### 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.

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.

## Minimal Surfaces and Gauss Curvature of Conoid in Fins-ler Spaces with (α, β)-Metrics

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.

- 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.

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.

- Article DOI: /IJAR01/ Journal Homepage:.
- Pdf A Bernstein Type Theorem On A Randers Space;
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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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Using equation 3. A Spray is isotropic , the equation 3. Then the equqtion 3. Assume that 3. We obtain the following: Lemma 3. Proposition 4. Assume that is Einstein. By definition of Weyl curvature tensor 2. Conversly, if 4. Full Text Available It was known for quite long time that a quaternion space can be generalized to a Clifford space , and vice versa; but how to find its neat link with more convenient metric form in the General Relativity theory, has not been explored extensively. We begin with a representation of group with non-zero quaternions to derive closed FLRW metric [1], and from there obtains Carmeli metric , which can be extended further to become 5D and 6D metric which we propose to call Kaluza-Klein-Carmeli metric.

We also note possible implications to quantum gravity. Further observations are of course recommended in order to refute or verify this proposition. Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces. Full Text Available The aim of this paper is to prove a common fixed point theorem for a pair of weakly compatible mappings in fuzzy metric space by using the CLRg property. An example is also furnished which demonstrates the validity of our main result. As an application to our main result, we present a fixed point theorem for two finite families of self mappings in fuzzy metric space by using the notion of pairwise commuting.

Our results improve the results of Sedghi, Shobe and Aliouche [A common fixed point theorem for weakly compatible mappings in fuzzy metric spaces , Gen. A common fixed point theorem for weakly compatible mappings in Menger probabilistic quasi metric space. Full Text Available In this paper, we prove a common fixed point theorem for finite number of self mappings in Menger probabilistic quasi metric space. Our result improves and extends the results of Rezaiyan et al. Forum 5 6 ] and Sastry et al. Forum 5 52 Given a graph embedded in a metric space , its dilation is the maximum over all distinct pairs of vertices of the ratio between their distance in the graph and the metric distance between them.

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Given such a graph G with n vertices and m edges and consisting of at most two connected components, we Common fixed points for generalized contractive mappings in cone metric spaces. Full Text Available The purpose of this paper is to establish coincidence point and common fixed point results for four maps satisfying generalized weak contractions in cone metric spaces. Also, an example is given to illustrate our results. Some common random fixed point theorems for contractive type conditions in cone random metric spaces. Full Text Available In this paper, we establish some common random fixed point theorems for contractive type conditions in the setting of cone random metric spaces.

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## Global Journal of Advanced Research on Clasical and Modern Geometry

Our results unify, extend and generalize many known results from the current existing literature. On reflexivity of random walks in a random environment on a metric space. In this paper, we consider random walks in random environments on a countable metric space when jumps of the walks of the fractions are finite. For the random walk, a sufficient condition of nonreflexivity is obtained. Examples for metric spaces Z d free groups and free product of finite numbers cyclic groups of the second order and some other metric spaces are considered.

Space -time decomposition of the Hilbert-Palatini action, written in a form which admits degenerate metrics , is considered. The canonical structure of the degenerate sector turns out to be awkward.