scholarly journals AdS geometry from CFT on a general conformally flat manifold

2018 ◽  
Vol 933 ◽  
pp. 262-274 ◽  
Author(s):  
Sinya Aoki ◽  
Shuichi Yokoyama
Keyword(s):  
Conformally Flat ◽  
Flat Manifold ◽  
Conformally Flat Manifold ◽  
Ads Geometry

scholarly journals REMARKS ON BIANCHI SUMS AND PONTRJAGIN CLASSES

2014 ◽  
Vol 97 (3) ◽  
pp. 365-382 ◽  
Author(s):  
MOHAMMED LARBI LABBI
Keyword(s):  
Riemannian Curvature ◽  
Conformally Flat ◽  
Flat Manifold ◽  
Equality Case ◽  
Conformally Flat Manifold

AbstractWe use the exterior and composition products of double forms together with the alternating operator to reformulate Pontrjagin classes and all Pontrjagin numbers in terms of the Riemannian curvature. We show that the alternating operator is obtained by a succession of applications of the first Bianchi sum and we prove some useful identities relating the previous four operations on double forms. As an application, we prove that for a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k$-conformally flat manifold of dimension $n\geq 4k$, the Pontrjagin classes $P_i$ vanish for any $i\geq k$. Finally, we study the equality case in an inequality of Thorpe between the Euler–Poincaré characteristic and the $k{\rm th}$ Pontrjagin number of a $4k$-dimensional Thorpe manifold.

Hypersurfaces in a conformally flat space

2021 ◽  
pp. 2150067
Author(s):  
Sabina Eyasmin
Keyword(s):  
Space Form ◽  
Sufficient Conditions ◽  
Shape Operator ◽  
Flat Space ◽  
Conformally Flat ◽  
Flat Manifold ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Conformally Flat Manifold ◽  
Proper Generalization

The hypersurface of a space is one of the most important objects in a space. Many authors studied the various geometric aspects of hypersurfaces in a space form. The notion of conformal flatness is one of the most primitive concepts in differential geometry. Again, conformally flat space is a proper generalization of a space form. In this paper, we study the geometry of hypersurfaces in a conformally flat manifold. Then we have investigated some sufficient conditions imposed on the shape operator for which the hypersurface satisfies various pseudosymmetric-type conditions imposed on its conformal curvature tensor.

KOBAYASHI CONFORMAL METRIC ON MANIFOLDS, CHERN-SIMONS AND η-INVARIANTS

1991 ◽  
Vol 02 (04) ◽  
pp. 361-382 ◽  
Author(s):  
BORIS N. APANASOV
Keyword(s):  
Conformal Structure ◽  
Conformally Flat ◽  
Flat Manifold ◽  
Conformal Invariant ◽  
Conformal Class ◽  
Conformal Metric ◽  
Conformally Flat Manifold ◽  
Euclidean Metrics ◽  
Chern Simons

The main aim of this paper is to present a canonical Riemannian smooth metric on a given uniformized conformal manifold (conformally flat manifold) which is compatible with the conformal structure. This metric is related to the Kobayashi construction for complex-analytic manifolds and gives a new conformal invariant. As an application, the paper studies the Chern-Simons functional and the η-invariant associated with the conformal class of conformally-Euclidean metrics on a closed hyperbolic 3-manifold.

CONFORMALLY FLAT CONTACT THREE-MANIFOLDS

2016 ◽  
Vol 103 (2) ◽  
pp. 177-189 ◽  
Author(s):  
JONG TAEK CHO ◽  
DONG-HEE YANG
Keyword(s):  
Constant Curvature ◽  
Smooth Function ◽  
Sasakian Manifold ◽  
Conformally Flat ◽  
Flat Manifold ◽  
Smooth Functions ◽  
Flat Contact

In this paper, we consider contact metric three-manifolds $(M;\unicode[STIX]{x1D702},g,\unicode[STIX]{x1D711},\unicode[STIX]{x1D709})$ which satisfy the condition $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}h=\unicode[STIX]{x1D707}h\unicode[STIX]{x1D711}+\unicode[STIX]{x1D708}h$ for some smooth functions $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$, where $2h=\unicode[STIX]{x00A3}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D711}$. We prove that if $M$ is conformally flat and if $\unicode[STIX]{x1D707}$ is constant, then $M$ is either a flat manifold or a Sasakian manifold of constant curvature $+1$. We cannot extend this result for a smooth function $\unicode[STIX]{x1D707}$. Indeed, we give such an example of a conformally flat contact three-manifold which is not of constant curvature.

Conformally flat kähler spaces

2020 ◽  
Author(s):  
O. Lesechko ◽  
O. Latysh ◽  
T. Spychak
Keyword(s):  
Conformally Flat

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 $L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Li ◽  
Shuxiang Feng ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Finiteness Theorem ◽  
Conformally Flat ◽  
Locally Conformally Flat ◽  
Flat Riemannian Manifold ◽  
Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

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