INVARIANT SUBMANIFOLDS IN GOLDEN RIEMANNIAN MANIFOLDS

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 suffi cient 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.


Introduction
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. In particular, 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. Invariant submanifolds are one of typical classes among all submanifolds of an ambient manifold. It is well known that in general, an invariant submanifold inherits almost all properties of the ambient manifold. Therefore, invariant submanifolds are an active and fruitful research …eld playing a signi…cant role in the development of modern di¤erential 126 M . GÖK, S. KELEŞ, E. KILIÇ geometry. Also, the papers related to invariant submanifolds have appeared in various ambient manifolds, such as almost contact Riemannian manifolds [3,4], normal contact metric manifolds [5], Sasakian manifolds [6], almost product Riemannian manifolds [7], CR-manifolds [8] etc.
Recently, C 1 -di¤erentiable manifolds endowed with golden structures, i.e., golden manifolds have become a popular topic in di¤erential geometry. In [9], M. C. Crâşm¼ areanu and C. E. Hreţcanu have shown that there exists a close relationship between golden and almost product structures. In this sense, F. Etayo, R. Santamaría and A. Upadhyay have analyzed almost golden Riemannian manifolds by use of the corresponding almost product structures in [10], where the concept of a golden manifold was de…ned as a C 1 -di¤erentiable manifold admitting an integrable golden structure. In [11], M. Gök, S. Keleş and E. K¬l¬ç have examined the Schouten and Vr¼ anceanu connections on golden manifolds. The di¤erent kind of classes of submanifolds in a golden Riemannian manifold have been de…ned according to the behaviour of their tangent bundles with respect to the action of the golden structure of the ambient manifold and studied by several geometers in [12,13,14,15,16]. Invariant submanifolds, which are one of important and known classes of submanifolds, have been investigated in a golden Riemannian manifold for the …rst time by C. E. Hreţcanu and M. C. Crâşm¼ areanu with the help of induced structures on them by the golden structure of the ambient manifold in [17] we can …nd their some fundamental properties. The authors have obtained a characterization for any submanifold in a golden Riemannian manifold to be invariant and proved that the Nijenhuis tensor of the induced structure vanishes identically on invariant submanifolds in the case that the ambient manifold is a locally decomposable golden Riemannian manifold. Also, an example of an invariant submanifold regarding a product of two spheres in an Euclidean space has been given in [18].
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.
The paper has three sections and is organized as follows: Section 2 is devoted to preliminaries containing basic de…nitions, concepts, formulas, notations and results for golden Riemannian manifolds and their submanifolds. Section 3 deals with an investigation of invariant submanifolds of a golden Riemannian manifold. We prove that any invariant submanifold of a locally decomposable golden Riemannian manifold is also locally decomposable. We obtain a characterization for any submanifold in a golden Riemannian manifold to be invariant. We …nd some necessary conditions for any invariant submanifold to be totally geodesic. Also, we get other results on invariant submanifolds. Lastly, we construct an induced structure on a product of hyperspheres in an Euclidean space as an example of a golden Riemannian structure.

Preliminaries
In this section, we recall some basic facts on golden Riemannian manifolds and their submanifolds.
A non-trivial C 1 -tensor …eld f of type (1; 1) on a C 1 -di¤erentiable manifold M is called a polynomial structure of degree n if it satis…es the algebraic equation where I is the identity (1; 1)-tensor …eld on M and f n 1 (p) ; f n 2 (p) ; : : : ; f (p) ; I are linearly independent for every point p 2 M . Also, the monic polynomial Q (x) is named the structure polynomial [19]. A polynomial structure of degree 2 with the structure polynomial Q (x) = x 2 x 1 on a C 1 -di¤erentiable real manifold M is called a golden structure. That is, the golden structure is a tensor …eld of type (1; 1) satisfying the algebraic equation 2 = + I.
(2) In this case, we say that M is a golden manifold. We denote by T M the Lie algebra of di¤erentiable vector …elds on M . If there exists a Riemannian metric g on M endowed with a golden structure such that g and verify the relation for any vector …elds X; Y 2 T M , then the pair g; is said to be a golden Riemannian structure and the triple M ; g; is called a golden Riemannian manifold. The eigenvalues of the golden structure are = 1+ being the roots of the algebraic equation x 2 x 1 = 0, where the former is the golden ratio [9,17,18].
Let M be an n-dimensional submanifold of codimension r, isometrically immersed in an m-dimensional golden Riemannian manifold M ; g; . We denote by T p M and T p M ? its tangent and normal spaces at a point p 2 M , respectively. Then the tangent space T p M admits the decomposition for each point p 2 M . The induced Riemannian metric on M is given by for any vector …elds X; Y 2 (T M ), where i is the di¤erential of the immersion i : M ! M . We consider a local orthonormal frame fN 1 ; : : : ; N r g of the normal bundle T M ? . For every tangent vector …eld X 2 (T M ), the vector …elds (i X) and (N ) on the ambient manifold M can be decomposed into tangential and normal components as follows: and respectively, where is a tensor …eld of type (1; 1) on M , 's are tangent vector …elds on M , u 's are di¤erential 1-forms on M and (a ) is a matrix of type r r of real functions on M for any ; 2 f1; : : : ; rg. Thus, we obtain a structure ; g; u ; " ; (a ) r r induced on M by the golden Riemannian structure g; .
We denote by r and r the Levi-Civita connections on M and M , respectively. Then the Gauss and Weingarten formulas of M in M are given, respectively, by and for any vector …elds X; Y 2 (T M ), where h 's are the second fundamental tensors corresponding to N 's, A 's are the shape operators in the direction of N 's and l 's are the 1-forms on M corresponding to the normal connection r ? for any ; 2 f1; : : : ; rg. Also, the following relations hold: and for any vector …elds X; Y 2 (T M ) [17].
As it is well known, the submanifold M is called totally geodesic if h = 0. Besides, the mean curvature vector H of M is de…ned by where fe 1 ; : : : ; e n g is orthonormal basis of the tangent space T p M at a point p 2 M . If the mean curvature vector H vanishes identically, then M is said to be a minimal submanifold. If h (X; Y ) = g (X; Y ) H for any vector …elds X; Y 2 (T M ), then M is named a totally umbilical submanifold [20].
The triple M ; g; is called a locally decomposable golden Riemannian manifold if the golden structure is parallel with respect to the Levi-Civita connection r, i.e., the covariant derivative r is identically zero. Also, under the assumption that the induced structure is a golden structure, the same de…nition can be applied to the submanifold (M; g; ) in terms of the Levi-Civita connection r of M .

Invariant Submanifolds of Golden Riemannian Manifolds
This section is mainly concerned with invariant submanifolds in golden Riemannian manifolds. We show that any invariant submanifold in a locally decomposable golden Riemannian manifold preserves the locally decomposability of the ambient manifold. We get an equivalent expression to the invariance of any submanifold in a golden Riemannian manifold. We give some necessary conditions for the totally geodesicity of invariant submanifolds. Besides, we obtain some results on invariant submanifolds.
As a beginning, we remember that the notion of an invariant submanifold in golden Riemannian manifolds. Any invariant submanifold M of a golden Riemannian manifold M ; g; is submanifold such that the golden structure of the ambient manifold M carries each tangent vector of the submanifold M into its corresponding tangent space in the ambient manifold M , in other words, Let M be an n-dimensional invariant submanifold of codimension r, isometrically immersed in an m-dimensional golden Riemannian manifold M ; g; . Then we have = 0 and u = 0 for any 2 f1; : : : ; rg. Hence, (5) and (6) can be expressed in the following forms: and respectively. on M by the golden Riemannian structure g; satis…es the following relations: and for any vector …elds X; Y 2 (T M ). and for any vector …elds X; Y 2 (T M ).
Theorem 4. Let M be an n-dimensional invariant submanifold of codimension r, isometrically immersed in an m-dimensional locally decomposable golden Riemannian manifold M ; g; . Then M is a locally decomposable golden Riemannian manifold whenever is non-trivial.
Theorem 9. Let M be an n-dimensional submanifold of codimension r, isometrically immersed in an m-dimensional golden Riemannian manifold M ; g; . Then the second fundamental tensors h 's are zero for any 2 f1; : : : ; t < rg if the following relations are veri…ed: Proof. By reason of (39), (40) and (41), we infer from (5) and (6) that Proof. We denote by fe 1 ; : : : ; e n g an orthonormal basis of the tangent space T p M at a point p 2 M . Since the submanifold M is totally umbilical, there are constants 's such that h = g for any 2 f1; : : : ; rg. Then (22) is given by for any vector …elds X; Y 2 (T M ). Putting X p = Y p = e i for any i 2 f1; : : : ; ng at the point p 2 M in (45), we have Summing over i in (46), we get which implies Multiplying (47) by the matrix element a and then summing over , we obtain Using (18), (48) takes the form from which we have Hence, substituting (49) into (47), we …nd On the other hand, on account of the fact that ftr( )g 2 6 = n fn + tr( )g, or equivalently ftr( )g 2 6 = 2 n 2 in the hypothesis, it follows from (50) that r X =1 a = 0.
We consider a tensor …eld of type (1; 1) de…ned by 2 . In this case, it is easy to show that h; i ; is a golden Riemannian structure and E 2(p+q) ; h; i ; is a golden Riemannian manifold. Because of the fact that E 2(p+q) = E p E p E q E q , we have the following four hyperspheres: .
We construct the product manifold S p 1 (r 1 ) S p 1 (r 2 ) S q 1 (r 3 ) S q 1 (r 4 ) in a similar way as in [18]. We denote it by M for simplicity. Its every point has the coordinates x i ; y i ; z j ; w j satisfying the equation where R 2 = r 2 1 + r 2 2 + r 2 3 + r 2 4 . Then M is a submanifold of codimension 4 in the Euclidean space E 2(p+q) and M is a submanifold of codimension 3 in the sphere S 2(p+q) 1 (R). Hence, there exist successive embeddings such that M ,! S 2(p+q) 1 (R) ,! E 2(p+q) .
Also, its tangent space T (x i ;y i ;z j ;w j ) M at a point x i ; y i ; z j ; w j is as follows: T (x i ;0 i ;0 j ;0 j ) S p 1 (r 1 ) T (0 i ;y i ;0 j ;0 j ) S p 1 (r 2 ) T (0 i ;0 i ;z j ;0 j ) S q 1 (r 3 ) T (0 i ;0 i ;0 j ;w j ) S q 1 (r 4 ) .
As it is seen, any tangent vector X i ; Y i ; Z j ; W j 2 T (x i ;y i ;z j ;w j ) E 2(p+q) belongs to T (x i ;y i ;z j ;w j ) M for every point x i ; y i ; z j ; w j 2 M if and only if In addition, since X i ; Y i ; Z j ; W j is a tangent vector on the sphere S 2(p+q) 1 (R), we have T (x i ;y i ;z j ;w j ) M T (x i ;y i ;z j ;w j ) S 2(p+q) 1 (R) for every point x i ; y i ; z j ; w j 2 M . Let us consider a local orthonormal basis fN 1 ; N 2 ; N 3 ; N 4 g for the normal space T (x i ;y i ;z j ;w j ) M ? at a point x i ; y i ; z j ; w j . Then we can choose the normal vectors N 1 , N 2 , N 3 and N 4 such that N 1 = 1 R x i ; y i ; z j ; w j , N 2 = 1 R r 2 r 1 x i ; r 1 r 2 y i ; r 4 r 3 z j ; r 3 r 4 w j , N 3 = 1 R r 3 r 1 x i ; r 4 r 2 y i ; r 1 r 3 z j ; r 2 r 4 w j and N 4 = 1 R r 4 r 1 x i ; r 3 r 2 y i ; r 2 r 3 z j ; r 1 r 4 w j .
We identify i X with X for any tangent vector X 2 T (x i ;y i ;z j ;w j ) M . From (6) Hence, using the matrix elements a 's given above, it follows from (51) that In this case, we have T (x i ;y i ;z j ;w j ) M ? T (x i ;y i ;z j ;w j ) M ? .
From (5), we can write the following relation: