Let be a space-like hypersurface without umbilical points in the Lorentz space form . We define the conformal metric and the conformal second fundamental form on the hypersurface, which determines the hypersurface up to conformal transformation of . We calculate the Euler-Lagrange equations of the volume functional of the hypersurface with respect to the conformal metric, whose critical point is called a Willmore hypersurface, and we give a conformal characteristic of the hypersurfaces with constant mean curvature and constant scalar curvature. Finally, we prove that if the hypersurface with constant mean curvature and constant scalar curvature is Willmore, then is a hypersurface in . Tongzhu Li and Changxiong Nie Copyright © 2013 Tongzhu Li and Changxiong Nie. All rights reserved. Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties Sun, 08 Sep 2013 13:55:40 +0000 Let be an integral and nondegenerate variety. Let be the minimal integer such that is the -secant variety of , that is, the minimal integer such that for a general there is with and , where is the linear span. Here we prove that for every there is a zero-dimensional scheme such that and ; we may take as union of points and tangent vectors of . E. Ballico Copyright © 2013 E. Ballico. All rights reserved. A Porism Concerning Cyclic Quadrilaterals Tue, 13 Aug 2013 08:45:08 +0000 We present a geometric theorem on a porism about cyclic quadrilaterals, namely, the existence of an infinite number of cyclic quadrilaterals through four fixed collinear points once one exists. Also, a technique of proving such properties with the use of pseudounitary traceless matrices is presented. A similar property holds for general quadrics as well as for the circle. Jerzy Kocik Copyright © 2013 Jerzy Kocik. All rights reserved. Certain Results on Ricci Solitons in -Sasakian Manifolds Mon, 15 Jul 2013 11:49:15 +0000 We study Ricci solitons in -Sasakian manifolds and show that it is a shrinking or expanding soliton and the manifold is Einstein with Killing vector field. Further, we prove that if is conformal Killilng vector field, then the Ricci soliton in 3-dimensional -Sasakian manifolds is shrinking or expanding but cannot be steady. S. R. Ashoka, C. S. Bagewadi, and Gurupadavva Ingalahalli Copyright © 2013 S. R. Ashoka et al. All rights reserved. Fekete-Szegö Coefficient Functional for Certain Subclasses of Close-to-Star Functions Thu, 27 Jun 2013 11:27:14 +0000 We introduce some subclasses of close-to-star functions defined by subordination and obtain sharp upper bounds of the functional , real, for an analytic function , , belonging to these sub-classes. B. S. Mehrok and Gagandeep Singh Copyright © 2013 B. S. Mehrok and Gagandeep Singh. All rights reserved. On a Hypersurface of a Finsler Space with Randers Change of Matsumoto Metric Wed, 26 Jun 2013 13:45:58 +0000 The present paper contains certain geometrical properties of a hypersurface of a Finsler space with Randers change of Matsumoto metric. M. K. Gupta, Abhay Singh, and P. N. Pandey Copyright © 2013 M. K. Gupta et al. All rights reserved. Hypersurfaces with Null Higher Order Anisotropic Mean Curvature Mon, 24 Jun 2013 14:36:22 +0000 Given a positive function on which satisfies a convexity condition, for , we define for hypersurfaces in the th anisotropic mean curvature function , a generalization of the usual th mean curvature function. We call a hypersurface anisotropic minimal if , and anisotropic -minimal if . Let be the set of points which are omitted by the hyperplanes tangent to . We will prove that if an oriented hypersurface is anisotropic minimal, and the set is open and nonempty, then is a part of a hyperplane of . We also prove that if an oriented hypersurface is anisotropic -minimal and its th anisotropic mean curvature is nonzero everywhere, and the set is open and nonempty, then has anisotropic relative nullity . Hua Wang and Yijun He Copyright © 2013 Hua Wang and Yijun He. All rights reserved. On a Subclass of Meromorphic Functions Defined by Hilbert Space Operator Wed, 19 Jun 2013 09:19:31 +0000 In this paper, we define a new operator on the class of meromorphic functions and define a subclass using Hilbert space operator. Coefficient estimate, distortion bounds, extreme points, radii of starlikeness, and convexity are obtained. Thomas Rosy and S. Sunil Varma Copyright © 2013 Thomas Rosy and S. Sunil Varma. All rights reserved. On Parallelism of Half-Lightlike Submanifolds of Indefinite Kenmotsu Manifolds Wed, 05 Jun 2013 16:08:55 +0000 We mainly investigate the parallelism of half-lightlike submanifolds of indefinite Kenmotsu manifolds. It is proved that a tangential half-lightlike submanifold of an indefinite Kenmotsu space form with semiparallel second fundamental form either satisfies or is -mixed geodesic. Wenjie Wang and Ximin Liu Copyright © 2013 Wenjie Wang and Ximin Liu. All rights reserved. Cyclic Branched Coverings Over Some Classes of (1,1)-Knots Thu, 16 May 2013 14:30:00 +0000 We construct a 4-parametric family of combinatorial closed 3-manifolds, obtained by glueing together in pairs the boundary faces of polyhedral 3-balls. Then, we obtain geometric presentations of the fundamental groups of these manifolds and determine the corresponding split extension groups. Finally, we prove that the considered manifolds are cyclic coverings of the 3-sphere branched over well-specified -knots, including torus knots and Montesinos knots. Agnese Ilaria Telloni Copyright © 2013 Agnese Ilaria Telloni. All rights reserved. CR-Submanifolds of Generalized -Space Forms Tue, 30 Apr 2013 08:13:05 +0000 We study sectional curvature, Ricci tensor, and scalar curvature of submanifolds of generalized -space forms. Then we give an upper bound for foliate -horizontal (and vertical) CR-submanifold of a generalized -space form and an upper bound for minimal -horizontal (and vertical) CR-submanifold of a generalized -space form. Finally, we give the same results for special cases of generalized -space forms such as -space forms, generalized Sasakian space forms, Sasakian space forms, Kenmotsu space forms, cosymplectic space forms, and almost -manifolds. Mahmood Jaafari Matehkolaee Copyright © 2013 Mahmood Jaafari Matehkolaee. All rights reserved. The Geometry of Tangent Bundles: Canonical Vector Fields Sun, 14 Apr 2013 09:20:34 +0000 A canonical vector field on the tangent bundle is a vector field defined by an invariant coordinate construction. In this paper, a complete classification of canonical vector fields on tangent bundles, depending on vector fields defined on their bases, is obtained. It is shown that every canonical vector field is a linear combination with constant coefficients of three vector fields: the variational vector field (canonical lift), the Liouville vector field, and the vertical lift of a vector field on the base of the tangent bundle. Tongzhu Li and Demeter Krupka Copyright © 2013 Tongzhu Li and Demeter Krupka. All rights reserved. Darboux Transforms of a Harmonic Inverse Mean Curvature Surface Sun, 07 Apr 2013 16:24:39 +0000 The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäcklund transform. For a given isothermic harmonic inverse mean curvature surface, its classical Darboux transform is a harmonic inverse mean curvature surface. Then a transform of a solution to the Painlevé III equation in trigonometric form is defined by a classical Darboux transform of a harmonic inverse mean curvature surface of revolution. Katsuhiro Moriya Copyright © 2013 Katsuhiro Moriya. All rights reserved. An Upper Bound for the Symmetric Tensor Rank of a Low Degree Polynomial in a Large Number of Variables Thu, 14 Mar 2013 19:15:39 +0000 Fix integers and . Let be a degree homogeneous polynomial in variables. Here, we prove that is the sum of at most -powers of linear forms (of course, this inequality is nontrivial only if .) E. Ballico Copyright © 2013 E. Ballico. All rights reserved. On an R-Randers th-Root Space Wed, 06 Feb 2013 15:40:44 +0000 We consider an n-dimensional Finsler space with the metric , where is an th-root metric and is a Riemannian metric. We call such space as an R-Randers th-root space. We obtain the expressions for the fundamental metric tensor, Cartan tensor, geodesic spray coefficients, and the coefficients of nonlinear connection in an R-Randers th-root space. Some other properties of such space have also been discussed. P. N. Pandey and Shivalika Saxena Copyright © 2013 P. N. Pandey and Shivalika Saxena. All rights reserved. Generalized Projectively Symmetric Spaces Mon, 04 Feb 2013 09:27:29 +0000 We study generalized projectively symmetric spaces. We first study some geometric properties of projectively symmetric spaces and prove that any such space is projectively homogeneous and under certain conditions the projective curvature tensor vanishes. Then we prove that given any regular projective s-space (, ), there exists a projectively related connection , such that (, ) is an affine s-manifold. Dariush Latifi and Asadollah Razavi Copyright © 2013 Dariush Latifi and Asadollah Razavi. All rights reserved.