On the Curvature Operator Spectrum of (Half)Conformally Flat Riemannian Metrics

2015 ◽  
Author(s):  
Evgeny Rodionov ◽  
Victor Slavskii ◽  
Olesia Khromova
Keyword(s):  
Riemannian Metrics ◽  
Curvature Operator ◽  
Conformally Flat ◽  
Operator Spectrum

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].

On the spectrum of the curvature operator of conformally flat Riemannian manifolds

2013 ◽  
Vol 87 (3) ◽  
pp. 279-281
Author(s):  
O. P. Gladunova ◽  
E. D. Rodionov ◽  
V. V. Slavskii
Keyword(s):  
Riemannian Manifolds ◽  
Curvature Operator ◽  
Conformally Flat

scholarly journals Alignment of time series with use of the generalized Legendre's transformation and methods of idempotenty mathematics

2021 ◽  
Vol 16 (3) ◽  
pp. 75-82
Author(s):  
Maria V. Kurkina ◽  
Sergey P. Semenov ◽  
Viktor V. Slavsky ◽  
Olga V. Samarina ◽  
Olga A. Petuhova ◽  
...  
Keyword(s):  
Time Series ◽  
Riemannian Metrics ◽  
Digital Processing ◽  
Conformally Flat ◽  
Idempotent Algebra ◽  
Program Complex ◽  
Tropical Mathematics ◽  
Main Tendency ◽  
And Mathematics ◽  
Under Construction

Alignment of time series [time-series smoothing] identification of the main tendency of development (временнго a trend) by "cleaning" of a time series of the accidental deviations distorting this tendency. At a research of time series of economy (bioinformation science) apply for detection of patterns [1-3]. In this work it is offered to use for this purpose Legendre's transformation well-known in physics and mathematics. Its direct application to poorly regular objects is difficult therefore in work its idempotent analog is defined previously and on its basis the concept of the TRACK for a time series is defined. In recent years within the international center "Cuofus Li" the new field of mathe-matics idempotent or "tropical" mathematics gained intensive development that is reflected in works of the academician V.P. Maslov and his pupils: G.L. Litvinov, A.N. Sobolevsky, etc. The purpose of this work to be beyond duality of the theory of linear vector spaces, using similar concepts of duality of conformally flat Riemannian geometry and of idempotent algebra. By analogy with the polar transformation of a conformally flat Riemannian metrics entered in E.D. Rodionov and V.V. Slavsky's works the abstract idempotent analog of transformation of Legendre is under construction. In the MATLAB system the program complex for calculation the TRACK of a time series is created. It is in-vestigated its opportunities for digital processing of time series.

AN IMPORTANT CLASS OF CONFORMALLY FLAT WEAK EINSTEIN FINSLER METRICS

2013 ◽  
Vol 24 (01) ◽  
pp. 1350003 ◽  
Author(s):  
GUANGZU CHEN ◽  
XINYUE CHENG
Keyword(s):  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Finsler Metrics ◽  
Important Class ◽  
Conformally Flat ◽  
Minkowski Metric ◽  
Minkowski Metrics

In this paper, we study conformally flat (α, β)-metrics in the form F = αϕ(β/α), where α is a Riemannian metric and β is a 1-form on a C∞ manifold M. We prove that if ϕ = ϕ(s) is a polynomial in s, the conformally flat weak Einstein (α, β)-metric must be either a locally Minkowski metric or a Riemannian metric. Moreover, we prove that conformally flat (α, β)-metrics with isotropic S-curvature are also either locally Minkowski metrics or Riemannian metrics.

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