Bülent Ünal

Doubly warped products

Abstract In this paper we study geodesic completeness of Riemannian doubly warped products and Lorentzian doubly warped products. We give necessary conditions for generalized Robertson–Walker space-times with doubly warped product spacial parts to be globally hyperbolic. We also state some results about Killing and conformal vector fields of doubly warped products.

Curvature of multiply warped products

Abstract In this paper, we study Ricci-flat and Einstein–Lorentzian multiply warped products. We also consider the case of having constant scalar curvatures for this class of warped products. Finally, after we introduce a new class of space–times called as generalized Kasner space–times, we apply our results to this kind of space–times as well as other relativistic space–times, i.e., Reissner–Nordstrom, Kasner space–times, Banados–Teitelboim–Zanelli and de Sitter black hole solutions.

Multiply warped products

Abstract In this paper, we first state covariant derivative formulas for multiply warped products and consider the geodesic equations for these spaces. Then we state some basic facts about causality of Lorentzian multiply products and study Cauchy surfaces and global hyperbolicity. Finally, we consider null, time-like and space-like geodesic completeness of Lorentzian multiply products and geodesic completeness of Riemannian multiply warped products.

Geodesic structure of standard static space–times

The geodesic structure of standard static space–times is studied and conditions are found which imply nonreturning and pseudoconvex geodesic systems. As a consequence, it is shown that if the Riemannian factor manifold F satisfies the nonreturning property and has a pseudoconvex geodesic system and if the warping function f:F→(0,∞) is bounded above then the standard static space–time f(a,b)×F is geodesically connected.

On a differential equation characterizing Euclidean spheres

Abstract A characterization of Euclidean spheres out of complete Riemannian manifolds is made by certain vector fields on complete Riemannian manifolds satisfying a partial differential equation on vector fields.

Characterizing Specific Riemannian Manifolds by Differential Equations

Some characterizations of certain rank-one symmetric Riemannian manifolds by the existence of nontrivial solutions to certain partial differential equations on Riemannian manifolds are surveyed.

Divergence theorems in semi-Riemannian geometry

Two forms of divergence theorem for semi-Riemannian manifolds are proven.

About curvature, conformal metrics and warped products

We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds (B, gB) and (F, gF) furnished with metrics of the form c2gB ⊕ w2gF and, in particular, of the type w2μgB ⊕ w2gF, where c, w:B → (0, ∞) are smooth functions and μ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If (B, gB) is Riemannian, the latter question involves nonlinear elliptic partial differential equations with concave–convex nonlinearities and singular partial differential equations of the Lichnerowicz–York-type among others.

The Effect of Split Attention in Surgical Education

Surgical education through simulation is an important area to improve the level of education and to decrease the risks, ethical considerations and cost of the educational environments. In the literature there are several studies conducted to better understand the effect of these simulation environments on learning. However among those studies the human-computer interaction point of view is very limited. Surgeons need to look at radiological images such as magnetic resonance images (MRI) to be sure about the location of the patient’s tumor during a surgical operation. Thus, they go back and forth between physically separated places (e.g. the operating table and light screen display for MRI volume sets). This study is conducted to investigate the effect of presenting different information sources in close proximity on human performance in surgical education. For this purpose, we have developed a surgical education simulation scenario which is controlled by a haptic interface. To better understand the effect of split attention in surgical education, an experimental study is conducted with 27 subjects. The descriptive results of study show that even the integrated group performed the tasks with a higher accuracy level (by traveling less distance, entering less wrong directions and hitting less walls), the results are not statistically significant. Accordingly, even there are some evidences about the effect of split attention on surgical simulation environments, the results of this study need to be validated by controlling students’ skill levels on controlling the haptic devices and 2D/3D space perception skills. The results of this study may guide the system developers to better design the HCI interface of their designs especially for the area of surgical simulation.