ISRN Geometry http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. Postulation of General Unions of Lines and Multiplicity Two Points in Sun, 23 Mar 2014 08:30:34 +0000 http://www.hindawi.com/journals/isrn.geometry/2014/120850/ We prove that a general union of prescribed numbers of lines and double points has maximal rank, except a few well-known exceptional cases. E. Ballico Copyright © 2014 E. Ballico. All rights reserved. Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes Tue, 18 Mar 2014 09:53:08 +0000 http://www.hindawi.com/journals/isrn.geometry/2014/974695/ We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle. Junhong Dong and Ximin Liu Copyright © 2014 Junhong Dong and Ximin Liu. All rights reserved. On the Class of -Metrics Mon, 10 Mar 2014 09:22:48 +0000 http://www.hindawi.com/journals/isrn.geometry/2014/596090/ We study a class of Finsler metrics in the form , where is a Riemannian metric, is a -form, and is called -metrics. We find the necessary and sufficient conditions under which the class of -metrics is locally projectively flat and Douglas metrics, respectively. Ahmad Alimohammadi Copyright © 2014 Ahmad Alimohammadi. All rights reserved. A Survey on Seifert Fiber Space Theorem Wed, 05 Mar 2014 00:00:00 +0000 http://www.hindawi.com/journals/isrn.geometry/2014/694106/ We review the history of the proof of the Seifert fiber space theorem, as well as its motivations in 3-manifold topology and its generalizations. Jean-Philippe Préaux Copyright © 2014 Jean-Philippe Préaux. All rights reserved. Submanifolds of Sasakian Manifolds with Certain Parallel Operators Sun, 02 Feb 2014 13:32:37 +0000 http://www.hindawi.com/journals/isrn.geometry/2014/821869/ We study submanifolds of Sasakian manifolds and obtain a condition under which certain naturally defined symmetric tensor field on the submanifold is to be parallel and use this result to obtain conditions under which a submanifold of the Sasakian manifold is an invariant submanifold. Amira A. Ishan Copyright © 2014 Amira A. Ishan. All rights reserved. Rank 1 Decompositions of Symmetric Tensors Outside a Fixed Support Sun, 29 Dec 2013 08:54:07 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/704072/ Let , , denote the degree Veronese embedding of . For any , let be the minimal cardinality of such that . Identifying with a homogeneous polynomial (or a symmetric tensor), corresponds to writing as a sum of powers with a linear form (or as a sum of -powers of vectors). Here we fix an integral variety and and study a similar decomposition with and minimal. For instance, if is a linear subspace, then we prove that and classify all such that . E. Ballico Copyright © 2013 E. Ballico. All rights reserved. -Homothetic Deformation of -Contact Manifolds Mon, 16 Dec 2013 14:40:20 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/392608/ We study -homothetic deformations of -contact manifolds. We prove that -homothetically deformed -contact manifold is a generalized Sasakian space form if it is conharmonically flat. Further, we find expressions for scalar curvature of -homothetically deformed -contact manifolds. H. G. Nagaraja and C. R. Premalatha Copyright © 2013 H. G. Nagaraja and C. R. Premalatha. All rights reserved. Postulation of a Union , , of a Given Zero-Dimensional Scheme and Several General Lines Thu, 12 Dec 2013 10:52:36 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/391721/ We study the Hilbert function of the union of a given zero-dimensional scheme , , and general lines of , proving, for example, that if , then has maximal rank. E. Ballico Copyright © 2013 E. Ballico. All rights reserved. Slant Curves in the Unit Tangent Bundles of Surfaces Thu, 05 Dec 2013 19:02:21 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/821429/ Let be a surface and let be the unit tangent bundle of endowed with the Sasaki metric. We know that any curve in consist of a curve in and as unit vector field along . In this paper we study the geometric properties and satisfying when is a slant geodesic. Zhong Hua Hou and Lei Sun Copyright © 2013 Zhong Hua Hou and Lei Sun. All rights reserved. On the Existence of a Point Subset with 3 or 6 Interior Points Thu, 05 Dec 2013 15:36:34 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/328095/ For any finite planar point set in general position, an interior point of the set is a point of the set such that it is not on the boundary of the convex hull of the set . For any positive integer , let be the smallest integer such that every finite planar point set with no three collinear points and with at least interior points has a subset for which the interior of the convex hull of the set contains exactly or interior points of the set . In this paper, we prove that . Banyat Sroysang Copyright © 2013 Banyat Sroysang. All rights reserved. A Rabbit Hole between Topology and Geometry Tue, 17 Sep 2013 13:13:37 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/379074/ Topology and geometry should be very closely related mathematical subjects dealing with space. However, they deal with different aspects, the first with properties preserved under deformations, and the second with more linear or rigid aspects, properties invariant under translations, rotations, or projections. The present paper shows a way to go between them in an unexpected way that uses graphs on orientable surfaces, which already have widespread applications. In this way infinitely many geometrical properties are found, starting with the most basic such as the bundle and Pappus theorems. An interesting philosophical consequence is that the most general geometry over noncommutative skewfields such as Hamilton's quaternions corresponds to planar graphs, while graphs on surfaces of higher genus are related to geometry over commutative fields such as the real or complex numbers. David G. Glynn Copyright © 2013 David G. Glynn. All rights reserved. On Twisted Products Finsler Manifolds Wed, 10 Jul 2013 11:09:26 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/732432/ On the product of two Finsler manifolds , we consider the twisted metric which is constructed by using Finsler metrics and on the manifolds and , respectively. We introduce horizontal and vertical distributions on twisted product Finsler manifold and study C-reducible and semi-C-reducible properties of this manifold. Then we obtain the Riemannian curvature and some of non-Riemannian curvatures of the twisted product Finsler manifold such as Berwald curvature, mean Berwald curvature, and we find the relations between these objects and their corresponding objects on and . Finally, we study locally dually flat twisted product Finsler manifold. E. Peyghan, A. Tayebi, and L. Nourmohammadi Far Copyright © 2013 E. Peyghan et al. All rights reserved. Nonexistence of Totally Contact Umbilical Slant Lightlike Submanifolds of Indefinite Cosymplectic Manifolds Mon, 03 Jun 2013 17:33:39 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/231869/ We study totally contact umbilical slant lightlike submanifolds of indefinite cosymplectic manifolds. We prove that there do not exist totally contact umbilical proper slant lightlike submanifolds in indefinite cosymplectic manifolds other than totally contact geodesic proper slant lightlike submanifolds. We also prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic space forms. Finally we give characterization theorems on minimal slant lightlike submanifolds. Rashmi Sachdeva, Rakesh Kumar, and Satvinder Singh Bhatia Copyright © 2013 Rashmi Sachdeva et al. All rights reserved. An Upper Bound for the Tensor Rank Thu, 23 May 2013 15:53:50 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/241835/ Let be a tensor of format , , over . We prove that has tensor rank at most . E. Ballico Copyright © 2013 E. Ballico. All rights reserved. On the M-Projective Curvature Tensor of -Contact Metric Manifolds Tue, 26 Mar 2013 18:39:59 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/932564/ The object of the present paper is to study some curvature conditions on -contact metric manifolds. R. N. Singh and Shravan K. Pandey Copyright © 2013 R. N. Singh and Shravan K. Pandey. All rights reserved. A Study on Ricci Solitons in Kenmotsu Manifolds Thu, 07 Mar 2013 09:50:34 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/412593/ We study and obtain results on Ricci solitons in Kenmotsu manifolds satisfying , , , and , where and are C-Bochner and pseudo-projective curvature tensor. C. S. Bagewadi, Gurupadavva Ingalahalli, and S. R. Ashoka Copyright © 2013 C. S. Bagewadi et al. All rights reserved. gs Continuous Function in Topological Space Tue, 26 Feb 2013 08:03:12 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/787014/ We introduce the different notions of a new class of continuous functions called generalized semi Lambda (gs) continuous function in topological spaces. Its properties and characterization are also discussed. S. Pious Missier and Vijilius Helena Raj Copyright © 2013 S. Pious Missier and Vijilius Helena Raj. All rights reserved. On the Classification of Almost Kenmotsu Manifolds of Dimension 3 Wed, 30 Jan 2013 16:49:06 +0000 http://www.hindawi.com/journals/isrn.geometry/2013/927159/ This paper deals with the classification of a 3-dimensional almost Kenmotsu manifold satisfying certain geometric conditions. Moreover, by applying our main classification theorem, we obtain some suffcient conditions for an almost Kenmotsu manifold of dimension 3 to be an Einstein-Weyl manifold. Yaning Wang and Ximin Liu Copyright © 2013 Yaning Wang and Ximin Liu. All rights reserved. On Generalized Sasakian-Space-Forms Thu, 27 Dec 2012 16:01:37 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/309486/ The purpose of the present paper is to characterize pseudoprojectively flat and pseudoprojective semisymmetric generalized Sasakian-space-forms. H. G. Nagaraja, G. Somashekhara, and Savithri Shashidhar Copyright © 2012 H. G. Nagaraja et al. All rights reserved. On the Conharmonic Curvature Tensor of Generalized Sasakian-Space-Forms Tue, 25 Dec 2012 15:29:11 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/876276/ The object of the present paper is to characterize generalized Sasakian-space-forms satisfying certain curvature conditions on conharmonic curvature tensor. In this paper we study conharmonically semisymmetric, conharmonically flat, -conharmonically flat, and conharmonically recurrent generalized Sasakian-space-forms. Also generalized Sasakian-space-forms satisfying and have been studied. U. C. De, R. N. Singh, and Shravan K. Pandey Copyright © 2012 U. C. De et al. All rights reserved. Singular Points of Reducible Sextic Curves Tue, 11 Dec 2012 09:46:06 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/680247/ There are 106 individual types of singular points for reducible complex sextic curves. David A. Weinberg and Nicholas J. Willis Copyright © 2012 David A. Weinberg and Nicholas J. Willis. All rights reserved. Quarter-Symmetric Nonmetric Connection on -Sasakian Manifolds Mon, 03 Dec 2012 12:57:40 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/659430/ The object of the present paper is to study a quarter-symmetric nonmetric connection on a -Sasakian manifold. In this paper we consider the concircular curvature tensor and conformal curvature tensor on a -Sasakian manifold with respect to the quarter-symmetric nonmetric connection. Next we consider second-order parallel tensor with respect to the quarter-symmetric non-metric connection. Finally we consider submanifolds of an almost paracontact manifold with respect to a quarter-symmetric non-metric connection. Abul Kalam Mondal and U. C. De Copyright © 2012 Abul Kalam Mondal and U. C. De. All rights reserved. On Submersion of CR-Submanifolds of l.c.q.K. Manifold Wed, 28 Nov 2012 08:10:51 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/309145/ We study submersion of CR-submanifolds of an l.c.q.K. manifold. We have shown that if an almost Hermitian manifold admits a Riemannian submersion of a CR-submanifold of a locally conformal quaternion Kaehler manifold , then is a locally conformal quaternion Kaehler manifold. Majid Ali Choudhary, Mahmood Jaafari Matehkolaee, and Mohd. Jamali Copyright © 2012 Majid Ali Choudhary et al. All rights reserved. Geodesic Lightlike Submanifolds of Indefinite Kenmotsu Manifolds Wed, 14 Nov 2012 10:00:27 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/804520/ The aim of the present paper is to study geodesic contact screen Cauchy Riemannian (SCR-) lightlike submanifolds, geodesic screen transversal lightlike, and geodesic transversal lightlike submanifolds of indefinite Kenmotsu manifolds. S. M. Khursheed Haider, Mamta Thakur, and Advin Copyright © 2012 S. M. Khursheed Haider et al. All rights reserved. Curvature Properties and -Einstein -Contact Metric Manifolds Mon, 05 Nov 2012 16:12:53 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/970682/ We study curvature properties in -contact metric manifolds. We give the characterization of -Einstein -contact metric manifolds with associated scalars. H. G. Nagaraja and C. R. Premalatha Copyright © 2012 H. G. Nagaraja and C. R. Premalatha. All rights reserved. Hemi-Slant Lightlike Submanifolds of Indefinite Kenmotsu Manifolds Thu, 13 Sep 2012 15:11:47 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/251213/ We introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold. We give an example of hemi-slant lightlike submanifold and establish two characterization theorems for the existence of such submanifolds. We prove some theorems which ensure the existence of minimal hemi-slant lightlike submanifolds and obtain a condition under which the induced connection ∇ on M is a metric connection. An example of proper minimal hemi-slant lightlike submanifolds is also given. S. M. Khursheed Haider, Mamta Thakur, and Advin Copyright © 2012 S. M. Khursheed Haider et al. All rights reserved. On Some πΏπ‘˜-Finite-Type Euclidean Hypersurfaces Thu, 06 Sep 2012 13:26:43 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/591296/ We study some πΏπ‘˜-finite-type Euclidean hypersurfaces. We classify πΏπ‘˜-1-type Euclidean hypersurfaces and πΏπ‘˜-null-2-type Euclidean hypersurfaces with at most two distinct principal curvatures. We also prove that, under some mild restrictions, there exists no πΏπ‘˜-null-3-type Euclidean hypersurface. Akram Mohammadpouri and S. M. B. Kashani Copyright © 2012 Akram Mohammadpouri and S. M. B. Kashani. All rights reserved. A Note on the Growth of Periodic Points for Commuting Toral Automorphisms Thu, 16 Aug 2012 14:03:01 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/165808/ In this note we study the growth of the number of periodic points for non-degenerate actions of commuting hyperbolic toral automorophisms. Mark Pollicott Copyright © 2012 Mark Pollicott. All rights reserved. Ricci Solitons in 𝛼-Sasakian Manifolds Wed, 08 Aug 2012 09:22:46 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/421384/ We study Ricci solitons in 𝛼-Sasakian manifolds. It is shown that a symmetric parallel second order-covariant tensor in a 𝛼-Sasakian manifold is a constant multiple of the metric tensor. Using this, it is shown that if ℒ𝑉𝑔+2𝑆 is parallel where 𝑉 is a given vector field, then (𝑔,𝑉,πœ†) is Ricci soliton. Further, by virtue of this result, Ricci solitons for 𝑛-dimensional 𝛼-Sasakian manifolds are obtained. Next, Ricci solitons for 3-dimensional 𝛼-Sasakian manifolds are discussed with an example. Gurupadavva Ingalahalli and C. S. Bagewadi Copyright © 2012 Gurupadavva Ingalahalli and C. S. Bagewadi. All rights reserved. On Fundamental Domains for Subgroups of Isometries Acting in ℍ𝑛 Wed, 08 Aug 2012 09:21:36 +0000 http://www.hindawi.com/journals/isrn.geometry/2012/685103/ Given 𝒫 a fundamental polyhedron for the action of 𝐺, a classical kleinian group, acting in 𝑛-dimensional hyperbolic space, and Ξ“, a finite index subgroup of 𝐺, one obtains a fundamental domain for Ξ“ pasting copies of 𝒫 by a Schreier process. It also generalizes the side pairing generating theorem for exact or inexact polyhedra. It is proved as well that the general Möbius group acting in ℝ𝑛 is transitive on “π‘˜-spheres”. Hence, describing the hyperbolic π‘˜-planes in the upper half space model intrinsically, and providing also an alternative proof of the transitive action on them. Some examples are given in detail, derived from the classical modular group and the Picard group. Antonio Lascurain Orive and Rubén Molina Hernández Copyright © 2012 Antonio Lascurain Orive and Rubén Molina Hernández. All rights reserved.