Almost Pseudo Ricci Symmetric Manifold Admitting Schouten Tensor

2021 ◽  
Vol 19 (2) ◽  
pp. 217-225
Author(s):  
Mohabbat Ali ◽  
Mohd Vasiulla
Keyword(s):  
Symmetric Manifold ◽  
Schouten Tensor

On Lagrangian Catenoids

2007 ◽  
Vol 50 (3) ◽  
pp. 321-333 ◽  
Author(s):  
David E. Blair
Keyword(s):  
General Problem ◽  
Lagrangian Submanifolds ◽  
Minimal Submanifold ◽  
Conformally Flat ◽  
Totally Geodesic ◽  
Schouten Tensor ◽  
Multiplicity One

AbstractRecently I. Castro and F.Urbano introduced the Lagrangian catenoid. Topologically, it is ℝ × Sn–1 and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ℂn is foliated by round (n – 1)-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ℂn. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

scholarly journals Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

2019 ◽  
Vol 74 (4) ◽  
Author(s):  
Marcos Tulio Carvalho ◽  
Mauricio Pieterzack ◽  
Romildo Pina
Keyword(s):  
Sufficient Conditions ◽  
Conformally Flat ◽  
Symmetric Tensors ◽  
Conformally Flat Manifolds ◽  
Dimensional Group ◽  
Schouten Tensor ◽  
Locally Conformally Flat ◽  
Flat Manifolds ◽  
Necessary And Sufficient ◽  
Complete Metrics

Abstract We consider the pseudo-Euclidean space $$({\mathbb {R}}^n,g)$$(Rn,g), with $$n \ge 3$$n≥3 and $$g_{ij} = \delta _{ij} \varepsilon _{i}$$gij=δijεi, where $$\varepsilon _{i} = \pm 1$$εi=±1, with at least one positive $$\varepsilon _{i}$$εi and non-diagonal symmetric tensors $$T = \sum \nolimits _{i,j}f_{ij}(x) dx_i \otimes dx_{j} $$T=∑i,jfij(x)dxi⊗dxj. Assuming that the solutions are invariant by the action of a translation $$(n-1)$$(n-1)- dimensional group, we find the necessary and sufficient conditions for the existence of a metric $$\bar{g}$$g¯ conformal to g, such that the Schouten tensor $$\bar{g}$$g¯, is equal to T. From the obtained results, we show that for certain functions h, defined in $$\mathbb {R}^{n}$$Rn, there exist complete metrics $$\bar{g}$$g¯, conformal to the Euclidean metric g, whose curvature $$\sigma _{2}(\bar{g}) = h$$σ2(g¯)=h.

scholarly journals Some Properties of Generalized Einstein Tensor for a Pseudo-Ricci Symmetric Manifold

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
S. Aynur Uysal ◽  
Hülya Bağdatlı Yılmaz
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Einstein Tensor ◽  
Symmetric Manifold ◽  
Necessary And Sufficient

The object of the paper is to study some properties of the generalized Einstein tensor GX,Y which is recurrent and birecurrent on pseudo-Ricci symmetric manifolds PRSn. Considering the generalized Einstein tensor GX,Y as birecurrent but not recurrent, we state some theorems on the necessary and sufficient conditions for the birecurrency tensor of GX,Y to be symmetric.

scholarly journals ON PSEUDO CYCLIC RICCI SYMMETRIC MANIFOLDS

2009 ◽  
Vol 02 (02) ◽  
pp. 227-237
Author(s):  
Absos Ali Shaikh ◽  
Shyamal Kumar Hui
Keyword(s):  
Riemannian Manifold ◽  
Einstein Manifold ◽  
Conformally Flat ◽  
Geometric Properties ◽  
Symmetric Manifold ◽  
Euclidean Manifold ◽  
Flat Riemannian Manifold

The object of the present paper is to introduce a type of non-flat Riemannian manifold called pseudo cyclic Ricci symmetric manifold and study its geometric properties. Among others it is shown that a pseudo cyclic Ricci symmetric manifold is a special type of quasi-Einstein manifold. In this paper we also study conformally flat pseudo cyclic Ricci symmetric manifolds and prove that such a manifold can be isometrically immersed in a Euclidean manifold as a hypersurface.

scholarly journals Geometry of the Wiman–Edge monodromy

2021 ◽  
pp. 1-29
Author(s):  
Matthew Stover
Keyword(s):  
Automorphism Group ◽  
Modular Group ◽  
Alternating Group ◽  
Projective Surface ◽  
Congruence Condition ◽  
Symmetric Manifold ◽  
New Information ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Surface Of General Type

The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group [Formula: see text]. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group [Formula: see text]. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group [Formula: see text]. In this note, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.

Sign in / Sign up

Export Citation Format

Share Document