Abstract
In this paper, we study invariant submanifolds of a golden Riemannian manifold with the aid of induced structures on them by the golden structure of the ambient manifold. We demonstrate that any invariant submanifold in a locally decomposable golden Riemannian manifold leaves invariant the locally decomposability of the ambient manifold. We give a necessary and sufficient condition for any submanifold in a golden Riemannian manifold to be invariant. We obtain some necessary conditions for the totally geodesicity of invariant submanifolds. Moreover, we find some facts on invariant submanifolds. Finally, we present an example of an invariant submanifold.
Highlights
The di¤erential geometry of submanifolds has occupied an important place in natural and engineering sciences since some particular types of submanifolds have been used as a geometric tool to solve many problems concerning these disciplines
We study invariant submanifolds of a golden Riemannian manifold with the aid of induced structures on them by the golden structure of the ambient manifold
We demonstrate that any invariant submanifold in a locally decomposable golden Riemannian manifold leaves invariant the locally decomposability of the ambient manifold
Summary
The di¤erential geometry of submanifolds has occupied an important place in natural and engineering sciences since some particular types of submanifolds have been used as a geometric tool to solve many problems concerning these disciplines. Invariant submanifolds have a key role in applied mathematics and theoretical physics as a method, such as for determining non-linear normal modes in non-linear systems [1] and constructing the reduced description for dissipative systems of reaction kinetics [2]. When considered from this point of view, invariant submanifolds have a special meaning in di¤erential geometry. The main purpose of this paper is to examine invariant submanifolds of a golden Riemannian manifold by means of induced structures on them by the golden structure of the ambient manifold. We construct an induced structure on a product of hyperspheres in an Euclidean space as an example of a golden Riemannian structure
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