Turkish Journal of Mathematics 

About: Turkish Journal of Mathematics is an academic journal published by Scientific and Technological Research Council of Turkey (TUBITAK). The journal publishes majorly in the area(s): Computer science & Nonlinear system. It has an ISSN identifier of 1300-0098. Over the lifetime, 2089 publications have been published receiving 13405 citations. The journal is also known as: Turkish journal of mathematics.

New Special Curves and Developable Surfaces

Abstract: We define new special curves in Euclidean 3-space which we call slant helices and conical geodesic curves Those notions are generalizations of the notion of cylindrical helices One of the results in this paper gives a classification of special developable surfaces under the condition of the existence of such a special curve as a geodesic As a result, we consider geometric invariants of space curves By using these invariants, we can estimate the order of contact with those special curves for general space curves All arguments in this paper are straight forward and classical However, there have been no papers which have investigated slant helices and conical geodesic curves so far as we know

Virtual neighborhoods and pseudo-holomorphic curves

Abstract: Since Gromov introduced his pseudo-holomorphic curve theory in the 80’s, pseudoholomorphic curve has soon become an eminent technique in symplectic topology. Many important theorems in this field have been proved by this technique, among them, the squeezing theorem [Gr], the rigidity theorem [E], the classification of rational and ruled symplectic 4-manifolds [M2], the proof of the existence of non-deformation equivalent symplectic structures [R2]. The pseudo-holomorphic curve also plays a critical role in a number of new subjects such as Floer homology theory,etc. In the meantime of this development, a great deal of efforts has been made to solidify the foundation of pseudo-holomorphic curve theory, for examples, McDuff’s transversality theorem for “cusp curves” [M1] and the various proofs of Gromov compactness theorem. In the early day of Gromov theory, Gromov compactness theorem was enough for its applications to symplectic topology. However, it was insufficient for its potential applications in algebraic geometry, where a good compactification is often very important. For example, it is particularly desirable to tie Gromov-compactness theorem to the DeligneMumford compactification of the moduli space of stable curves. Gromov’s original proof is geometric. Afterwards, many works were done to prove Gromov compactness theorem in the line of Uhlenbeck bubbling off. It was succeed by Parker-Wolfson [PW] and Ye [Ye]. One outcome of their work was a more delicate compactification of the moduli space of pseudo-holomorphic maps. But it didn’t attract much attention until several years later when Kontsevich and Manin [KM] rediscovered this new compactification in algebraic geometry and initiated an algebro-geometric approach to the same theory. Now this new compactification becomes known as the moduli space of stable maps. The moduli space of stable maps is one of the basic ingredients of this paper. During last several years, pseudo-holomorphic curve theory entered a period of rapid expansion. We has witnessed its intensive interactions with algebraic geometry, mathematical physics and recently with new Seiberg-Witten theory of 4-manifolds [T2]. One should mention that those recent activities in pseudo-holomorphic curve theory did not come from the internal drive of symplectic topology. It was influenced mostly by mathematical physics, particularly, Witten’s theory of topological sigma model. Around 1990, there were many activities in string theory about “quantum cohomology” and mirror symmetry. The core of quantum cohomology theory is so called “counting the numbers of

Surgery diagrams for contact 3-manifolds

Abstract: In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact (\pm 1)-surgeries, and used this to prove that any (closed) contact 3-manifold can be obtained from the standard contact structure on S3 by a sequence of such contact (\pm 1)-surgeries. In the present paper, we give a shorter proof of that result and a more explicit algorithm for turning a contact r-surgery into (\pm 1)-surgeries. We use this to give explicit surgery diagrams for all contact structures on S3 and S1 \times S2, as well as all overtwisted contact structures on arbitrary closed, orientable 3-manifolds. This amounts to a new proof of the Lutz-Martinet theorem that each homotopy class of 2-plane fields on such a manifold is represented by a contact structure.

Characterizations of slant helices in Euclidean 3-space

Abstract: In this paper we investigate the relations between a general helix and a slant helix. Moreover, we obtain some differential equations which they are characterizations for a space curve to be a slant helix. Also, we obtain the slant helix equations and its Frenet aparatus.