Metallic structures on Riemannian manifolds (original) (raw)
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In this thesis, we study the basics of metallic structure, which is polynomial structure with structure polynomial on manifolds using metallic ratio, which is the generalization of Golden proportion or golden ratio. We study some mathematical terminologies, study some theorems and proof where possible of the metallic structures. We also study some examples of metallic structure. Also, we study some properties of the metallic Riemannian metric. Hence, study the-sectional and-Bisectional curvature of metallic Riemannian manifold.
Applications of the Golden Ratio on Riemannian Manifolds
2009
The Golden Ratio is a fascinating topic that continually generates new ideas. The main purpose of the present paper is to point out and find some applications of the Golden Ratio and of Fibonacci numbers in Differential Geometry. We study a structure defined on a class of Riemannian manifolds, called by us a Golden Structure. A Riemannian manifold endowed with a Golden Structure will be called a Golden Riemannian manifold. Precisely, we say that an (1,1)-tensor field P on a m-dimensional Riemannian manifold ( M, g ) is a Golden Structure if it satisfies the equation P 2 = P + I (which is similar to that satisfied by the Golden Ratio φ )w hereI stands for the (1,1) identity tensor field. First, we establish several properties of the Golden Structure. Then we show that a Golden Structure induces on every invariant submanifold a Golden Structure, too. This fact is illustrated on a product of spheres in an Euclidean space.
On metallic Riemannian structures
TURKISH JOURNAL OF MATHEMATICS, 2015
The paper is devoted to the study of metallic Riemannian structures. An integrability condition and curvature properties for these structures by means of a Φ-operator applied to pure tensor fields are presented. Examples of these structures are also given.
A New Class of Golden Riemannian Manifold
International electronic journal of geometry, 2020
In this paper, we introduce a new class of almost Golden Riemannian structures and study their essential examples as well as their fundamental properties. Next, we investigate a particular type belonging to this class and we establish some basic results for Riemannian curvature tensor and the sectional curvature. Concrete examples are given.
On Submanifolds in a Riemannian Manifold with Golden Structure
2019
A golden Riemannian structure (J,g)(J,g)(J,g) on a Riemannian manifold is given by a tensor field JJJ of type (1,1)(1,1)(1,1) satisfying the golden section relation J2=J+I,J^{2}=J+I,J2=J+I, and a pure Riemannian metric ggg, that is a metric satisfying g(JX,Y)=g(X,JY).g(JX,Y)=g(X,JY).g(JX,Y)=g(X,JY). We investigate some fundamental properties of the induced structure on submanifolds immersed in golden Riemannian manifolds. We obtain effective relations for some induced structures on submanifolds of codimension 2. We also construct an example on submanifold of a golden Riemannian manifold.
The (α, p)-Golden Metric Manifolds and Their Submanifolds
The notion of Golden structure was introduced 15 years ago by the present authors and know a constant interest by several geometers. Now, we propose a new generalization apart of that called metallic structure and considered also by the authors. By adding a compatible Riemannian metric we focus on the study of submanifolds in this setting.
Induced structures on Golden Riemannian manifolds
Beiträge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry, 2018
We introduce the notion of Golden Riemannian manifolds of type (r, s) and starting from a Golden Riemannian structure, we construct some remarkable classes of the induced structures on Riemannian manifold. Concret examples are given.
Types of Submanifolds in Metallic Riemannian Manifolds: A Short Survey
Mathematics
We provide a brief survey on the properties of submanifolds in metallic Riemannian manifolds. We focus on slant, semi-slant and hemi-slant submanifolds in metallic Riemannian manifolds and, in particular, on invariant, anti-invariant and semi-invariant submanifolds. We also describe the warped product bi-slant and, in particular, warped product semi-slant and warped product hemi-slant submanifolds in locally metallic Riemannian manifolds, obtaining some results regarding the existence and nonexistence of non-trivial semi-invariant, semi-slant and hemi-slant warped product submanifolds. We illustrate all these by suitable examples.