Geometry
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New Representations of Spherical Indicatricies of Bertrand Curves in Minkowski 3Space
Wed, 28 Jan 2015 11:26:23 +0000
http://www.hindawi.com/journals/geometry/2015/509058/
We obtained a new representation for timelike Bertrand
curves and their Bertrand mate in 3dimensional Minkowski space. By using
this representation, we expressed new representations of spherical indicatricies
of Bertrand curves and computed their curvatures and torsions. Furthermore
in case the indicatricies of a Bertrand curve are slant helices, we investigated
some new characteristic features of these curves.
İsmail Aydemir and Fırat Yerlikaya
Copyright © 2015 İsmail Aydemir and Fırat Yerlikaya. All rights reserved.

Hypersurface Family with a Common Isoasymptotic Curve
Wed, 31 Dec 2014 08:15:45 +0000
http://www.hindawi.com/journals/geometry/2014/623408/
We handle the problem of finding a hypersurface family from a given asymptotic curve in . Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.
Ergin Bayram and Emin Kasap
Copyright © 2014 Ergin Bayram and Emin Kasap. All rights reserved.

Proving and Generalizing Desargues’ TwoTriangle Theorem in 3Dimensional Projective Space
Thu, 18 Dec 2014 00:10:06 +0000
http://www.hindawi.com/journals/geometry/2014/276108/
With the use of only the incidence axioms we prove and generalize Desargues’ twotriangle Theorem in threedimensional 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 triangleconnected 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
http://www.hindawi.com/journals/geometry/2014/715679/
The structure equations for a twodimensional 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
http://www.hindawi.com/journals/geometry/2014/931319/
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
http://www.hindawi.com/journals/geometry/2014/835394/
This is a review paper of uptodate research done on the existence of unique null curves, screen distributions, LeviCivita connection, symmetric Ricci tensor, and scalar curvature for a large variety of lightlike submanifolds of semiRiemannian (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 PseudoRiemannian Submersions
Wed, 07 May 2014 09:51:48 +0000
http://www.hindawi.com/journals/geometry/2014/616487/
We introduce the notion of paracomplex paracontact pseudoRiemannian submersions from almost paraHermitian 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 HalfSphere
Thu, 20 Mar 2014 11:24:41 +0000
http://www.hindawi.com/journals/geometry/2014/582367/
In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition
on the standard 3dimensional halfsphere. 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
http://www.hindawi.com/journals/geometry/2014/953702/
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 Hyperkähler Manifolds
Sun, 16 Feb 2014 07:36:12 +0000
http://www.hindawi.com/journals/geometry/2014/243236/
We prove that if is a positive holomorphic line bundle on a compact hyperkähler 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
http://www.hindawi.com/journals/geometry/2014/968064/
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 KählerEinstein 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
http://www.hindawi.com/journals/geometry/2013/589362/
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
http://www.hindawi.com/journals/geometry/2013/694169/
We introduce a metric notion of Ricci curvature for manifolds and study its convergence properties. We also prove a fitting version of the BonnetMyers 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 AlphaConvex Functions
Tue, 08 Oct 2013 13:21:32 +0000
http://www.hindawi.com/journals/geometry/2013/348251/
The objective of the present paper is to obtain the sharp upper bound of for pvalent αconvex 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
http://www.hindawi.com/journals/geometry/2013/484508/
We study a family of closed connected orientable 3manifolds 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) 2bridge 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 2bridge knots, have Heegaard genus 2 and are 2fold coverings of the 3sphere branched over wellspecified links.
Alberto Cavicchioli, Fulvia Spaggiari, and Agnese Ilaria Telloni
Copyright © 2013 Alberto Cavicchioli et al. All rights reserved.

The LongTime Behavior of the Ricci Tensor under the Ricci Flow
Thu, 19 Sep 2013 08:43:17 +0000
http://www.hindawi.com/journals/geometry/2013/235436/
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 3dimensional 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
http://www.hindawi.com/journals/geometry/2013/549602/
Let be a spacelike 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 EulerLagrange 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
http://www.hindawi.com/journals/geometry/2013/614195/
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 zerodimensional 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
http://www.hindawi.com/journals/geometry/2013/483727/
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
http://www.hindawi.com/journals/geometry/2013/573925/
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 3dimensional 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.

FeketeSzegö Coefficient Functional for Certain Subclasses of ClosetoStar Functions
Thu, 27 Jun 2013 11:27:14 +0000
http://www.hindawi.com/journals/geometry/2013/642142/
We introduce some subclasses of closetostar functions defined by subordination and obtain sharp upper bounds of the functional , real, for an analytic function , , belonging to these subclasses.
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
http://www.hindawi.com/journals/geometry/2013/842573/
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
http://www.hindawi.com/journals/geometry/2013/718272/
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
http://www.hindawi.com/journals/geometry/2013/671826/
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 HalfLightlike Submanifolds of Indefinite Kenmotsu Manifolds
Wed, 05 Jun 2013 16:08:55 +0000
http://www.hindawi.com/journals/geometry/2013/615819/
We mainly investigate the parallelism of halflightlike submanifolds of indefinite Kenmotsu manifolds. It is proved that a tangential halflightlike 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
http://www.hindawi.com/journals/geometry/2013/549198/
We construct a 4parametric family of combinatorial closed 3manifolds, obtained by glueing together in pairs the boundary faces of polyhedral 3balls. 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 3sphere branched over wellspecified knots, including torus knots and Montesinos knots.
Agnese Ilaria Telloni
Copyright © 2013 Agnese Ilaria Telloni. All rights reserved.

CRSubmanifolds of Generalized Space Forms
Tue, 30 Apr 2013 08:13:05 +0000
http://www.hindawi.com/journals/geometry/2013/654780/
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) CRsubmanifold of a generalized space form and an upper bound for minimal horizontal (and vertical) CRsubmanifold 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
http://www.hindawi.com/journals/geometry/2013/364301/
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
http://www.hindawi.com/journals/geometry/2013/902092/
The notion of a generalized harmonic inverse mean curvature surface in the Euclidean fourspace 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
http://www.hindawi.com/journals/geometry/2013/715907/
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.