Ricci Solitons conformally equivalent to left invariant metrics

In this paper we study the geometry of Riemannian metrics conformally equivalent to invariant metrics on Lie groups. Then we give a necessary and sufficient condition for these metrics to be Ricci solitons. Using this condition, many explicit examples of shrinking, steady and expanding Ricci solitons are given. Finally, we give an example of Ricci solitons which is not conformally equivalent to a left invariant Riemannian metric

On the left invariant Randers and Matsumoto metrics of Berwald type on 3-dimensional Lie groups

In this paper we identify all simply connected 3-dimensional real Lie groups which admit Randers or Matsumoto metrics of Berwald type with a certain underlying left invariant Riemannian metric. Then we give their flag curvatures formulas explicitly

Randers Ricci soliton homogeneous nilmanifolds

Let $F$ be a left invariant Randers metric on a simply connected nilpotent Lie group $N$, induced by a left invariant Riemannian metric ${\hat{\textbf{\textit{a}}}}$ and a vector field $X$ which is $I_{\hat{\textbf{\textit{a}}}}(M)$-invariant. If the Ricci flow equation has a unique solution then, $(N,F)$ is a Ricci soliton if and only if $(N,F)$ is a semialgebraic Ricci soliton

On the Existence of Homogeneous Geodesics in Homogeneous Kropina Spaces

Recently, it is shown that each regular homogeneous Finsler space $M$ admits at least one homogeneous geodesic through any point $o\in M$. The purpose of this article is to study the existence of homogeneous geodesics on singular homogeneous $(\alpha,\beta)$-spaces, specially, homogeneous Kropina spaces. We show that any homogeneous Kropina space admits at least one homogeneous geodesic through any point. It is shown that, under some conditions, the same result is true for any $(\alpha,\beta)$-homogeneous space. Also, in the case of homogeneous Kropina space of Douglas type, a necessary and sufficient condition for a vector to be a geodesic vector is given. Finally, as an example, homogeneous geodesics of $3$-dimensional non-unimodular real Lie groups equipped with a left invariant Randers metric of Douglas type are investigated

Symplectic Connections Induced by the Chern Connection

Let $(M,\omega)$ be a symplectic manifold and $F$ be a Finsler structure on $M$. In the present paper we define a lift of the symplectic two-form $\omega$ on the manifold $TM\backslash 0$, and find the conditions that the Chern connection of the Finsler structure $F$ preserves this lift of $\omega$. In this situation if $M$ admits a nowhere zero vector field then we have a non-empty family of Fedosov structures on $M$

Left invariant lifted (α,β)(\alpha,\beta)-metrics of Douglas type on tangent Lie groups

In this paper we study lifted left invariant $(\alpha,\beta)$-metrics of Douglas type on tangent Lie groups. Let $G$ be a Lie group equipped with a left invariant $(\alpha,\beta)$-metric of Douglas type $F$, induced by a left invariant Riemannian metric $g$. Using vertical and complete lifts, we construct the vertical and complete lifted $(\alpha,\beta)$-metrics $F^v$ and $F^c$ on the tangent Lie group $TG$ and give necessary and sufficient conditions for them to be of Douglas type. Then, the flag curvature of these metrics are studied. Finally, as some special cases, the flag curvatures of $F^v$ and $F^c$ in the cases of Randers metrics of Douglas type, and Kropina and Matsumoto metrics of Berwald type are given

Riemannian Geometry of Two Families of Tangent Lie Groups

Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article we study the Riemannian geometry of tangent bundle of two families of Lie groups. The first one is the family of special Lie groups considered by J. Milnor and the second one is the class of Lie groups with one-dimensional commutator groups. The Levi-Civita connection, sectional and Ricci curvatures have been investigated

Automatic Detection of Diabetes Diagnosis using Feature Weighted Support Vector Machines based on Mutual Information and Modified Cuckoo Search

Diabetes is a major health problem in both developing and developed countries and its incidence is rising dramatically. In this study, we investigate a novel automatic approach to diagnose Diabetes disease based on Feature Weighted Support Vector Machines (FW-SVMs) and Modified Cuckoo Search (MCS). The proposed model consists of three stages: Firstly, PCA is applied to select an optimal subset of features out of set of all the features. Secondly, Mutual Information is employed to construct the FWSVM by weighting different features based on their degree of importance. Finally, since parameter selection plays a vital role in classification accuracy of SVMs, MCS is applied to select the best parameter values. The proposed MI-MCS-FWSVM method obtains 93.58% accuracy on UCI dataset. The experimental results demonstrate that our method outperforms the previous methods by not only giving more accurate results but also significantly speeding up the classification procedure

Extended Mixture of MLP Experts by Hybrid of Conjugate Gradient Method and Modified Cuckoo Search

This paper investigates a new method for improving the learning algorithm of Mixture of Experts (ME) model using a hybrid of Modified Cuckoo Search (MCS) and Conjugate Gradient (CG) as a second order optimization technique. The CG technique is combined with Back-Propagation (BP) algorithm to yield a much more efficient learning algorithm for ME structure. In addition, the experts and gating networks in enhanced model are replaced by CG based Multi-Layer Perceptrons (MLPs) to provide faster and more accurate learning. The CG is considerably depends on initial weights of connections of Artificial Neural Network (ANN), so, a metaheuristic algorithm, the so-called Modified Cuckoo Search is applied in order to select the optimal weights. The performance of proposed method is compared with Gradient Decent Based ME (GDME) and Conjugate Gradient Based ME (CGME) in classification and regression problems. The experimental results show that hybrid MSC and CG based ME (MCS-CGME) has faster convergence and better performance in utilized benchmark data sets.Comment: 13 pages, 2 figure

Left invariant Ricci solitons on five-dimensional nilmanifolds

In 2002, using a variational method, Lauret classified five-dimensional nilsolitons. In this work, using the algebraic Ricci soliton equation, we obtain the same classification. We show that, among ten classes of five-dimensional nilmanifolds, seven classes admit Ricci soliton structure. In any case, the derivation which satisfies the algebraic Ricci soliton equation is computed