A Bernstein type theorem on a Randers space

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Then, Riccis work was used to formulate Einstein theory of gravitation [3]. Hence, Finsler metric is said to be Einstein if the Ricci scalar is a function of alone. In Riemannian space if and are pointwise projectively related Riemannian metric on manifold of dimensional 3, then is of constant curvature if and only if is of constant curvature.

Moreover, the authors , Z. Shen, N.

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Sadeghzadeh, A. Razavi and B. Razaei were studied the projectively related Einstein Finsler metrics 13 , Corresponding Author:- Narasimhamurthy S. Address:- Department of P. In , the author Y. Shen and L. Zhao were proved that the Randers metrics is projectively flat if and only if is projectively flat and is closed with constant flag curvature. In[22], Z. Shen found out that two pointwise projectively equivalent Einstein Finsler metric and on a dimensional compact manifold have same sign Einstein constants.

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In addition, if two pointwise projectively related Einstein metrics are complete with negative Einstein constants then one of them is a multiple of the other. And the authors Z.

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Shen, Yibing and Yaoyoug were got the results, the two Einstein Randers metrics are projectively related then and are Einstein metrics with non positive scalar curvature and and have non positive Ricci curvature[13]. So, it is natural to study projectively related Einstein Randers metrics, which is just the purpose of this paper. If is projectively related to of non zero Ricci scalar. Then, i is Einstein if and only if it is a constant co-efficients of , when is not projectively flat. If and are projectively related then they are homothetic. Let be an -dimensional manifold.

Each element of has the form , where and. The symmetric tensor defined by,. Every Finsler metric including a spray:. Where the matrix means the inverse of matrix and the coefficients , and -curvature of the. The Riemannian Curvature has the following properties.


  1. Article DOI: 10.21474/IJAR01/2459: Journal Homepage:.
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If is a constant, then is said to be of constant curvature. Therefore, the Ricci scalar function is positive homogeneous of degree 0 in. This means that , depends on the direction of the flag pole but not its length. Ricci flat manifolds are Riemannian manifolds whose Ricci tensor vanishes.

In physics they are important because they represent vacuum solution to Einsteins equations. Definition 2. In , A. Rapcsak [6] proved the following:Lemma 2.

A Finsler metric is pointwise projective to if and only if The study of Weyl curvature of spray as an important projective invariant. The Weyls projective invariant is constructed from the Riemannian curvature. Define 1.

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By this direction the author Z. Shen has proved that [17]. Theorem 2. As a result of Busemann-Mayer theorem ; the authors M. Sepasi and B. Bidabada were proved the following [14], Corollary2.


  1. Article DOI: /IJAR01/ Journal Homepage:.
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Weyl projective curvature of randers metric:In general it is much more difficult to compute the Weyl projective curvature tensor. Lemma 3. Where denote the Weyl curvature tensor of. We assume that is killing form with constant length and since Weyl curvature tensor is a projective invariant. Consider the another spray as. Here, we see that and are projectively equivalent. Now,we compute the Weyl curvature tensor using. Where denote the covariant derivatives of with respect to.


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