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.

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Applications of the Golden Ratio on Riemannian Manifolds

Our aim in this paper is to focus on some applications in differential geometry of the metallic means family (a generalization of the golden mean) and generalized Fibonacci sequences, using a class of polynomial structures defined on Riemannian manifolds. We search for properties of the induced structure on a submanifold by metallic Riemannian structures and we find a necessary and sufficient condition for a submanifold to be also a metallic Riemannian manifold in terms of invariance. Also, the totally geodesic, minimal and respectively totally umbilical hypersurfaces in metallic Riemannian manifolds are analyzed and the Euclidean space and its hypersphere is treated as example.

Metallic structures on Riemannian manifolds

The aim of our paper is to focus on some properties of slant and semi-slant submanifolds of metallic Riemannian manifolds. We give some characterizations for submanifolds to be slant or semi-slant submanifolds in metallic or Golden Riemannian manifolds and we obtain integrability conditions for the distributions involved in the semi-slant submanifolds of Riemannian manifolds endowed with metallic or Golden Riemannian structures. Examples of semi-slant submanifolds of the metallic and Golden Riemannian manifolds are given.

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Slant and Semi-Slant Submanifolds in Metallic Riemannian Manifolds

The aim of our paper is to focus on some properties of submanifolds in Riemannian manifolds endowed with endomorphisms that generalize the Golden Riemannian structure, named metallic Riemannian structures. We focus on the properties of the structure induced on submanifolds, named by us $\Sigma$-metallic Riemannian structures, especialy regarding the normality of this types of structure. Examples of structures induced on a sphere of codimension 1 by some metallic Riemannian structures defined on an Euclidean space are given.

Submanifolds in metallic Riemannian manifolds

In this paper, we defined and studied a new class of submanifolds of a locally Riemannian product manifold, i.e., skew semi-invariant submanifolds. We give two sufficient conditions for submanifolds to be skew semi-invariant submanifolds. Moreover, we discussed the sectional curvature of skew semi-invariant submanifolds and obtained many interesting results.

Skew semi-invariant submanifolds of a locally product manifold.

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Gheorghe Pitis

Papers

On some submanifolds of a locally product manifold