IJRRAS 26 (3) ● March 2016 www.arpapress.com/Volumes/Vol26Issue3/IJRRAS_26_3_04.pdf 136 EULER-LAGRANGIAN EQUATIONS WITH KÄHLER-EINSTEIN METRIC AND EQUAL KÄHLER ANGLES ON FANO MANIFOLDS Zeki Kasap Faculty of Education, Department of Elementary Education, Pamukkale University, Denizli / Turkey. Email: zekikasap@hotmail.com ABSTRACT It is well-known that a classical field theory deals with the general idea of a quantity and it is a function of time and space. Classical field theory interested in classical mechanics. Classical mechanics explains the motion style of object with Euler-Lagrange equations. It is well-known that a classical field theory explain the study of how one or more physical .elds interact with matter which is used quantum and classical mechanics of physics branches. This manuscript set forth an attempt to introduce Lagrangian formalism for mechanical systems using with Kähler -Einstein metrics on Fano manifolds which represent an interesting multidisciplinary field of research. In this study, we will obtain the geodesic equations of moving objects on Fano manifolds. As a result of this study, partial differential equations will be obtained for movement of objects in space and solutions of these equations will be made using the Maple computation program. In addition to, the geometrical-physical results related to on Kähler-Einstein mechanical systems are also given. Keywords: Kähler-Einstein, Lagrangian, Mechanical System, Fano Varieties. MSC (2000) : 32Q15, 32Q20, 37N20, 51P05, 53C15, 53C25, 53C07, 58J60, 70H03, 83C05. 1. INTRODUCTION A Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important problem for this area is the existence of Kähler–Einstein metrics for compact Kähler manifolds. In the case in which there is a Kähler metric, the Ricci curvature is proportional to the Kähler metric. Therefore, the first Chern class is either negative, or zero, or positive. Kähler–Einstein metric (or Einstein metric) is a Kähler metric on a complex manifold ܯ ௡ whose Ricci curvature tensor is proportional to the metric tensor. This proportionality is an analog of the Einstein field equation in the general theory of relativity. A Kähler–Einstein manifold (or Einstein manifold) is a complex manifold equipped with a Kähler–Einstein metric. In this case the Ricci curvature tensor, considered as an operator on the tangent space, is just multiplication by a constant. De Leon presented many studies about Lagrangian dynamics, mechanics, formalisms, systems and equations [1]. Tian submitted an expository paper on Kähler metrics of positive scalar curvature [2]. Vries shown that the Lagrangian motion equations have a very simple interpretation in relativistic quantum mechanics [3]. Paracomplex analogue of the Euler-Lagrange equations was obtained in the framework of para-Kählerian manifold and the geometric results on a paracomplex mechanical systems were found by Tekkoyun [4]. Electronic origins, molecular dynamics simulations, computational nanomechanics, multiscale modelling of materials fields were contributed by Liu [5]. Chen et al. provided that any compact complex surface with  ଵ >Ͳ admits an Einstein metric which is conformally related to a Kähler metric [6]. Spotti investigated how Fano manifolds equipped with a Kähler-Einstein metric can degenerate as metric spaces and some of the relations of this question with algebraic geometry [7]. Le Brun showed that  ଶ #ʹ ଶ admits an Einstein metric [8]. Heier carry out Nadel’s method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler–Einstein metric [9]. Li solved a folklore conjecture, it is often referred as the Yau-Tian-Donaldson conjecture, on Fano manifolds without nontrivial holomorphic vector fields [10]. Coevering gave many examples of Kähler-Einstein strictly pseudo convex manifolds on bundles and resolutions [11]. Ro̆ ek had studied the relationship between the curvature of a Kähler-Einstein manifold with Kähler potential ܭ and the curvature of the base manifold [12]. Nadel introduced a coherent sheaf of ideals and showed that it satisfies various global algebrogeometric conditions, including a cohomology vanishing theorem [13]. Chen and Tian proved that if ܯis a Kähler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature [14]. Koiso and Sakane had considered  ଵ ሺℂሻ-bundles over compact Kähler-Einstein manifolds to obtain non-homogeneous Kähler-Einstein manifolds with positive Ricci tensor [15]. Alekseevskya et al. examined that a para- Kähler manifold can be defined as a pseudo-Riemannian manifold ሺ ,ܯሻ with a parallel skew-symmetric paracomplex structures ܭ[16]. Salavessa and Valli considered → ܯ :ܨ a minimal submanifold ܯof real dimension ʹ, immersed into a Kähler–Einstein manifold  of complex dimension ʹ, and scalar curvature  [17].