A Note on Contact Manifolds and Applications

Differential Geometry and its Applications, 2012

Generalized contact pairs were introduced in Poon and Wade (2011) [25]. In this paper, we carry out a detailed study of geometric properties of these structures. First, we give geometric conditions expressing the integrability of a generalized contact pair. Then, we use them to obtain insights into the characteristic foliation of a generalized contact manifold. Finally we show that, locally, any smooth manifold endowed with a generalized contact pair is equivalent to the product of an almost cosymplectic manifold whose associated 2-form is closed by a generalized complex manifold.

2011

The present paper deals with difierent geometrical properties of hyperbolic contact manifold of a quasi sasakian manifold. In the end we studies the properties of the Integrability of the distributions on a hyperbolic contact manifold of a quasi sasakian manifold.

2013

Although contact geometry and topology was briefly discussed in the book by V.I. Arnol'd "Mathematical Methods of Classical Mechanics" (Springer-Verlag, 1989, 2nd edition), it still remains the domain of research in pure mathematics, e.g. see recent monograph by H. Geiges "An Introduction to Contact Topology" (Cambridge Univ. Press, 2008). Some attempts to use contact geometry in physics were made in the well-written monograph "Contact Geometry and Nonlinear Differential Equations" (Cambridge Univ. Press, 2007). Unfortunately, even excellent style of this monograph happens to be not sufficient to attract enough attention of physics community to this type of problematics as the GOOGLE search results indicate. To our knowledge, this book is the first serious attempt to change the existing status quo. In it we demonstrate that, in fact, all branches of theoretical physics can be rewritten in the language of contact geometry and topology. From mechanics, thermodynamics and electrodynamics to optics, gauge fields and gravity. From physics of liquid crystals to quantum mechanics and quantum computers, etc. The book is written in the style of the famous Landau-Lifshitz (L-L) multivolume course in theoretical physics. This means that its readers are expected to have solid background in theoretical physics (at least at the level of the L-L course). No prior knowledge of specialized mathematics is required. All needed new mathematical information is given in the context of discussed physical problems. As in the L-L course some problems/exercises are formulated along the way. As in this course, these are always supplemented by either solutions or by hints (with exact references). Unlike the L-L course, though, some theorems (sometimes with complete proofs) and remarks are also presented. This is done with the purpose of facilitating interest of our readers in deeper study of subject matters discussed in the text. v Applications of Contact Geometry and Topology in Physics Downloaded from www.worldscientific.com by 99.111.80.192 on 06/06/13. For personal use only.

ISRN Geometry, 2013

We studyDa-homothetic deformations ofK-contact manifolds. We prove thatDa-homothetically deformedK-contact manifold is a generalized Sasakian space form if it is conharmonically flat. Further, we find expressions for scalar curvature ofDa-homothetically deformedK-contact manifolds.

Cubo, 2010

The object of the present paper is to study a type of contact metric manifolds, called N (k)contact metric manifolds admitting a non-null concircular and torse forming vector field. Among others it is shown that such a manifold is either locally isometric to the Riemannian product E n+1 (0) × S n (4) or a Sasakian manifold. Also it is shown that such a contact metric manifold can be expressed as a warped product I×ep * M , where ( * M , * g ) is a 2n-dimensional manifold. El objetivo del presente artículo es estudiar un tipo de variedades métricas de contacto, llamadas N (k)-variedades métricas de contacto admitiendo un campo de vectores concircular y forma torse. Es demostrado también que tales variedades son o localmente isométricas a productos Riemannianos E n+1 (0) × S n (4) o una variedade Sasakian. Es demostrado que tales variedades métricas de contacto pueden ser expresadas como un producto deformado I×ep * M , donde ( * M , * g ) es una variedad 2n-dimensional. 12, 1 (2010)

SUT Journal of Mathematics

The object of this paper is to study K-contact manifolds with quasiconformal curvature tensor. We characterised K-contact manifolds satisfying certain curvature conditions on the quasi-conformal curvature tensor.

JOURNAL OF ADVANCES IN MATHEMATICS

In this paper, we study the geometry of the contact pseudo-slant submanifolds of a Sasakian manifold. We derive the integrability conditions of distributions in the definition of a contact pseudo-slant submanifold. The notions contact pseudo-slant product is defined, and the necessary and sufficient conditions for a submanifold to contact pseudo-slant product is given. Also, a non-trivial example is used to demonstrate that the method presented in this paper is effective.

2007

In the present study, we considered 3-dimensional generalized (•;")-contact metric manifolds. We proved that a 3-dimensional gener- alized (•;")-contact metric manifold is not locally `-symmetric in the sense of Takahashi. However such a manifold is locally `-symmetric pro- vided thatand " are constants. Also it is shown that if a 3-dimensional generalized (•;") -contact metric manifold is Ricci-symmetric, then it is a (•;")-contact metric manifold. Further we investigated certain condi- tions under which a generalized (•;")-contact metric manifold reduces to a (•;")-contact metric manifold. Then we obtain several necessary and su-cient conditions for the Ricci tensor of a generalized (•;")-contact metric manifold to be ·-parallel. Finally, we studied Ricci-semisymmetric generalized (•;")-contact metric manifolds and it is proved that such a manifold is either ∞at or a Sasakian manifold.

Contact geometry is the odd-dimensional analogue of symplectic geometry with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter it originated in questions of classical and analytical mechanics. Contact geometry has, as does symplectic geometry, broad applications in mathematical physics, geometrical optics, classical mechanics, analytical mechanics, mechanical systems, thermodynamics, geometric quantization and applied mathematics such as control theory. On the other hand, one way of solving problems in classical mechanics is with the help of the Euler-Lagrange and the Hamilton equations. In this study, Euler-Lagrange mechanical equations as representing the motion of the body were found on contact 5-manifolds. Also, closed solutions of the differential equations found in this study are solved by symbolic computation program.

2003

We prove that the dimension of the 1-nullity distributionN(1) on a closed Sasakian manifold M of rank l is at least equal to 2l−1 provided thatM has an isolated closed characteristic. The result is then used to provide some examples ofK-contact manifolds which are not Sasakian. On a closed, 2n+ 1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifoldM is isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k-nullity, Weinstein conjecture, and minimal unit vector fields. 2000 Mathematics Subject Classification: 53D35. 1. Introduction. Contact