CLASSES OF GRADIENT RICCI SOLITONS

2011 ◽  
Vol 08 (04) ◽  
pp. 783-796 ◽  
Author(s):  
GABRIEL BERCU ◽  
MIHAI POSTOLACHE
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Wide Class ◽  
Positive Function ◽  
Polar Coordinates ◽  
Ricci Solitons ◽  
Positive Sectional Curvature ◽  
Gradient Ricci Solitons ◽  
Research Article

We introduce a study of Riemannian manifold M = ℝ2 endowed with a metric of diagonal type of the form [Formula: see text], where g is a positive function, of C∞-class, depending on the variable x2 only. We emphasize the role of metric [Formula: see text] in determining manifolds having negative, null or positive sectional curvature. Within this framework, we find a wide class of gradient Ricci solitons (see, Theorems 4 and 7) and specialize these results to discuss some 2D and 4D case studies. The present study can be thought as a natural continuation of those included in monograph [22] by Constantin Udrişte, and to those in the research article [12] by Richard S. Hamilton (the result in Proposition 8 is precisely the famous "Hamilton cigar" in polar coordinates).

CLASSIFICATION OF STEADY GRADIENT RICCI SOLITONS ON TWO-MANIFOLDS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250049 ◽  
Author(s):  
GABRIEL BERCU ◽  
MIHAI POSTOLACHE
Keyword(s):  
Riemannian Manifold ◽  
Case Studies ◽  
Wide Class ◽  
Positive Function ◽  
Ricci Solitons ◽  
Gradient Ricci Solitons

In our very recent published work [Int. J. Geom. Meth. Mod. Phys.8(4) (2011) 783–796], we considered the Riemannian manifold M = ℝ2 endowed with the warped metric ḡ(x, y) = diag (g(y), 1), where g is a positive function, of C∞-class, depending on the variable y only. Within this framework, we found a wide class of 2D gradient Ricci solitons and specialized our results to discuss some case studies. This research is a natural continuation, providing classification results for the subclass of steady gradient Ricci solitons.

scholarly journals Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

2018 ◽  
Vol 2020 (5) ◽  
pp. 1346-1365 ◽  
Author(s):  
Jason DeVito ◽  
Ezra Nance
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Lie Group ◽  
Positive Curvature ◽  
Positive Sectional Curvature ◽  
Compact Symmetric Space ◽  
Rank 2 ◽  
Set Of Points ◽  
Positively Curved

Abstract A Riemannian manifold is said to be almost positively curved if the set of points for which all two-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented two-planes in $\mathbb{R}^{7}$ admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group $\mathbf{G}_{2}$ and the octonions, so do not obviously generalize to any other Grassmannians.

scholarly journals η-Ricci Solitons and Gradient Ricci Solitons on f-Kenmotsu Manifolds

2021 ◽  
pp. 19-34
Author(s):  
Mohd Siddiqi
Keyword(s):  
Ricci Solitons ◽  
Gradient Ricci Solitons ◽  
Research Article ◽  
Kenmotsu Manifolds ◽  
Metric Connection

The aim of the present research article is to study the f-kenmotsu manifolds admitting the η-Ricci Solitons and gradient Ricci solitons with respect to the semi-symmetric non metric connection.

scholarly journals On the growth of fundamental groups of nonpositive curvature manifolds

1996 ◽  
Vol 54 (3) ◽  
pp. 483-487 ◽  
Author(s):  
Yi-Hu Yang
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Sectional Curvature ◽  
Exponential Growth ◽  
Compact Riemannian Manifold ◽  
Nonpositive Curvature ◽  
Fundamental Groups ◽  
Positive Sectional Curvature ◽  
Classic Result ◽  
Negative Sectional Curvature

Milnor's classic result that the fundamental group of a compact Riemannian manifold of negative sectional curvature has exponential growth is generalised to the case of negative Ricci curvature and non-positive sectional curvature.

scholarly journals ON LOCALLY CONFORMALLY FLAT GRADIENT SHRINKING RICCI SOLITONS

2011 ◽  
Vol 13 (02) ◽  
pp. 269-282 ◽  
Author(s):  
XIAODONG CAO ◽  
BIAO WANG ◽  
ZHOU ZHANG
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Exponential Growth ◽  
Integral Identity ◽  
Ricci Solitons ◽  
Riemannian Curvature ◽  
Conformally Flat ◽  
Gradient Ricci Solitons ◽  
Riemannian Curvature Tensor ◽  
Locally Conformally Flat

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.

scholarly journals On the δ-Pinching Function of the Sectional Curvature of a Compact Connected Lie Group G with a Bi-Invariant Riemannian Metric and a Vectorial Torsion Connection

2020 ◽  
pp. 117-120
Author(s):  
E.D. Rodionov ◽  
O.P. Khromova
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Topological Structure ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Positive Sectional Curvature ◽  
Levi Civita Connection ◽  
Connected Lie Group

One of the important problems of Riemannian geometry is the problem of establishing connections between curvature and the topology of a Riemannian manifold, and, in particular, the influence of the sign of sectional curvature on the topological structure of a Riemannian manifold. Of particular importance in these studies is the question of the influence of d-pinching of Riemannian metrics of positive sectional curvature on the geometric and topological structure of the Riemannian manifold. This question is most studied for the homogeneous Riemannian case. In this direction, the classification of homogeneous Riemannian manifolds of positive sectional curvature, obtained by M. Berger, N. Wallach, L. Bergeri, as well as a number of results on d- pinching of homogeneous Riemannian metrics of positive sectional curvature, is well known. In this paper, we investigate Riemannian manifolds with metric connection being a connection with vectorial torsion. The Levi-Civita connection falls into this class of connections. Although the curvature tensor of these connections does not possess the symmetries of the Levi-Civita connection curvature tensor, it seems possible to determine sectional curvature. This paper studies the d-pinch function of the sectional curvature of a compact connected Lie group G with a biinvariant Riemannian metric and a connection with vectorial torsion. It is proved that it takes the values d(||V ||)∈(0,1].

scholarly journals $\eta$-RICCI SOLITONS AND GRADIENT RICCI SOLITONS ON $\delta$- LORENTZIAN TRANS-SASAKIAN MANIFOLDS

2021 ◽  
pp. 529
Author(s):  
Mohd Danish Siddiqi ◽  
Mehmet Akif Akyol
Keyword(s):  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Sasakian Manifolds ◽  
Constant Multiple ◽  
Metric Tensor ◽  
3 Dimensional ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Research Article

The objective of the present research article is to study the $\delta$-Lorentzian trans-Sasakian manifolds conceding the $\eta$-Ricci solitons and gradient Ricci soliton. We shown that a symmetric second order covariant tensor in a $\delta$-Lorentzian trans-Sasakian manifold is a constant multiple of metric tensor. Also, we furnish an example of $\eta$-Ricci soliton on 3-diemsional $\delta$-Lorentzian trans-Sasakian manifold is provide in the region where $\delta$-Lorentzian trans-Sasakian manifold is expanding. Furthermore, we discuss some results based on gradient Ricci solitons on $3$-dimensional $\delta$- Lorentzian trans-Sasakian manifold.

Role of Self Help Groups A Study with Specific Reference to Vellore District

GIS Business ◽  
2019 ◽  
Vol 14 (6) ◽  
pp. 243-252
Author(s):  
Dr. M.A. Bilal Ahmed ◽  
Dr. S. Thameemul Ansari
Keyword(s):  
Leadership Role ◽  
Three Dimensions ◽  
Specific Reference ◽  
Government Organizations ◽  
Research Article ◽  
Self Help ◽  
The People ◽  
The World ◽  
Self Help Groups

SHG is a movement which came to being in the early 1969. Prof. Muhammed Younus, a great economist of Bangladesh took initiative in setting up Self Help Groups and these SHGs were gradually spread all over the world. This social movement unites the people hailing from poor background. Those who are joining this group feel socially and economically responsible to one another. In India, there are some likeminded bodies and stakeholders of some government organizations play pivotal role towards the formation of SHG In this research article, role of SHGs in Vellore district is studies under the three dimensions of Cognitive role, leadership role and role towards entrepreneurship.

براہوئی ادب ‘ ادبی ادارہ او نوشتوک آتا کڑد

Al-Burz ◽  
2018 ◽  
Vol 10 (1) ◽  
pp. 132-142
Author(s):  
Nilofer Usman ◽  
Dr.Liaquat Ali Sani ◽  
Yousaf Rodeni
Keyword(s):  
Literary History ◽  
Research Method ◽  
Research Paper ◽  
Vital Role ◽  
Modern Poetry ◽  
Descriptive Research ◽  
Research Article ◽  
Language Literature ◽  
Literary Circles

This research article describes the role of Brahui literary circles, which have played a vital role for the preservation and promotion of Brahui Language, Literature and build a literary tendency. This paper also shows how the internal disagreement between learned established new literary circles. Few prominent personalities like  Noor Muhammad Parwana, Nawab Ghaus Bakhsh Raisani, Babo Abudl Rehman Kurd, Abdul Rehman Brahui, Syed Kamal al-Qadri and others have initiated this work in Brahui literary history. Now more the two dozen registered and non-registered Brahui literary originations working for betterment of Brahui literature. Every origination has set their separate Moto and vision, few of them promote Brahui Modern poetry few have introduced new literary tendencies, few have urged that criticism is better for new thoughts and new trend in Brahui literature. This research paper helps to understand the different periods in Brahui literature in context of Brahui originations. A descriptive research method will have been adopted to conclude this paper.

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