Types of Submanifolds in Metallic Riemannian Manifolds: A Short Survey

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.


Introduction
The notion of Golden structure on a Riemannian manifold was introduced by C. E. Hretcanu and M. Crasmareanu in ([1]). Then, the properties of submanifolds in Golden Riemannian manifolds were studied in ( [2,3]) using the corespondents of a Golden structure related to an almost product structure. The metallic structure introduced in ([4]) is a generalization of the Golden structure. Different types of submanifolds in metallic and Golden Riemannian manifolds were studied in ( [5][6][7]), obtaining different integrability conditions for the distributions involved in these types of submanifolds. The metallic (and in particular Golden) warped product Riemannian manifold was studied in ( [8][9][10]).
In this paper we provide a brief survey on the properties of metallic structures defined on Riemannian manifolds, stating the definitions and some properties of these structures, related to generalized secondary Fibonacci sequences (in Section 2). Then, in Section 3, we discuss about some properties of the structures induced on submanifolds, called by us, Σ-metallic Riemannian structures, especially regarding the normality of these types of structures and, in particular, we present some properties of invariant and anti-invariant submanifolds in metallic Riemannian manifolds ( [11]). In Section 4, we define and discuss about properties of slant submanifolds in metallic Riemannian manifolds. In Section 5, we treat bi-slant submanifolds in metallic Riemannian manifolds and, in particular, for semi-slant and hemi-slant submanifolds, we obtain some integrability conditions of the distributions involved. In Section 6, we give some properties of warped product submanifolds in metallic Riemannian manifolds and provide examples of warped product semi-invariant, semi-slant and respectively hemi-slant submanifolds in these types of manifolds. and so on. The most remarkable element from (MMF) is the Golden mean, known from ancient times as an expression of harmony of many constructions, paintings and in music. It also appears as an expression of the objects from the natural world (flowers, trees, fruits) that possess pentagonal symmetry ( [18]).
One can remark that Golden mean is determined by the ratio of two consecutive classical Fibonacci numbers and the silver mean is determined by the ratio of two consecutive Pell numbers ( [19]). The bronze mean plays an important role in studying topics such as dynamical systems and quasicrystals and the subtle mean is significant in the theory of Cantorian fractal-like micro-space-time E ∞ , being involved, in a fundamental way, in noncommutative geometry and four manifold theory ( [20,21]).
The members of (MMF), named also metallic numbers, have found many applications in differential geometry, as it can be seen in the next properties.
A metallic structure on a differentiable manifold is a particular case of a polynomial structure introduced by S. I. Goldberg, K. Yano and N. C. Petridis in ( [22,23]).
Precisely, a polynomial structure J of degree 2, defined on a differentiable manifold M, is called a metallic structure if it satisfies the equality J 2 = p · J + q · I, where I is the identity endomorphism on TM and p, q ∈ N * . The pair (M, J) is called a metallic manifold.
Another important (for our study) is the polynomial structure F of degree 2 on M, i.e., F 2 = I, called an almost product structure. In this case, (M, F) is called an almost product manifold.
The metallic structure and the almost product structure are closely related shown below.
Proposition 1 ( [4]). (i) Any metallic structure J induces two almost product structures, given by (ii) Any almost product structure F induces two metallic structures, given by is called a metallic Riemannian manifold if the differentiable manifold M is endowed with a metallic structure J and a Riemannian metric g such that g(JX, Y) = g(X, JY), for any X, Y ∈ Γ(TM).
We remark that in a metallic Riemannian manifold (M, g, J), we have for any X, Y ∈ Γ(TM).
Recall that an almost product Riemannian structure (g, F) on M is a pair (g, F), where g is a Riemannian metric and F is a g-compatible almost product structure on M, i.e., for any X, Y ∈ Γ(TM). In this case, (M, g, F) is called almost product Riemannian manifold.
If the almost product structure F is a Riemannian one, then the induced metallic structures J 1 and J 2 are metallic Riemannian structures.

Proposition 2 ([4]).
A metallic structure J on M has the following properties: (i) for every integer number n ≥ 1: where (G n ) n∈N * is the generalized secondary Fibonacci sequence with G 0 = 0 and G 1 = 1; (ii) J is an isomorphism on T x M, for every x ∈ M. It follows that J is invertible and its inversê J = J −1 = 1 q · J − p q · I is not a metallic structure, but it is still polynomial, more precisely, a quadratic one: q ·Ĵ 2 + p ·Ĵ − I = 0; (iii) the eigenvalues of J are the metallic number σ p,q and p − σ p,q .
In particular, for p = q = 1 one gets the Golden structure ( [24,25]), namely a (1, 1)tensor field J which satisfies J 2 = J + I. In this case, (M, J) is called a Golden manifold. Every Golden structure defines two almost product structures and any almost product structure defines two Golden structures ( [2,24]). The eigenvalues of J are the Golden Ratio φ and 1 − φ. Notice that the Golden structures appear in pairs, i.e., if J is a Golden structure, thenJ = I − J is also a Golden structure.
Moreover, the power n ≥ 1 of the Golden structure J on M can be written using the Fibonacci sequence ( f n ) n∈N * ( [24]) By using an explicit expression for the Fibonacci sequence, namely, the Binet's formula from ( [18]) we can obtain an expression of the power n ≥ 1 of the Golden structure J by means of the Golden number φ ( [24]).

Submanifolds in Metallic Riemannian Manifolds
In ( [11]), C. E. Hretcanu and A. M. Blaga discussed about some properties of the structure induced by a metallic Riemannian structure on a submanifold, called Σ-metallic Riemannian structure.
Let M m be an m-dimensional Riemannian manifold with the metric g and let Γ(TM) be the set of all vector fields on M.
If M is a submanifold of dimension m ∈ N * , isometrically immersed in a metallic Riemannian manifold (M, g, J), T x M is the tangent space of M in a point for any x ∈ M. Denoting by i * the differential of the immersion i : M → M and by X the vector field i * X, for X ∈ Γ(TM), the induced Riemannian metric g on M is given by g(X, Y) = g(i * X, i * Y), for any X, Y ∈ Γ(TM). It can be easily checked that the maps T and n are g-symmetric ( [5]) g(TX, Y) = g(X, TY), g(nU, V) = g(U, nV), for any X, Y ∈ Γ(TM) and U, V ∈ Γ(T ⊥ M). Moreover: for any X ∈ Γ(TM) and U ∈ Γ(T ⊥ M).
If r ∈ N * is the codimension of the submanifold M in the Riemannian manifold (M, g), then we can fix a local orthonormal basis {N 1 , ..., N r } of T ⊥ M. Hereafter we assume that the indices α, β, γ run over the range {1, ..., r}.
Then the vectors JX and JN α can be decomposed into the tangential and normal components ( [4]) for any X ∈ Γ(TM), where ξ α =: tN α are vector fields on M, u α are 1-forms on M, (a αβ ) r is an r × r matrix of differentiable functions on M, α, β ∈ {1, ..., r}, and We can remark that Theorem 1 ( [11]). The structure (T, g, u α , ξ α , (a αβ ) r ) induced on a submanifold M of codimension r by the metallic Riemannian structure (g, J) on M satisfies the relations (1)-(4). Thus, it is a Σ-metallic Riemannian structure.
On the Riemannian manifold (M, g) and its submanifold (M, g), we consider the Levi-Civita connections ∇ and ∇, respectively. For any X, Y ∈ Γ(TM) and V ∈ Γ(T ⊥ M), the Gauss and Weingarten formulas are given by where h is the second fundamental form, A V is the shape operator of M and ∇ ⊥ is the normal connection. Moreover: for any X, Y ∈ Γ(TM) and V ∈ Γ(T ⊥ M).
We can define the notion of locally metallic Riemannian manifold by analogy with a locally product manifold, as follows.

Definition 3.
We say that a metallic Riemannian manifold (M, g, J) is a locally metallic Riemannian manifold if J is parallel with respect to the Levi-Civita connection ∇ on M, i.e., ∇J = 0.

Proposition 3 ([11]
). If M is a submanifold in a locally metallic Riemannian manifold (M, g, J) with ∇ the Levi-Civita connection on M and ∇ ⊥ the normal connection, then, for any X, Y ∈ Γ(TM), we have

Proposition 5 ([11]
). Let M be a submanifold of codimension r in a locally metallic Riemannian manifold (M, g, J) and let (T, g, u α , ξ α , (a αβ ) r ) be the Σ-metallic Riemannian structure induced by the metallic Riemannian structure (g, J). Then, for any X, Y ∈ Γ(TM), we get

Proposition 6 ([11]
). Let M be a submanifold of codimension r in a locally metallic Riemannian manifold (M, g, J) and let (T, g, u α , ξ α , (a αβ ) r ) be the Σ-metallic Riemannian structure induced by the metallic Riemannian structure (g, J). Then, for any X, Y ∈ Γ(TM), we get If N J is the Nijenhuis tensor field of J, defined for any X, Y ∈ Γ(TM), by then it verifies ( [26]) Thus, we remark that in a locally metallic Riemannian manifold (M, g, J), we have N J = 0 and the metallic structure (g, J) can be called integrable. Now we shall define a normal Σ-metallic Riemannian structure.

Definition 4.
A Σ-metallic Riemannian structure induced on a submanifold M of codimension r in a metallic Riemannian manifold (M, g, J) is said to be normal if We denote by B α := TA α − A α T and remark that g(B α X, Y) = −g(X, B α Y), for any X, Y ∈ Γ(TM).

Theorem 2 ([11]
). If (T, g, u α , ξ α , (a αβ ) r ) is the Σ-metallic Riemannian structure induced on a submanifold M of codimension r in a locally metallic Riemannian manifold (M, g, J), then, for any X, Y ∈ Γ(TM), we have where l αβ are the coefficients of the normal connection ∇ ⊥ . Remark 1 ( [11]). In the conditions of the previous theorem, if T commutes with the Weingarten operators A α , for any α ∈ {1, ..., r}, i.e., B α = 0, then the Nijenhuis tensor field of T vanishes on M.
By a direct computation, for any X, Y ∈ Γ(TM), we get The components N (1) , N (2) , N (3) and N (4) of the Nijenhuis tensor field of T can be computed using the similar idea from ( [27]), as follows.
We shall further provide conditions such that the induced Σ-metallic Riemannian structure to be normal.
We remark that, if the Σ-metallic Riemannian structure induced on M is normal and the normal connection ∇ ⊥ of M vanishes identically, i.e., l αβ = 0, then ( for any X, Y ∈ Γ(TM).

Theorem 3 ([11]
). Let M be a submanifold of codimension r ≥ 2 in a locally metallic Riemannian manifold (M, g, J). If the normal connection ∇ ⊥ vanishes identically and M is a non-invariant submanifold with respect to the metallic structure J, then the vector fields {ξ 1 , ..., ξ r } are linearly independent. Theorem 4 ([11]). Let M be a submanifold of codimension r ≥ 1 in a locally metallic Riemannian manifold (M, g, J). If the normal connection ∇ ⊥ vanishes identically and T commutes with every Weingarten operator A α , then the induced Σ-metallic Riemannian structure on M is normal. Theorem 5 ([11]). Let M be a submanifold of codimension r ≥ 1 in a locally metallic Riemannian manifold (M, g, J). If the normal connection ∇ ⊥ vanishes identically and M is a non-invariant submanifold with respect to the metallic structure J, then the induced Σ-metallic Riemannian structure on M is normal if and only if T commutes with the Weingarten operator A α , for any α ∈ {1, ..., r}.
Corollary 1 ( [11]). If M is a non-invariant totally umbilical (or totally geodesic) submanifold of codimension r ≥ 1 in a locally metallic Riemannian manifold (M, g, J) such that the normal connection ∇ ⊥ vanishes identically, then the Σ-metallic Riemannian structure induced on M is normal.
We can observe that the matrix A := (a αβ ) r of the Σ-structure induced on an invariant submanifold M by the metallic Riemannian structure (g, J) from M is a metallic matrix, that is a matrix which verifies where I r is the identically matrix of order r. If A := (a αβ ) r is a metallic matrix, then ∑ r γ=1 a αγ a γβ = pa αβ + qδ αβ and we obtain u β (ξ α ) = 0, which implies that T 2 ξ α = pTξ α + qξ α and Jξ α = Tξ α , for any α ∈ {1, ..., r}.

Anti-Invariant Submanifolds a Metallic Riemannian Manifold
Proposition 11 ([5]). Let M be an anti-invariant submanifold of codimension r isometrically immersed in a locally metallic Riemannian manifold (M, g, J). Then, for any X, Y ∈ Γ(TM):

Slant Submanifolds in Metallic Riemannian Manifolds
Let M be a submanifold of codimension r ∈ N * isometrically immersed in a metallic Riemannian manifold (M, g, J). For any X ∈ Γ(TM), one obtains (from the Cauchy-Schwartz inequality) g(JX, TX) ≤ JX · TX . Therefore, we can consider a function θ : Γ(TM) → [0, π 2 ], such that: for any x ∈ M and any nonzero X x ∈ T x M. If JX x = 0, then the angle θ(X x ) between JX x and T x M is called the Wirtinger angle of X. g(TX, TY) = cos 2 θ · g(X, p · TY + q · Y), g(NX, NY) = sin 2 θ · g(X, p · TY + q · Y).

Remark 6. From ([5]) we obtain
Like in the Riemannian product case ( [28]), we can define a slant distribution in a metallic Riemannian manifold. (P D T) 2 X = λ · (p · P D TX + q · X), for any X ∈ Γ(D). Moreover, the slant angle θ D of D satisfies λ = cos 2 θ D .

Bi-Slant Submanifolds in Metallic Riemannian Manifolds
The differential geometry of slant submanifolds has shown an increasing development in the early 1990's when B.-Y. Chen defined slant submanifolds in complex manifolds ( [29]). Particular cases of bi-slant submanifolds, such as semi-invariant submanifolds in locally product Riemannian manifolds were studied in ( [30,31]), semi-slant submanifolds were studied by J. L. ). Moreover, slant and semi-slant submanifolds in almost product Riemannian manifolds were studied in ( [28,30]). The hemi-slant submanifolds (called, also, pseudo-slant submanifolds) in locally decomposable Riemannian manifolds were studied by M. Atçeken et al. ( [35]) and in locally product Riemannian manifolds were studied by H. M. Taştan and F. Ozdem in ( [36]). If M is a bi-slant submanifold of (M, g, J), then T(D 1 ) ⊆ D 1 and T(D 2 ) ⊆ D 2 .
We provide an example of a bi-slant submanifold in a metallic Riemannian manifold.

Semi-Slant Submanifolds in Metallic Riemannian Manifolds
Semi-slant submanifolds in a metallic Riemannian manifold are particular cases of bi-slant submanifolds, which can be defined in a similar manner as semi-slant submanifolds in a locally product Riemannian manifold ( [28]).

Definition 8 ([6]).
A semi-slant submanifold M in a metallic Riemannian manifold (M, g, J) is a submanifold which has two orthogonal differentiable distributions D and D θ , such that: TM admits the orthogonal decomposition TM = D ⊕ D θ ; 2.
the distribution D is invariant; 3.

Remark 7.
Let M be a semi-slant submanifold in a metallic Riemannian manifold (M, g, J) with TM = D ⊕ D θ . If θ = π 2 , then M is a semi-invariant submanifold of M.

Proposition 18 ([6]). A necessary and sufficient condition for a submanifold M in a metallic
Riemannian manifold (M, g, J) to be a semi-slant submanifold in M is to exist a constant λ ∈ [0, 1), such that D 0 = {X ∈ Γ(TM) | T 2 X = λ(p · TX + q · X)} is a differentiable distribution, and NX = 0, for any X ∈ Γ(TM) orthogonal to D 0 .
Proposition 19 ([6]). Let M be a semi-slant submanifold in a locally metallic Riemannian manifold (M, g, J). Then: (i) a necessary and sufficient condition for the integrability of the distribution D is: for any X, Y ∈ Γ(D) and V ∈ Γ(T ⊥ M); (ii) a necessary and sufficient condition for the integrability of the distribution D θ is: for any Z, W ∈ Γ(D θ ). A nV X = TA V X = A V TX.

Definition 9.
Let M be a semi-slant submanifold in a metallic Riemannian manifold (M, g, J). We say that M is a D − D θ mixed totally geodesic if h(X, Z) = 0, for any X ∈ Γ(D) and any Z ∈ Γ(D θ ). Proposition 21 ([6]). A necessary and sufficient condition for the semi-slant submanifold M in a locally metallic Riemannian manifold (M, g, J) to be a D − D θ mixed totally geodesic submanifold is A V X ∈ Γ(D) and A V Z ∈ Γ(D θ ), for any X ∈ Γ(D), Z ∈ Γ(D θ ) and V ∈ Γ(T ⊥ M). ([6]). If M is a proper semi-slant submanifold in a locally metallic Riemannian manifold (M, g, J), then M is a D − D θ mixed totally geodesic submanifold if one of the following conditions are true, for any X ∈ Γ(D), Z ∈ Γ(D θ ) and V ∈ Γ(T ⊥ M): (i) (∇ X N)Z = 0 and h(X, Z) is not an eigenvector of the tensor field n with the eigenvalue − q p ; (ii) A nV X = TA V X = A V TX and h(X, Z) is not an eigenvector of the tensor field n. Proposition 23 ([6]). If M is a D − D θ mixed totally geodesic proper semi-slant submanifold in a locally metallic Riemannian manifold (M, g, J), then (∇ X N)Y = 0, for any X ∈ Γ(D) and Y ∈ Γ(D θ ).

Hemi-Slant Submanifolds in Metallic Riemannian Manifolds
Hemi-slant submanifolds in a metallic Riemannian manifold are particular cases of bislant submanifolds, which can be defined in a similar manner as hemi-slant submanifolds in a locally product Riemannian manifold ( [36]).

Definition 10 ([7]).
A hemi-slant submanifold M in a metallic Riemannian manifold (M, g, J) is a submanifold which has two orthogonal differentiable distributions D θ and D ⊥ , such that: 1.
the distribution D ⊥ is anti-invariant; 3.
Definition 11. Let M be a hemi-slant submanifold in a metallic Riemannian manifold (M, g, J). We say that M is a D θ − D ⊥ mixed totally geodesic if h(X, Z) = 0, for any X ∈ Γ(D θ ) and any Z ∈ Γ(D ⊥ ).

Proposition 26 ([7]). A necessary and sufficient condition for a submanifold M in a metallic
Riemannian manifold (M, g, J) to be a hemi-slant submanifold in M is to exist a constant λ ∈ [0, 1), such that D 0 = {X ∈ Γ(TM) | T 2 X = λ(p · TX + q · X)} is a differentiable distribution, and TX = 0, for any X ∈ Γ(TM) orthogonal to D 0 . Proposition 27 ([7]). Let M be a hemi-slant submanifold in a locally metallic Riemannian manifold (M, g, J). Then: (i) the distribution D θ is integrable and, for any X, Y ∈ Γ(D θ ), we get (ii) a necessary and sufficient condition for the integrability of the distribution D ⊥ is: for any Z, W ∈ Γ(D ⊥ ). A nV X = TA V X = A V TX.

Proposition 29 ([7]).
A necessary and sufficient condition for the hemi-slant submanifold M in a locally metallic Riemannian manifold (M, g, J) to be a D θ − D ⊥ mixed totally geodesic submanifold Proposition 30 ([7]). If M is a proper hemi-slant submanifold in a locally metallic Riemannian manifold (M, g, J) and (∇ X N)Y = 0, for any X, Y ∈ Γ(TM), then M is a D θ − D ⊥ mixed totally geodesic submanifold in M.
In this section we present some results regarding the existence and nonexistence of non-trivial semi-invariant, semi-slant and hemi-slant warped product submanifolds in locally metallic Riemannian manifolds and we provide examples.
Let (M 1 , g 1 ) and (M 2 , g 2 ) be two Riemannian manifolds and denote by π 1 and π 2 the projection maps from the product manifold M 1 × M 2 onto M 1 and M 2 , respectively. Definition 12 ([45]). If g is the Riemannian metric on M 1 × M 2 defined by g := π * 1 g 1 + ( f • π 1 ) 2 π * 2 g 2 , where f : M 1 → (0, ∞) is a differentiable function on M 1 , then M 1 × f M 2 =: (M 1 × M 2 , g) is called the warped product Riemannian manifold of M 1 and M 2 , having the warping function f . Moreover, M 1 × f M 2 is called trivial if f is constant. In this case, it is just a Riemannian product M 1 × M 2 , where M 2 is equipped with the metric f 2 g 2 (which is homothetic to g 2 ).
Let M ⊥ and M be the integral manifolds of the distributions D 1 and D 2 , respectively. Thus, M ⊥ × f M is a warped product semi-invariant submanifold in the metallic Riemannian manifold (R 5 , ·, · , J), with the metric g := g M ⊥ + f 2 g M , where g M ⊥ := pσ q + 2 d f 2 and g M := dα 2 + dβ 2 .

Proposition 36 ([10]).
If M × f M θ is a proper warped product semi-slant submanifold in a locally metallic Riemannian manifold (M, g, J), then it is a trivial warped product Riemannian manifold.
We provide an example of a non-trivial warped product semi-slant submanifold M θ × f M in a metallic Riemannian manifold.