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.

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.

A CLASSIFICATION AND A NON-EXISTENCE THEOREM FOR CONFORMALLY FLAT HYPERSURFACES IN EUCLIDEAN 4-SPACE

2005 ◽  
Vol 16 (01) ◽  
pp. 53-85 ◽  
Author(s):  
YOSHIHIKO SUYAMA
Keyword(s):  
Existence Theorem ◽  
Fundamental Form ◽  
Riemannian Metrics ◽  
Equivalence Classes ◽  
Classification Theorem ◽  
Conformally Flat ◽  
Conformal Class ◽  
First Fundamental Form ◽  
Conformal Equivalence ◽  
Conformally Flat Hypersurface

We study generic conformally flat hypersurfaces in the Euclidean 4-space satisfying a certain condition on the conformal class of the first fundamental form. We first classify such hypersurfaces by determining all conformal-equivalence classes of generic conformally flat hypersurfaces satisfying the condition. Next, as an application of the classification theorem, we give some examples of flat Riemannian metrics which are not conformal to the first fundamental form of any generic conformally flat hypersurface. These flat Riemannian metrics seem to provide counter-examples to Hertrich–Jeromin's claim [3, 5].

GAPLESS EDGE EXCITATIONS IN THE FRACTIONAL QUANTUM HALL EFFECT AND CONFORMAL INVARIANCE

1992 ◽  
Vol 06 (19) ◽  
pp. 3189-3204
Author(s):  
A. A. BELOV ◽  
K. D. CHALTIKIAN ◽  
YU. E. LOZOVIK
Keyword(s):  
Quantum Hall Effect ◽  
Quantum Hall ◽  
Fractional Quantum Hall Effect ◽  
Conformal Invariant ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
Invariant Model ◽  
Theoretical Construction ◽  
Chern Simons

We propose the conformal-invariant model for the description of the edge excitations (EE) in the FQH states using field-theoretical construction with current-current interactions. We discuss the classification of the topological orders (TO) of a given model. Using Chern-Simons ~ Wess-Zumino equivalence, we determine the possible types of the excitations in the FQHE — namely, we calculate their charges and statistics.

scholarly journals Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure

2021 ◽  
Author(s):  
S. Berjawi ◽  
E. V. Ferapontov ◽  
B. S. Kruglikov ◽  
V. S. Novikov
Keyword(s):  
Three Dimensional ◽  
Lax Pair ◽  
Conformal Structure ◽  
Second Order ◽  
Contact Transformation ◽  
Conformal Class ◽  
Weyl Geometry ◽  
Symmetric Connection ◽  
Integrable Pdes ◽  
Monge Ampere

AbstractEinstein–Weyl geometry is a triple $$({\mathbb {D}},g,\omega )$$ ( D , g , ω ) where $${\mathbb {D}}$$ D is a symmetric connection, [g] is a conformal structure and $$\omega $$ ω is a covector such that $$\bullet $$ ∙ connection $${\mathbb {D}}$$ D preserves the conformal class [g], that is, $${\mathbb {D}}g=\omega g$$ D g = ω g ; $$\bullet $$ ∙ trace-free part of the symmetrised Ricci tensor of $${\mathbb {D}}$$ D vanishes. Three-dimensional Einstein–Weyl structures naturally arise on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, covector $$\omega $$ ω is a somewhat more mysterious object, recovered from the Einstein–Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge–Ampère type), the covector $$\omega $$ ω is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of g and $$\omega $$ ω provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein–Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein–Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.

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.

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