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Slant and Semi-Slant Submanifolds in Metallic Riemannian Manifolds.

1. Introduction

Since B.Y. Chen defined slant submanifolds in complex manifolds ([1,2]) in the early 1990s, the differential geometry of slant submanifolds has shown an increasing development. Then, many authors have studied slant submanifolds in different kind of manifolds, such as slant submanifolds in almost contact metric manifolds (A. Lotta ([3])), in Sasakian manifolds (J.L. Cabrerizo et al. ([4, 5])), in para-Hermitian manifold (P. Alegre, A. Carriazo ([6])), and in almost product Riemannian manifolds (B. Sahin ([7]), M. Atfeken ([8, 9])).

The notion of slant submanifold was generalized by semislant submanifold, pseudo-slant submanifold, and bi-slant submanifold, respectively, in different types of differentiable manifolds. The semi-slant submanifold of almost Hermitian manifold was introduced by N. Papagiuc ([10]). A. Cariazzo et al. ([11]) defined and studied bi-slant immersion in almost Hermitian manifolds and pseudo-slant submanifold in almost Hermitian manifolds. The pseudo-slant submanifolds in Kenmotsu or nearly Kenmotsu manifolds ([12, 13]), in LCS-manifolds ([14]), or in locally decomposable Riemannian manifolds ([15]) were studied by M. Atceken et al. Moreover, many examples of semi-slant, pseudoslant, and bi-slant submanifolds were built by most of the authors.

Semi-slant submanifolds are particular cases of bi-slant submanifolds, defined and studied by A. Cariazzo ([11]). The geometry of slant and semi-slant submanifolds in metallic Riemannian manifolds is related by the properties of slant and semi-slant submanifolds in almost product Riemannian manifolds, studied in ([7, 8,16]).

The notion of Golden structure on a Riemannian manifold was introduced for the first time by C.E. Hretcanu and M. Crasmareanu in ([17]). Moreover, the authors investigated the properties of a Golden structure related to the almost product structure and of submanifolds in Golden Riemannian manifolds ([18, 19]). Examples of Golden and productshaped hypersurfaces in real space forms were given in ([20]). The Golden structure was generalized as metallic structures, defined on Riemannian manifolds in ([21]). A.M. Blaga studied the properties of the conjugate connections by a Golden structure and expressed their virtual and structural tensor fields and their behavior on invariant distributions. Also, she studied the impact of the duality between the Golden and almost product structures on Golden and product conjugate connections ([22]). The properties of the metallic conjugate connections were studied by A.M. Blaga and C.E. Hretcanu in ([23]) where the virtual and structural tensor fields were expressed and their behavior on invariant distributions was analyzed.

Recently, the connection adapted on the almost Golden Riemannian structure was studied by F. Etayo et al. in ([24]). Some properties regarding the integrability of the Golden Riemannian structures were investigated by A. Gezer et al. in ([25]).

The metallic structure J is a polynomial structure, which was generally defined by S.I. Goldberg et al. in ([26, 27]), inspired by the metallic number given by [[sigma].sub.p,q] = (p + [square root of [p.sup.2] + 4q)/2, which is the positive solution of the equation [x.sup.2]-px-q = 0, for positive integer values of p and q. These apq numbers are members of the metallic means family or metallic proportions (as generalizations of the Golden number [phi] = (1 + [square root of 5])/2 = 1.618 ...), introduced by Vera W. de Spinadel ([28]). Some examples of the members of the metallic means family are the Silver mean, the Bronze mean, the Copper mean, the Nickel mean, and many others.

The purpose of the present paper is to investigate the properties of slant and semi-slant submanifolds in metallic (or Golden) Riemannian manifolds. We have found a relation between the slant angles [theta] of a submanifold M in a Riemannian manifold ([bar.M], [bar.g]) endowed with a metallic (or Golden) structure J and the slant angle 9 of the same submanifold M of the almost product Riemannian manifold (M,g,F). Moreover, we have found some integrability conditions for the distributions which are involved in such types of submanifolds in metallic and Golden Riemannian manifolds. We have also given some examples of semi-slant submanifolds in metallic and Golden Riemannian manifolds.

2. Preliminaries

First of all we review some basic formulas and definitions for the metallic and Golden structures defined on a Riemannian manifold.

Let [bar.M] be an m-dimensional manifold endowed with a tensor field J of type (1,1). We say that the structure J is a metallic structure if it verifies

[J.sup.2] = pJ + ql, (1)

for p, q [member of] [N.sup.*], where I is the identity operator on the Lie algebra [GAMMA](T[bar.M]) of vector fields on [bar.M]. In this situation, the pair ([bar.M], J) is called metallic manifold.

If p = q = 1 one obtains the Golden structure ([17]) determined by a (1,1)-tensor field J which verifies [J.sup.2] = J + I. In this case, ([bar.M], J) is called Golden manifold.

Moreover, if ([bar.M], [bar.g]) is a Riemannian manifold endowed with a metallic (or a Golden) structure J, such that the Riemannian metric [bar.g] is J-compatible, i.e.,

[bar.g](JX,Y) = [bar.g](X,JY), (2)

for any X, Y [member of] [GAMMA](T[bar.M]), then ([bar.g],J) is called a metallic (or a Golden) Riemannian structure and ([bar.M], [bar.g], J) is a metallic (or a Golden) Riemannian manifold.

We can remark that

[bar.g](JX, JY) = [bar.g] ([J.sup.2]X, Y) = p[bar.g] (JX, Y) + q[bar.g](X, Y), (3)

for any X, V [member of] [GAMMA](T[bar.M]).

Any metallic structure J on [bar.M] induces two almost product structures on this manifold ([21]):

[mathematical expression not reproducible]. (4)

Conversely, any almost product structure F on [bar.M] induces two metallic structures on [bar.M] ([21]):

[mathematical expression not reproducible]. (5)

If the almost product structure F is a Riemannian one, then [J.sub.1] and [J.sub.2] are also metallic Riemannian structures. Also, on a metallic manifold ([bar.M], /) there are two complementary distributions [D.sub.1] and [D.sub.2] corresponding to the projection operators P and Q ([21]), given by

[mathematical expression not reproducible] (6)

and the operators P and Q verify the following relations:

P + Q = I, [P.sup.2] = P, [Q.sup.2] = Q, PQ = QP = 0 (7)

and

JP = PJ = (p - [[sigma].sub.p,q)] P, JQ = QJ = [[sigma].sub.p,q)]Q. (8)

In particular, if p = q = 1, we obtain that every Golden structure J on [bar.M] induces two almost product structures on this manifold and conversely, an almost product structure F on [bar.M] induces two Golden structures on [bar.M] ([17, 19]).

3. Submanifolds of Metallic Riemannian Manifolds

In the next issues we assume that [bar.M] is an m -dimensional submanifold, isometrically immersed in the m-dimensional metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J) with m,m [member of] [N.sup.*] and m > m'. Wedenote by [T.sup.[perpendicular].sub.x]M the tangent space of [bar.M] in a point x [member of] [bar.M] and by Tf[bar.M] the normal space of [bar.M] in x. The tangent space [T.sub.x][bar.M] of [bar.M] can be decomposed into the direct sum: [T.sub.x][bar.M] = [T.sub.x][bar.M][direct sum][T.sup.[perpendicular].sub.x][bar.M], for any x [member of] [bar.M]. Let [i.sub.*] be the differential of the immersion i : [bar.M][right arrow][bar.M]. The induced Riemannian metric g on Mis given by [bar.g](X, Y) = [bar.g]([i.sub.*]X, [i.sub.*]Y), for any X, Y [member of] [GAMMA](TM), where [GAMMA](TM) denotes the set of all vector fields of M. For the simplification of the notations, in the rest of the paper we shall note by X the vector field [i.sub.*]X, for any X [member of] [GAMMA](TM).

We consider the decomposition into the tangential and normal parts of JX and JV, for any X [member of] [GAMMA](TM) and V [member of] [GAMMA]([T.sup.[perpendicular]]M), are given by

(i) JX = TX + NX, (ii) JV = tV + nV, (9)

where T : [GAMMA](TM) [GAMMA](TM),N : [GAMMA](TM) T([T.sup.[perpendicular]]M),t : T([T.sup.[perpendicular]]M) [GAMMA](TM) and n : [GAMMA]([T.sup.[perpendicular]]M) T([T.sup.[perpendicular]]M), with

[T.sub.x] := [(JX).sup.T],

NX := [(JX).sup.[perpendicular]],

tV := [(JV).sup.T],

nV := [(JV).sup.[perpendicular]]. (10)

We remark that the maps T and n are [bar.g]-symmetric ([29]):

(i) [bar.g]([T.sub.x],Y) = [bar.g](X,TY),

(ii) [bar.g](nU,V) = [bar.g](U,nV) (11)

and

[bar.g](NX,U) = [bar.g](X,tU), (12)

for any X,Y [member of] [GAMMA](TM) and U,V [member of] T([T.sup.[perpendicular]]M).

For an almost product structure F, the decompositions into tangential and normal parts of FX and FV, for any X [member of] [GAMMA](TM) and V [member of] T([T.sup.[perpendicular]]M), are given by([7])

(i) FX = fX + [omega]X,

(ii) FV = BV + CV, (13)

where f : [GAMMA](TM) [right arrow] [GAMMA](TM), [omega] : [GAMMA](TM) [right arrow]([T.sup.[perpendicular]]M),

B : [GAMMA]([T.sup.[perpendicular]]M) [right arrow][GAMMA](TM), C : [GAMMA]([T.sup.[perpendicular]]M) T([T.sup.[perpendicular]]M), with

fX := [(FX).sup.T],

wX := [(FX).sup.[perpendicular]],

BV := [(FV).sup.T],

CV := [(FV).sup.[perpendicular]]. (14)

The maps J and C are [bar.g]-symmetric ([16]):

[bar.g](fX,Y) = [bar.g](X, fY), (15)

[bar.g](CU,V) = [bar.g](U,CV)

[bar.g]([omega]X,V) = [bar.g](X,BV), (16)

for any X, Y [member of] [GAMMA](TM) and U,V [member of] [GAMMA]([T.sup.[perpendicular]]M).

Remark 1. Let ([bar.M], [bar.g]) be a Riemannian manifold endowed with an almost product structure F and let J be the metallic structure induced by F on [bar.M]. If M is a submanifold in the almost product Riemannian manifold ([bar.M], [bar.g], F), then

[mathematical expression not reproducible] (17)

[mathematical expression not reproducible], (18)

for any X [member of] [GAMMA](TM) and V [member of] [GAMMA]([T.sup.[perpendicular]]M).

Remark 2. Let ([bar.M], [bar.g]) be a Riemannian manifold endowed with an almost product structure F and let J be the Golden structure induced by F on [bar.M]. If M is a submanifold in the almost product Riemannian manifold ([bar.M], [bar.g], F), then

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible], (20)

for any X [member of] [GAMMA](TM) and 7 [member of] T([T.sup.[perpendicular]]M).

Let r = m - m' be the codimension of M in [bar.M] (where r, m, m [member of] [N.sup.*]). We fix a local orthonormal basis {[N.sub.1], ..., [N.sub.r]} of the normal space [T.sup.[perpendicular].sub.x]M. Hereafter we assume that the indices [alpha], [beta], gamma] run over the range {1, ..., r}.

For any x [member of] M and X [member of] [T.sub.x]M, the vector fields J([i.sub.*]X) and J[N.sub.[alpha]] can be decomposed into tangential and normal components ([21]):

[mathematical expression not reproducible], (21)

where ([alpha] [member of] {1, ..., r}), T is an (1,1)-tensor field on M, [[xi].sub.[alpha]] are vector fields on M, [u.sub.[alpha]] are 1-forms on M, and [u.sub.[alpha[beta]] is an rxr matrix of smooth real functions on M.

Using (9) and (21), we remark that

[mathematical expression not reproducible]. (22)

Theorem 3. The structure [summation] = (T, g, [u.sub.[alpha]], [[xi].sub.[alpha]], [([a.sub.[alpha[beta]).sub.r]) induced on the submanifold M by the metallic Riemannian structure ([bar.g], J) on [bar.M] satisfies the following equalities ([30]):

[mathematical expression not reproducible], (23)

[mathematical expression not reproducible], (24)

[mathematical expression not reproducible], (25)

[mathematical expression not reproducible] (26)

for any X [member of] [GAMMA](TM), where [[degrees].sub.[alpha][beta]] is the Kronecker delta and p, q are positive integers ([21]).

A structure [summation] = (T, g, [u.sub.[alpha]], [[xi].sub.[alpha]], [([a.sub.[alpha[beta]).sub.r]) induced on the submanifold M by the metallic Riemannian structure ([bar.g], J) defined on [bar.M] (determined by the (1, l)-tensor held T on M, the vector fields [[xi].sub.[alpha]] on M, the 1-forms [u.sub.[alpha]] on M, and the rxr matrix [([a.sub.[alpha][beta]]).sub.r] of smooth real functions on M) which verifies the relations (23), (24), (25), and (26) is called [SIGMA]-metallic Riemannian structure ([30]).

For p = q = 1, the structure [summation] = (T, g, [u.sub.[alpha]], [[xi].sub.[alpha]], [([a.sub.[alpha[beta]).sub.r]) is called [SIGMA]-Golden Riemannian structure.

Remark 4. If [summation] = (T, g, [u.sub.[alpha]], [[xi].sub.[alpha]], [([a.sub.[alpha[beta]).sub.r]) is the induced structure on the submanifold M by the metallic (or Golden) Riemannian structure ([bar.g],J) on M, then M is an invariant submanifold with respect to J if and only if (M, T, g) is a metallic (or Golden) Riemannian manifold, whenever T is nontrivial ([21]).

Let [bar.[nabla]] and [nabla] be the Levi-Civita connections on ([bar.M], [bar.g]) and (M, g), respectively. The Gauss and Weingarten formulas are given by

(i) [[bar.[nabla]].sub.X]Y = [[nabla].sub.X]Y + h(X,Y),

(ii) [[bar.[nabla]].sub.X]V = -[A.sub.V]Y + [nabla](X,Y), (27)

for any X,Y [member of] [GAMMA](TM) and V [member of] [GAMMA]([T.sup.[perpendicular]]M), where h is the second fundamental form, Av is the shape operator of M.Thesecond fundamental form h and the shape operator Av are related by

[bar.g](h(X,Y),V) = [bar.g]([A.sub.V]X,Y). (28)

Remark 5. Using a local orthonormal basis [[N.sub.1], ..., [N.sub.r]] of the normal space [T.sup.[perpendicular]], where r is the codimension of [bar.M] in [mathematical expression not reproducible], for any [alpha] [member of] {1, ..., r}, we obtain

[mathematical expression not reproducible], (29)

for any X, 7 [member of] [GAMMA](TM).

Remark 6. For a [member of] {1, ..., r}, the normal connection [nabla][perpendicular to]/X[N.sub.[alpha]] has the decomposition [nabla][perpendicular to]/X[N.sub.[alpha]] = [[summation].sup.r.sub.B=1] [l.sub.[alpha][beta]](X)[N.sub.[beta]], for any X [member of] [GAMMA](TM), where [([l.sub.[alpha][beta]).sub.r] is an rxr matrix of 1-forms on M. Moreover, from g([N.sub.[alpha], [N.sub.[beta]]) = [[delta].sub.[alpha[beta]], we obtain ([30]): [bar.g]([nabla][perpendicular to]/X[N.sub.[alpha], [N.sub.[beta]]) + [bar.g]([N.sub.[alpha]], [nabla] [perpendicular to]/X[N.sub.[beta]]) = 0, which is equivalent to [l.sub.[alpha][beta]] = -[l.sub.[beta][alpha]], for any [alpha], [beta] [member of] {1, ..., r} and X [member of] [GAMMA](TM).

The covariant derivatives of the tangential and normal parts of JX and JV are given by

[mathematical expression not reproducible], (30)

and

[mathematical expression not reproducible], (31)

for any X, Y [member of] [GAMMA](TM) and V [member of] [GAMMA]([T.sup.[perpendicular to]]M). From [bar.g](JX, 7) [bar.g](X, JY), it follows that

[bar.g](([[bar.[nabla]].sub.X]J)Y,Z) = [bar.g](Y, ([[bar.[nabla]].sub.X]J)Z), (32)

for any X, Y, Z [member of] [GAMMA](TM). Moreover, if M is an isometrically immersed submanifold of the metallic Riemannian manifold ([bar.M], [bar.g], J), then ([23])

[bar.g](([[bar.[nabla]].sub.X]T)Y,Z) = [bar.g](Y, ([[bar.[nabla]].sub.X]T)Z), (33)

for any X, Y, Z [member of] [GAMMA](TM).

Using an analogy of a locally product manifold ([31]), we can define locally metallic (or locally Golden) Riemannian manifold as follows ([30]).

Definition 7. If ([bar.M], [bar.g], J) is a metallic (or Golden) Riemannian manifold and J is parallel with respect to the Levi-Civita connection [bar.[nabla]] on [bar.M] (i.e., [bar.[nabla]]J = 0), we say that ([bar.M], [bar.g], J) is a locally metallic (or locally Golden) Riemannian manifold.

Proposition 8. If M is a submanifold ofa locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J), then

T([X,Y]) = [[nabla].sub.X]TY-[[nabla].sub.X]TX-[A.sub.NY] X + [A.sub.NX] Y (34)

and

N([X,Y]) = h (X, TY) - h (TX, Y) + [nabla][perpendicular to]/X NY [nabla][perpendicular to]/Y NX, (35)

for any X,Y [member of] [GAMMA](TM), where [nabla] is the Levi-Civita connection on M.

Proof. From ([bar.M], [bar.g], J) locally metallic (or locally Golden) Riemannian manifold, we have ([[bar.[nabla]].sub.X]/)Y = 0, for any X, Y [member of] [gamma](TM).

Thus, [[bar.[nabla]].sub.X](TY + NY) = J{[[nabla].sub.X]Y + h(X,Y)}, which is equivalent to

[mathematical expression not reproducible], (36)

Taking the normal and the tangential components of this equality, we get

N ([[nabla].sub.X]Y) - [nabla][perpendicular to]/X NY = h (X, TY) - nh (X, Y) (37)

and

[[nabla].sub.X]TY - T([[nabla].sub.X]Y) = [A.sub.NY]X + th(X,Y). (38)

Interchanging X and Y and subtracting these equalities, we obtain the tangential and normal components of [X, Y] = [[nabla].sub.X] - [[nabla].sub.Y]X, which give us (34) and (35).

From (30), (31), (37), and (38) we obtain the following.

Proposition 9. If M is a submanifold of a locally metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J), then the covariant derivatives of T and N verify

[mathematical expression not reproducible], (39)

and

[mathematical expression not reproducible], (40)

for any X, Y [member of] [GAMMA](TM) and V [member of] T([T.sup.[perpendicular]]M).

Proposition 10. If M is an n-dimensional submanifold of codimension r in a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J), then the structure [summation] = (T, g, [u.sub.[alpha]], [[Xi].sub.[alpha]], [([a.sub.[alpha][beta]]).sub.r] induced on M by the metallic (or Golden) Riemannian structure ([bar.g], J) has the following properties ([30]):

[mathematical expression not reproducible], (41)

[mathematical expression not reproducible], (42)

for any X,Y [member of] [GAMMA](TM).

Proof. From [bar.[nabla]]J = 0 we obtain YXJY = J(VXY), for any X,Y 6 [GAMMA](TM). Using (27)(i), (29), and (21)(ii), we get

[mathematical expression not reproducible], (43)

for any X,Y [member of] [GAMMA](TM). Identifying the tangential and normal components, respectively, of the last two equalities, we get (41) and (42).

Using (34), (35), (41), and (42), we obtain the following.

Proposition 11. If M is a submanifold of a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J), then

[mathematical expression not reproducible] (44)

[mathematical expression not reproducible], (45)

for any X,Y [member of] [GAMMA](TM), where V is the Levi-Civita connection on M.

4. Slant Submanifolds in Metallic or Golden Riemannian Manifolds

Let M be an m'-dimensional submanifold, isometrically immersed in an m-dimensional metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J), where m, m' [member of] [N.sup.*] and m > m . Using the Cauchy-Schwartz inequality ([6]), we have

[absolute value of [bar.g](JX,TX)][less than or equal to][parallel]JX[parallel] * [parallel]TX[parallel], (46)

for any X [member of] [GAMMA](TM). Thus, there exists a function d : [GAMMA](TM) [right arrow] [0, [pi]], such that

[mathematical expression not reproducible], (47)

for any x [member of] M and any nonzero tangent vector [X.sub.x] [member of] [T.sub.x]M. The angle [theta]([X.sub.x]) between J[X.sub.x] and [T.sub.x]M is called the Wirtinger angle of X and it verifies

[mathematical expression not reproducible]. (48)

Definition 12 (see [29]). A submanifold M in a metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J) is called slant submanifold if the angle d([X.sub.x]) between J[X.sub.x] and [T.sub.x]M is constant, for any x [member of] M and [X.sub.x] [member of] [T.sub.x]M. In such a case, [theta] =: [theta]([X.sub.x]) is called the slant angle of M in [bar.M], and it verifies

[mathematical expression not reproducible]. (49)

The immersion i: M [right arrow] [bar.M] is named slant immersion of M in [bar.M].

Remark 13. The invariant and anti-invariant submanifolds in the metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J) are particular cases of slant submanifolds with the slant angle [theta] = 0 and [theta] = [pi]/2, respectively. A slant submanifold M in [bar.M], which is neither invariant nor anti-invariant, is called proper slant submanifold and the immersion i : M [right arrow] [bar.M] is called proper slant immersion.

Proposition 14. ([29]) Let M be an isometrically immersed submanifold of the metallic Riemannian manifold ([bar.M], [bar.g], J). If M is a slant submanifold with the slant angle 9, then, for any X, Y [member of] [GAMMA](TM) we get

[bar.g](TX], TY) = [cos.sup.2][theta][p[bar.g](X, TY) + q[bar.g](X, Y)] (50)

[bar.g] (NX, NY) = [sin.sup.2][theta][p[bar.g] (X, TY) + q[bar.g] (X, Y)]. (51)

Moreover, we have

[T.sup.2] = [cos.sup.2]O (pT + qI), (52)

where I is the identity on [GAMMA](TM) and

[nabla]([T.sup.2]) = p[cos.sup.2][theta]([nabla]T). (53)

Remark 15. Let I be the identity on [GAMMA](TM). From (23) and (52), we have

[mathematical expression not reproducible]. (54)

Proposition 16. If M is an isometrically immersed slant submanifold of the Golden Riemannian manifold ([bar.M], [bar.g], J) with the slant angle 9, then

[bar.g] (TX, TY) = [cos.sup.2][theta][p[bar.g] (X, TY) + [bar.g] (X, Y)]. (55)

[bar.g] (NX, NY) = [sin.sup.2][theta][p[bar.g] (X, TY) + [bar.g] (X, Y)]. (56)

for any X, Y [member of] [GAMMA](TM). If I is the identity on [GAMMA](TM), we have

[T.sup.2] = [cos.sup.2][theta] (T + I), [nabla] ([T.sup.2]) = [cos.sup.2][theta] ([nabla]T), (57)

[sin.sup.2][theta](T + I) = [r.summation over ([alpha]=1)][u.sub.[alpha]] [cross product][[xi].sub.[alpha]]. (58)

Definition 17 (see [8]). A submanifold M in an almost product Riemannian manifold ([bar.M], [bar.g], F) is a slant submanifold if the angle 0([X.sub.x]) between J[X.sub.x] and [T.sub.x]M is constant, for any x [member of] M and [X.sub.x] [member of] [T.sub.x]M. In such a case, [??] =: [??]([X.sub.x]) is called the slant angle of the submanifold M in M and it verifies

[mathematical expression not reproducible]. (59)

Proposition 18 (see [16]). If M is a slant submanifold isometrically immersed in an almost product Riemannian manifold ([bar.M], [bar.g], F) with the slant angle [??] then, for any X, Y [member of] [GAMMA](TM), we get

[mathematical expression not reproducible]. (60)

In the next proposition we find a relation between the slant angles 9 of the submanifold M in the metallic Riemannian manifold ([bar.M], [bar.g], J) and the slant angle 9 of the submanifold M in the almost product Riemannian manifold ([bar.M], [bar.g], F).

Theorem 19. Let M be a submanifold in the Riemannian manifold ([bar.M], [bar.g]) endowed with an almost product structure F on [bar.M] and let J be the induced metallic structure by F on ([bar.M], [bar.g]). If M is a slant submanifold in the almost product Riemannian manifold ([bar.M], [bar.g], F) with the slant angle 9 and F j= -I (I is the identity on [GAMMA](TM)) and J = ((2[[sigma].sub.p,q] - p)/2)F + (p/2)I, then M is a slant submanifold in the metallic Riemannian manifold [bar.M], [bar.g], J) with slant angle 9 given by

[mathematical expression not reproducible]. (61)

Proof. From (17)(ii), we obtain [bar.g](NX,NY) = [((2[[sigma].sub.p,q] - p).sup.2]/ 4)[bar.g]([omega]X, [omega]Y), for any X, Y [member of] [GAMMA](TM). From (51) and (60)(ii) and [bar.g](X, JY) = [bar.g](X, TY), we get

[(2[[sigma].sub.p,q] - p).sup.2]/4 [bar.g](X, Y)[sin.sup.2][??] = [p[bar.g] (X, JY) + q[bar.g] (X, Y)] [sin.sup.2][theta], (62)

for any X, Y [member of] [GAMMA](TM). Using J = (p/2)I + ((2[[sigma].sub.p,q] - p)/2)F, we have

[mathematical expression not reproducible], (63)

for any X, Y [member of] [GAMMA](TM). Replacing Y by FY and using [F.sup.2]Y = 7, for any Y [member of] [GAMMA](TM), we obtain

[mathematical expression not reproducible], (64)

for any X,Y [member of] [GAMMA](TM). Summing equalities (63) and (64), we obtain

[mathematical expression not reproducible], (65)

for any X,Y [member of] [GAMMA](TM). Using q + p[[sigma].sub.p,q] = [[[sigma].sup.2].sub.p,q], FY [not equal to] -7, and [theta], [??][member of] [0, [pi]) in (65), we get (61).

In particular, for p = q = 1, we obtain the relation between slant angle [theta] of the immersed submanifold M in a Golden Riemannian manifold ([bar.M], [bar.g], J) and the slant angle [??] of M immersed in the almost product Riemannian manifold ([bar.M], [bar.g], F).

Proposition 20. Let M be a submanifold in the Riemannian manifold ([bar.M], [bar.g]) endowed with an almost product structure F on [bar.M] and let J be the induced Golden structure by F on ([bar.M], [bar.g]). If M is a slant submanifold in the almost product Riemannian manifold ([bar.M], [bar.g], F) with the slant angle [??] and F [not equal to] -I (I is the identity on [GAMMA](TM)) and J = ((2[phi] - 1)/2)F + (1/2)7, then M is a slant submanifold in the Golden Riemannian manifold ([bar.M], [bar.g], J) with slant angle [theta] given by

sin [theta] = 2[phi] - 1/2[phi] sin [??], (66)

where [phi] = (1 + [square root of 5])/2 is the Golden number.

5. Semi-Slant Submanifolds in Metallic or Golden Riemannian Manifolds

We define the slant distribution of a metallic (or Golden) Riemannian manifold, using a similar definition as for Riemannian product manifold ([7,16]).

Definition 21. Let M be an immersed submanifold of a metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J). A differentiable distribution D on M is called a slant distribution if the angle [[theta].sub.D] between [JX.sub.x] and the vector subspace [D.sub.x] is constant, for any x [member of] M and any nonzero vector field [X.sub.x] [member of] T([D.sub.x]). The constant angle [[theta].sub.D] is called the slant angle of the distribution D.

Proposition 22. Let D be a differentiable distribution on a submanifold M of a metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J). The distribution D is a slant distribution if and only if there exists a constant [lambda] [member of] [0,1] such that

[([P.sub.D]T).sup.2] X = [lambda] (p[P.sub.D]TX + qX) , (67)

for any X [member of] [GAMMA](D), where [P.sub.D] is the orthogonal projection on D. Moreover, if [[theta].sub.D] is the slant angle of D, then it satisfies [lambda] = [cos.sup.2] [[theta].sub.D].

Proof. If the distribution D is a slant distribution on M, by using

[mathematical expression not reproducible], (68)

we get [bar.g]([P.sub.D][T.sub.x], [P.sub.D][T.sub.x]) = [cos.sup.2][[theta].sub.D] [bar.g](p[P.sub.D][T.sub.x] + qX, X), for any X [member of] [GAMMA](D) and we obtain (67).

Conversely, if there exists a constant [lambda] [member of] [0, 1] such that (67) holds for any X [member of] [GAMMA](D), we obtain [bar.g](JX, [P.sub.D][T.sub.x]) = [bar.g](X, [JP.sub.D][T.sub.x]) = [bar.g](X, [([P.sub.D]T).sup.2]X) and [bar.g](JX, [P.sub.D][T.sub.x]) = [lambda][bar.g](X, p[P.sub.D][T.sub.x] + qX) = X[bar.g](X,pJ[T.sub.x] + qX) = X[bar.g](XJ2X) = [lambda][bar.g](JX, JX). Thus, cos [[theta].sub.D] = [lambda]([parallel]JX[parallel]/[parallel][P.sub.D][T.sub.x] [parallel]), and using cos [theta] = [parallel][P.sub.D]TX[parallel]/[parallel]JX[parallel] we get [cos.sup.2][[theta].sub.D] = [lambda]. Thus, [cos.sup.2] [[theta].sub.D] is constant and D is a slant distribution on M.

Definition 23. Let M be an immersed submanifold in a metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J). We say that M is a bi-slant submanifold of [bar.M] if there exist two orthogonal differentiable distributions [D.sub.1] and [D.sub.2] on M such that TM = [D.sub.1] [direct sum] [D.sub.2] and [D.sub.1], [D.sub.2] are slant distributions with the slant angles [[theta].sub.1] and [[theta].sub.2], respectively.

For a differentiable distribution [D.sub.1] on M, we denote by [D.sub.2] := [D.sub.[perpendicular]] the orthogonal distribution of [D.sub.1] in M (i.e., TM = [D.sub.1] [direct sum] [D.sub.2]). Let [P.sub.1] and [P.sub.2] be the orthogonal projections on [D.sub.1] and [D.sub.2]. Thus, for any X [member of] [GAMMA](TM), we can consider the decomposition of X = [P.sub.1]X + [P.sub.2]X, where [P.sub.1]X [member of] [GAMMA]([D.sub.1]) and [P.sub.2]X [member of] T([D.sub.2]).

If M is a bi-slant submanifold of a metallic Riemannian manifold ([bar.M], [bar.g], J) with the orthogonal distribution [D.sub.1] and [D.sub.2] and the slant angles [[theta].sub.1] and [[theta].sub.2], respectively, then JX = [P.sub.1]TX + [P.sub.2]TX + NX = [TP.sub.1]X + [TP.sub.2]X + [NP.sub.1]X + [NP.sub.2]X, for any X [member of] [GAMMA](TM). In a similar manner as in ([16]), we can prove the following.

Proposition 24. If M is a bi-slant submanifold in a metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J), with the slant angles [[theta].sub.1] = [[theta].sub.2] = [theta] and g(JX, Y) = 0, for any X [member of] [GAMMA]([D.sub.1]) and Y [member of] [GAMMA]([D.sub.2]), then M is a slant submanifold in the metallic Riemannian manifold ([bar.M], [bar.g], J) with the slant angle [THETA].

Proof. From [bar.g](JX, Y) = [bar.g](TX, Y) = 0, for any X [member of] [GAMMA]([D.sub.1]) and Y [member of] [GAMMA]([D.sub.2]), it follows that [bar.g](X, JY) = [bar.g](X, TY) = 0. Thus, we obtain [T.sub.x] [member of] [GAMMA]([D.sub.1]), for any X [member of] [GAMMA]([D.sub.1]) and TY [member of] [GAMMA]([D.sub.2]), for any Y [member of] [GAMMA]([D.sub.2]). Moreover, using the projections of any X [member of] [GAMMA](TM) on [GAMMA]([D.sub.1]) and [GAMMA]([D.sub.2]), respectively, we obtain the decomposition X = [P.sub.1]X + [P.sub.2]X, where [P.sub.1]X [member of] [GAMMA]([D.sub.1])) and [P.sub.2]X [member of] [GAMMA]([D.sub.2]).

From [bar.g]([TP.sub.i]X, [TP.sub.i](X) = [cos.sup.2][[theta].sub.i]g([JP.sub.i]X, [JP.sub.i]X) (for i [member of] {1, 2}) and using [[theta].sub.1] = [[theta].sub.2] = [theta], we obtain [bar.g](TX, Tx)/[bar.g](JX, JX) = [cos.sup.2][theta], for any X [member of] [GAMMA](TM). Thus, M is a slant submanifold in the metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J) with the slant angle [theta].

If M is a bi-slant submanifold of a manifold [bar.M], for particular values of the angles [[theta].sub.1] = 0 and [[theta].sub.2] [not equal to] 0, we obtain the following.

Definition 25. An immersed submanifold M in a metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J) is a semi-slant submanifold if there exist two orthogonal distributions [D.sub.1] and [D.sub.2] on M such that

(1) TM admits the orthogonal direct decomposition TM = [D.sub.1] [direct sum] [D.sub.2];

(2) The distribution [D.sub.1] is invariant distribution (i.e., J([D.sub.1]) = [D.sub.1]);

(3) The distribution [D.sub.2] is slant with angle [theta] = 0.

Moreover, if dim ([D.sub.1]) x dim ([D.sub.2]) [not equal to] 0, then M is a proper semi-slant submanifold.

Remark 26. If M is a semi-slant submanifold of a metallic Riemannian manifold ([bar.M], [bar.g], J) with the slant angle [theta] of the distributions [D.sub.2], then we get that

(1) M is an invariant submanifold if dim ([D.sub.2]) = 0;

(2) M is an anti-invariant submanifold if dim ([D.sub.1]) = 0 and [theta] = [pi]/2;

(3) M is a semi-invariant submanifold if [D.sub.2] is anti-invariant (i.e., [theta] = [pi]/2).

If M is a semi-slant submanifold in a metallic (or Golden) Riemannian manifold ([bar.M], [bar.g], J) then, for any X [member of] [GAMMA](TM),

JX = [TP.sub.1] X + [TP.sub.2] x [NP.sub.1] x

= [P.sub.1] TX + [P.sub.2] TX + [NP.sub.2] X, (69)

(i) [JP.sub.1]X = [TP.sub.1]X,

(ii) [NP.sub.1]X = 0,

(iii) [TP.sub.2] X [member of] [GAMMA]([D.sub.2]). (70)

Moreover, we have [bar,g]([JP.sub.2]X, [TP.sub.2] X) = cos [theta](X) [parallel][TP.sub.2]X[parallel] x [parallel][TP.sub.2]X[parallel] and the cosine of the slant angle [theta](X) of the distribution [D.sub.2] is constant, for any nonzero X [member of] [GAMMA](TM). If [theta](X) =: [theta], for any nonzero X [member of] [GAMMA](TM) we get

[mathematical expression not reproducible]. (71)

Proposition 27. If M is a semi-slant submanifold of the metallic Riemannian manifold ([bar.M], [bar.g], J) with the slant angle 9 of the distribution [D.sub.2] then, for any X, Y [member of] [GAMMA](TM), we get

[mathematical expression not reproducible], (72)

[mathematical expression not reproducible]. (73)

Proof. Taking X + Y in (71) we have [bat.g]([TP.sub.2] X, [TP.sub.2] Y) = [cos.sup.2] [theta][bar.g]([JP.sub.2]X, [JP.sub.2] Y) = [cos.sup.2] [theta][p[bar.g]([JP.sub.2]X, P[P.sub.2]Y) + q[bar.g]([P.sub.2]X, [P.sub.2]Y)], for any X, Y [member of] [GAMMA](TM) and using (70)(iii) we get (72). From (70)(ii) we get [TP.sub.2]X = [JP.sub.2]X - NX, for any X [member of] [GAMMA](TM). Thus, we obtain [bar.g]([TP.sub.2]X, [TP.sub.2]Y) = g([JP.sub.2]X, [JP.sub.2]Y) - [bar.g](NX, NY), for any X,Y [member of] [GAMMA](TM) and it implies (73).

Remark 28. A semi-slant submanifold M of a Golden Riemannian manifold ([bar.M], [bar.g], J) with the slant angle [theta] of the distribution [D.sub.2] verifies

[mathematical expression not reproducible], (74)

[mathematical expression not reproducible]. (75)

for any X, Y [member of] [GAMMA](TM).

Proposition 29. Let M be a semi-slant submanifold of a metallic Riemannian manifold ([bar.M], [bar.g], J) with the slant angle 9 of the distribution [D.sub.2]. Then

[([TP.sub.2]).sup.2] = [cos.sup.2] [theta](p[TP.sub.2] + qI), (76)

where I is the identity on [GAMMA]([D.sub.2]) and

[nabla]([([TP.sub.2]).sup.2]) = p[cos.sup.2][theta][nabla]([TP.sub.2]). (77)

Proof. Using [bar.g]([TP.sub.2]X, [TP.sub.2]Y) = [bar.g]([([TP.sub.2]).sup.2]X, [P.sub.2]Y), for any X, Y [member of] [GAMMA](TM) and (72), we obtain (76). Moreover, we have ([[nabla].sub.x][([TP.sub.2]).sup.2])Y = [cos.sup.2][theta](p([[nabla].sub.x][TP.sub.2])Y + q([[nabla].sub.x]I)Y) = p [cos.sup.2][theta][[nabla].sub.x] ([P.sub.2]T)Y, for any X [member of] [GAMMA]([D.sub.2]) and Y [member of] [GAMMA](TM). For the identity I on [GAMMA]([D.sub.2]) we have ([[nabla].sub.x]I)[P.sub.2]Y = 0; thus, we get (77).

Remark 30. A semi-slant submanifold M of a Golden Riemannian manifold ([bar.M], [bar.g], J) with the slant angle [theta] of the distribution [D.sub.2] verifies

[([TP.sub.2]).sup.2] = [cos.sup.2] [theta] ([TP.sub.2] + I), (78)

where I is the identity on [GAMMA]([D.sub.2]) and

[nabla] ([([TP.sub.2]).sup.2]) = [cos.sup.2] [theta][nabla] ([TP.sub.2]). (79)

Proposition 31. Let M be an immersed submanifold of a metallic Riemannian manifold ([bar.M], [bar.g], J). Then M is a semislant submanifold in [bar.M] if and only ifexists a constant [lambda] [member of] [0, 1) such that D = {X [member of] [GAMMA](TM)|[T.sup.2]X = [lambda](p[T.sub.x] + qX)} is a distribution and NX = 0, for any X [member of] [GAMMA](TM) orthogonal to D, where p and q are given in (1).

Proof. If we consider M a semi-slant submanifold of the metallic Riemannian manifold ([bar.M], [bar.g], J) then, in (72) we put [lambda] = [cos.sup.2][theta] [member of] [0, 1). Thus, we obtain [T.sup.2]X = X(p[T.sub.x] + qX) and we get [D.sub.2] [subset not equal to] D. For a nonzero vector field X [member of] [GAMMA](D), let X = [X.sub.1] + [X.sub.2], where [X.sub.1] = [P.sub.1]X [member of] [GAMMA]([D.sub.1]) and [X.sub.2] = [P.sub.2]X [member of] [GAMMA]([D.sub.2]). Because [D.sub.1] is invariant, then [JX.sub.1] = [TX.sub.1] and using the property of the metallic structure (1), we obtain p[T X.sub.1] + q[X X.sub.1] = Pj[X X.sub.1] + q[X X.sub.1] = [J.sup.2][X.sub.1] = [T.sup.2][X.sub.1] = [lambda](p[TX.sub.1] + q[X.sub.1]), which implies (p[TX.sub.1] + q[X.sub.1])([lambda] - 1) = 0. Because X [member of] [0, 1), we obtain [TX.sub.1] = -(q/p)[X.sub.1] and we get [X.sub.1] = 0([q.sup.2]/[p.sup.2] [not equal to] 0 because p and q are nonzero natural numbers). Thus, we obtain X [member of] [GAMMA]([D.sub.2]) and D [subset not equal to] [D.sub.2], which implies D = [D.sub.2]. Therefore, [D.sub.1] = [D.sup.[perpendicular].

Conversely, if there exists a real number X [member of] [0, 1) such that we have [T.sup.2]X = [lambda](p[T.sub.x] + qX), for any X [member of] [GAMMA](D), it follows that [cos.sup.2]([theta](X)) = [lambda] which implies that [theta](X) = arccos([square root of [LAMBDA]]) does not depend on X. We can consider the orthogonal direct sum TM = D [direct sum] [D.sup.[perpendicular]]. For Y [member of] [GAMMA]([D.sup.[perpendicular]]) := [GAMMA]([D.sub.1]) and X [member of] [GAMMA](D) (with D := [D.sub.2]), we have [bar.g](X, [J.sup.2]Y) = [bar.g](X, T(JY)) = [bar.g]([T.sub.x], JY) = [bar.g]([T.sub.x], TY) = [bar.g] ([T.sup.2]X, Y) = [lambda][p[bar.g]([T.sub.x], Y) + q[bar.g](X, Y)]. From [bar.g](X, [J.sup.2]Y) = p[bar.g](X, JY) + q[bar.g](X, Y) and [bar.g](X, Y) = 0, we obtain [bar.g](X, JY) = [lambda][bar.g](X, TY) and this implies (1 - [lambda])TY [member of] [GAMMA]([D.sup.[perpendicular]]) and TY [member of] [GAMMA]([D.sup.[perpendicular]]). Thus, JY [member of] [GAMMA]([D.sup.[perpendicular]]), for any X [member of] [GAMMA]([D.sup.[perpendicular]]) and we obtain that [D.sup.[perpendicular]] is an invariant distribution.

Remark 32. An immersed submanifold M of the Golden Riemannian manifold ([bar.M], [bar.g], J) is a semi-slant submanifold in [bar.M] if and only if there exists a constant [lambda] [member of] [0,1) such that

D = {X [member of] [GAMMA] (TM) | [T.sup.2] X = [lambda] (TX + X)} (80)

is a distribution and NX = 0, for any X [member of] [GAMMA](TM) orthogonal to D.

Examples 1. Let [R.sup.7] be the Euclidean space endowed with the usual Euclidean metric <*, *>. Let f : M [right arrow] [R.sup.7] be the immersion given by

f{u, [t.sub.1], [t.sub.2]}

= (w cos [t.sub.1] u sin [t.sub.1], u cos [t.sub.2], u sin [t.sub.2], u, [t.sub.1], [t.sub.2]), (81)

where M := {(u, [t.sub.1], [t.sub.2])/u > 0, [t.sub.1], [t.sub.2] [member of] [0, [pi]/2]}.

We can find a local orthonormal frame on TM given by

[mathematical expression not reproducible]. (82)

We define the metallic structure J : [R.sup.7] [right arrow] [R.sup.7] given by

[mathematical expression not reproducible], (83)

for i [member of] {1, 2, 3, 4} and j [member of] {1, 2, 3} where [sigma] := [[sigma].sub.p,q] = (p +

[square root of [p.sup.2] + 4q])/2 is the metallic number (p, q [member of] [N.sup.*]) and [bar.[sigma]] = p-[sigma].

We can verify that [bar.[nabla]]J = 0 and we obtain that ([R.sup.7], <x, x}, J) is a locally metallic Riemannian manifold.

Moreover, we have J[Z.sub.2] = [sigma][Z.sub.2], J[Z.sub.3] = [bar.[sigma]] [Z.sub.3], and

[mathematical expression not reproducible]. (84)

We can verify that [mathematical expression not reproducible],

[mathematical expression not reproducible]. (85)

On the other hand, we have <J[Z.sub.1], [Z.sub.j]> = [sigma] + 2[bar.[sigma]] and <J[Z.sub.i], [Z.sub.j]> = 0, for any i [not equal to] j, where i, j [member of] {1, 2, 3}. We remark that

[mathematical expression not reproducible]. (86)

We define the distributions [D.sub.1] = span{[Z.sub.2], [Z.sub.3]} and [D.sub.2] = span{[Z.sub.1]}. We have J([D.sub.1]) [subset] [D.sub.1] (i.e., [D.sub.1] is an invariant distribution with respect to J). The Riemannian metric tensor of [D.sub.1] [direct sum] [D.sub.2] is given by g = 3[du.sup.2 + ([u.sup.2 + 1)([dt.sup.2.sub.1] + [dt.sup.2.sub.2]). Thus, [D.sub.1] [direct sum] [D.sub.2] is a warped product semi-slant submanifold in the locally metallic Riemannian manifold ([R.sup.7], <x, x>, J) with the

slant angle arccos(([sigma] + 2[bat.[sigma]])/ [square root of 3([[sigma].sup.2] + 2[[bar.[sigma]].sup.2])).

If J is the Golden structure J : [R.sup.7] [right arrow] [R.sup.7] given by

[mathematical expression not reproducible], (87)

for i [member of] {1, 2, 3, 4} and j [member of] {1, 2, 3}, where [phi] := (1 + [square root of 5])/2 is the Golden number and [bar.[phi]] = 1 - [phi], in the same manner we obtain

[mathematical expression not reproducible]. (88)

We define the distributions [D.sub.1] = span{[Z.sub.2], [Z.sub.3]} and [D.sub.2] = span{[Z.sub.1]}. We obtain that [D.sub.1] [direct sum] [D.sub.2] is a warped product semi-slant submanifold in the locally Golden Riemannian manifold ([R.sup.7], <x, x>, J), with the slant angle arccos(([phi] + 2[bar.[phi]])/ [square root of 3([[phi].sup.2] + 2 [[bar.[phi]].sup.2])).

Examples 2. Let M := {(u, [[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.n])/u > 0, [[alpha].sub.i] [member of] [0, [pi]/2], i [member of] {1, ..., n}} and f: M [right arrow] [R.sup.3n+1] is the immersion given by

f(u, [[alpha].sub.1], ..., [[alpha].sub.n]) = (u cos [[alpha].sub.1], ..., u cos [[alpha].sub.n], u sin [[alpha].sub.1], ..., u

sin [[alpha].sub.n], [[alpha].sub.1], ..., [[alpha].sub.n], u). (89)

We can find a local orthonormal frame of the submanifold TM in [R.sup.3n+1], spanned by the vectors:

[mathematical expression not reproducible], (90)

for any i [member of] {1, ..., n}. We remark that [[parallel][Z.sub.0][parallel].sup.2] = n + 1, [[parallel][Z.sub.i][parallel].sup.2] = [u.sup.2] + 1, for any i [member of] {1, ..., n}, [Z.sub.0] [perpendicular] [Z.sub.i], for any i [member of] {1, ..., n}, and [Z.sub.i] [perpendicular] [Z.sub.j], for i = j, where i, j [member of] {1, ..., n}.

Let J : [R.sup.3n+1] [right arrow] [R.sup.3n+1] be the (1, 1)-tensor field defined by

[mathematical expression not reproducible], (91)

where [sigma] := [[sigma].sub.p,q] is the metallic number and [bar.[sigma]] = p - [sigma]. It is easy to verify that J is a metallic structure on [R.sup.3n+1] (i.e., [J.sup.2] = pJ + qI). The metric [bar.g], given by the scalar product <x, x> on [R.sup.3n+1], is J compatible and ([R.sup.3n+1], [bar.g], J) is a metallic Riemannian manifold.

Also, J[Z.sub.0] = [sigma] [[summation].sup.n.sub.j=1] (cos [[alpha].sub.j]([partial derivative]/[partial derivative][x.sub.j]) + sin [[alpha].sub.j]([partial derivative]/[partial derivative][x.sub.n+j])) + [sigma]([partial derivative]/[partial derivative][x.sub.3n+1]) and, for any i [member of] {1, ..., n}, we get J[Z.sub.i] = [sigma](-u sin [[alpha].sub.i]([partial derivative]/[partial derivative][x.sub.i]) + u cos [[alpha].sub.i]([partial derivative]/[partial derivative][x.sub.n+i]) + [partial derivative]/[partial derivative][x.sub.2n+i]) = [sigma][Z.sub.i]. We can verify that J[Z.sub.0] is orthogonal to span{[Z.sub.1], ..., [Z.sub.n]} and cos [mathematical expression not reproducible].

If we consider the distributions [D.sub.1] = span{[Z.sub.i]/i [member of] {1, ..., n}} and [D.sub.2] = span{[Z.sub.0]}, then [D.sub.1] [direct sum] [D.sub.2] is a semislant submanifold in the metallic Riemannian manifold ([R.sup.3n+1], <x, x), J), with the Riemannian metric tensor g = (n + 1)[du.sup.2] + ([u.sup.2] + 1) [[summation].sup.n.sub.j=1] d[alpha].sup.2.sub.j].

6. On the Integrability of the Distributions of Semi-Slant Submanifolds

In this section we investigate the conditions for the integrability of the distributions of semi-slant submanifolds in metallic (or Golden) Riemannian manifolds.

Theorem 33. If M is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J), then

(i) the distribution [D.sub.1] is integrable if and only if

([[nabla].sub.Y] [u.sub.[alpha]]) (X) = ([[nabla].sub.X] [u.sub.[alpha]]) (Y), (92)

for any X, Y [member of] [GAMMA]([D.sub.1]);

(ii) the distribution [D.sub.2] is integrable if and only if

[mathematical expression not reproducible], (93)

for any X,Y [member of] [GAMMA]([D.sub.2]).

Proof, (i) For X, Y [member of] [GAMMA]([D.sub.1]), we have X = [P.sub.1]X and Y = [P.sub.1]Y. The distribution [D.sub.1] is integrable if and only if [X, Y] [member of] [GAMMA]([D.sub.1]), which is equivalent to N([X, Y]) = 0, for any X, Y [member of] [GAMMA]([D.sub.1]). From J([D.sub.1]) [subset] [D.sub.1] we obtain NX = NY = 0 and from (22)(i) we get [u.sub.[alpha]](X)[l.sub.[alpha][beta]](Y) = [u.sub.[alpha]](Y) [l.sub.[alpha][beta]](X) = 0. Thus, using (45) we have the distribution [D.sub.1] is integrable if and only if (92) holds.

(ii) For X, Y [member of] [GAMMA]([D.sub.2]), we have X = [P.sub.2]X, Y = [P.sub.2]Y. The distribution [D.sub.2] is integrable if and only if [X, Y] [member of] [GAMMA]([D.sub.2]), which is equivalent to [P.sub.1]T([X, Y]) = 0. Thus, from (44), we obtain that the distribution [D.sub.2] is integrable if and only if (93) holds.

Remark 34. If M is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J), then

(i) the distribution [D.sub.1] is integrable if and only if

h(X, TY) = h([T.sub.x], Y), (94)

for any X, Y [member of] [GAMMA]([D.sub.1]);

(ii) the distribution [D.sub.1] is integrable if and only if the shape operator of M satisfies

[JA.sub.V] X = [A.sub.V] JX, (95)

for any X [member of] [GAMMA]([D.sub.1]) and V [member of] [GAMMA]([T.sup.[perpendicular]]M);

(iii) the distribution [D.sub.2] is integrable if and only if

[P.sub.1] ([[nabla].sub.X]TY - [[nabla].sub.Y]Tx) = [P.sub.1] ([A.sub.NY]X - [A.sub.NX]Y), (96)

for any X, Y [member of] [GAMMA]([D.sub.2]).

Proof. (i) For any X, Y [member of] [GAMMA]([D.sub.1]), we have [X, Y] [member of] [GAMMA]([D.sub.1]) if and only if N([X, Y]) = 0 and from (35), we obtain (94).

(ii) For any X, Y [member of] [GAMMA]([D.sub.1]) and any V [member of] [GAMMA]([T.sup.[perpendicular]]M), from (28) and (2) we have

g([JA.sub.V]X - [A.sub.V] JX, Y)

= g(h(X, JY) - h(JX, Y), V). (97)

From (35) and NX = NY = 0 (because J([D.sub.1]) [subset] [D.sub.1])we have

g ([JA.sub.V]X - [A.sub.V]JX, Y) = g(N ([X, Y]), V) = 0, (98)

for any X, Y [member of] [GAMMA]([D.sub.1]) and any V [member of] [GAMMA]([T.sup.[perpendicular]]M). Thus, we have [X, Y] [member of] [GAMMA]([D.sub.1]).

(iii) For any X, Y [member of] [GAMMA]([D.sub.2]), we have X = [P.sub.2] X and Y = [P.sub.2]Y. The distribution [D.sub.2] is integrable if and only if [X, Y] [member of] [GAMMA]([D.sub.2]), which is equivalent to T([X, Y]) [member of] [GAMMA]([D.sub.2]) or [P.sub.1]T([X, Y]) = 0. Thus, from (34), we obtain that [D.sub.2] is integrable if and only if (96) holds.

Theorem 35. Let M be a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J). If [nabla]T = 0, then the distributions [D.sub.1] and [D.sub.2] are integrable.

Proof. First of all, we consider X,Y [member of] [GAMMA]([D.sub.1]) and we prove [X, Y] [member of] [GAMMA]([D.sub.1]). For any Y [member of] [GAMMA]([D.sub.1]), we get NY = 0 and using [nabla]T = 0 in (39)(i) we obtain th(X, Y) = 0, for any X, Y [member of] [GAMMA]([D.sub.1]), which implies Jh(X, Y) = nh(X, Y). From

[bar.g](th (X, Y), Z) = [bar.g](Jh (X, Y), Z)

= [bar.g](h(X, Y), JZ) (99)

and (69) we get [bar.g](h(X, Y), [NP.sub.2]Z) = 0, for any X, Y [member of] [GAMMA]([D.sub.1]) and Z [member of] [GAMMA](TM). Thus, from (1) and (2) we get

[bar.g](Jh (X, Y), JZ) = [bar.g]([J.sup.2]h (X, Y), Z)

= p[bar.g](Jh(X, Y), Z)

+ q[bar.g](h (X, Y), Z) = 0, (100)

for any X, Y [member of] and Z [member of] [GAMMA](TM). Moreover, for Z = [[nabla].sub.X]Y, we obtain

[mathematical expression not reproducible], (101)

which implies

[mathematical expression not reproducible]. (102)

On the other hand, from (73) and (102) we have

[mathematical expression not reproducible]. (103)

Using (102), (12) and JY = TY, for any Y [member of] [GAMMA]([D.sub.1]), we obtain

[mathematical expression not reproducible]. (104)

Thus, from (102) and (104) we have

[mathematical expression not reproducible]. (105)

From [mathematical expression not reproducible].

By using [mathematical expression not reproducible], we have

[mathematical expression not reproducible], (106)

which implies [bar.g](J([P.sub.2][[nabla].sub.X]Y), J([P.sub.2][[nabla].sub.X]Y)) = 0. Thus, we get J([P.sub.2][[nabla].sub.X]Y) = 0 and we obtain [P.sub.2][[nabla].sub.X]Y = 0. In conclusion, [[nabla].sub.X]Y [member of] [GAMMA]([D.sub.1]) for any X, F [member of] [GAMMA]([D.sub.1]) and this implies [X, 7] [member of] [GAMMA]([D.sub.1]). Thus, the distribution [D.sub.1] is integrable.

Moreover, because [D.sub.2] is orthogonal to [D.sub.x] and ([bar.M], [bar.g]) is a Riemannian manifold, we obtain the integrability of the distribution [D.sub.2].

In the next propositions, we consider semi-slant submanifolds in the locally metallic (or locally Golden) Riemannian manifolds and we find some conditions for these submanifolds to be [D.sub.1]- [D.sub.2] mixed totally geodesic (i.e., h(X, Y) = 0, for any X [member of] [GAMMA]([D.sub.1]) and Y [member of] [GAMMA]([D.sub.2])), in a similar manner as in the case of semi-slant submanifolds in locally product manifolds ([16]).

Proposition 36. If M is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J), then M is a [D.sub.1] - [D.sub.2] mixed totally geodesic submanifold if and only if [A.sub.V]X [member of] [GAMMA]([D.sub.1]) and [A.sub.V]Y [member of] [GAMMA]([D.sub.2]), for any X [member of] [GAMMA]([D.sub.1]), Y [member of] [GAMMA]([D.sub.2]) and V [member of] [gamma]([T.sup.[perpendicular]]M).

Proof. From (28) we remark that M is a [D.sub.1] - [D.sub.2] mixed totally geodesic submanifolds in the locally metallic (or locally Golden) Riemannian manifolds if and only if g([A.sub.V] X, Y) = g([A.sub.V] Y, X) = 0, for any X [member of] [GAMMA]([D.sub.1]), Y [member of] [GAMMA]([D.sub.2]) and V [member of] [GAMMA]([T.sup.[perpendicular]]M), which is equivalent to [A.sub.V]X [member of] [GAMMA]([D.sub.1]) and [A.sub.V] Y [member of] [GAMMA]([D.sub.2]).

Proposition 37. Let M be a proper semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J). If M is a [D.sub.1] - [D.sub.2] mixed totally geodesic submanifold, then ([[bar.[nabla]].sub.X] N)Y = 0, for any X [member of] [GAMMA]([D.sub.1]), and Y [member of] [GAMMA]([D.sub.2]).

Proof. If M is a [D.sub.1] - [D.sub.2] mixed geodesic submanifold, then nh(X, 7) = h(X, TY) and using (39)(ii), we obtain ([[bar.[nabla]].sub.X]N)Y = 0, for any X [member of] [GAMMA]([D.sub.1]), Y [member of] [GAMMA]([D.sub.2]).

Theorem 38. Let M be a proper semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J). If([[bar.[nabla]].sub.X]N)Y = 0, for any X [member of] [GAMMA]([D.sub.1]), Y [member of] [GAMMA]([D.sub.2]), and h(X, Y) is not an eigenvector of the tensor field n with the eigenvalue -q/p, then M is a [D.sub.1] - [D.sub.2] mixed totally geodesic submanifold.

Proof. If M is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J) and ([[bar.[nabla]].sub.X]N)Y = 0, then from (39)(ii), we get

[n.sup.2]h(X, Y) = nh (X, TY) = h (X, [T.sup.2]y), (107)

for any X [member of] [GAMMA]([D.sub.1]) and Y [member of] [GAMMA]([D.sub.2]), where nV := [(JV).sup.[perpendicular]] for any V [member of] [GAMMA]([T.sup.[perpendicular]]M). By using TY = [TP.sub.2]Y, for any Y [member of] [GAMMA]([D.sub.2]) and (76), we obtain

[n.sup.2]h(X, Y) = [cos.sup.2] [theta] (pnh (X, Y) + qh (X, 7)), (108)

where [theta] is the slant angle of the distribution [D.sub.2]. Using TX = T[P.sub.1]X = JX, for any X [member of] [GAMMA]([D.sub.1]), we obtain

[n.sup.2]h (X, Y) = h ([T.sup.2]X, Y) = h([J.sup.2]X, y)

= h(pJX + qX,Y))

= ph (TX, Y) + qh (X, Y). (109)

Thus, we get

[n.sup.2]h (X, Y) = pnh (X, Y) + qh (X, Y), (110)

for any X [member of] [GAMMA]([D.sub.1]) and Y [member of] [GAMMA]([D.sub.2]). From (108), (110) and [cos.sup.2][theta] [not equal to] 0 ([D.sub.2] is a proper semi-slant distribution), we remark that nh(X, Y) = -(q/p)h(X,Y) and this implies h(X,Y) = 0, for any X [member of] [GAMMA]([D.sub.1]) and Y [member of] T([D.sub.2]), because h(X, Y) is not an eigenvector of n with the eigenvalue -q/p. Thus, M is a [D.sub.1] - [D.sub.2] mixed totally geodesic submanifold in the locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J).

In a similar manner as in ([8], Theorem 4.8), we get the following.

Proposition 39. Let M be a semi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J). Then N is parallel if and only if the shape operator A verifies

[A.sub.nV]X = [TA.sub.V]X = [A.sub.V]TX, (111)

for any X [member of] [GAMMA](TM) and V [member of] [GAMMA]([T.sup.[perpendicular]]M).

Proof. From (2), we get [bar.g](nh(X, Y), V) = [bar.g](Jh(X, Y), V) = [bar.g](h(X, Y), nV), for any X, Y [member of] [GAMMA](TM), Y [member of] [GAMMA]([T.sup.[perpendicular]]M). Thus, by using (39)(ii), we obtain

[bar.g](([[bar.[nabla]].sub.X]N)Y, V) = [bar.g](h(X, Y), nV

- [bar.g](h(X, TY), V)

= [bar.g]([A.sub.nV]) X,Y) - [bar.g]([A.sub.V]X, TY), (112)

for any X, Y [member of] [GAMMA](TM), V [member of] [GAMMA]([T.sup.[perpendicular]]M) and we have

[bar.g](([[bar.[nabla]].sub.X]N)Y, V) = [bar.g]([A.sub.nV]) X - [TA.sub.V]X, Y)

= [bar.g]([A.sub.nV])Y - [A.sub.V]TY, X), (113)

for any X, Y [member of] [GAMMA](TM), V [member of] [GAMMA]([T.sup.[perpendicular]]M). Thus, from (113) we obtain (111).

Theorem 40. Let M be a proper semi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold ([bar.M], [bar.g], J). If the shape operator A verifies [A.sub.nV]X = [TA.sub.V]X = [A.sub.V]TX, for any X [member of] [GAMMA](TM), V [member of] [GAMMA]([T.sup.[perpendicular]]M) and h(X, Y) is not an eigenvector of the tensor field n with the eigenvalue -q/p then, M is a [D.sub.1] - [D.sub.2] mixed totally geodesic submanifold.

Proof. If [A.sub.nV]X = [TA.sub.V]X = [A.sub.V]TX, for any X [member of] [GAMMA](TM), V [member of] [GAMMA]([T.sup.[perpendicular]]M) then, from (113) we obtain ([[bar.[nabla]].sub.X]N)Y = 0 for any X, Y [member of] [GAMMA](TM) and using the Theorem 38, we obtain that M is a [D.sub.1] - [D.sub.2] mixed totally geodesic submanifold.

https://doi.org/10.1155/2018/2864263

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Financial support is provided by Project 2009-1-RO1-GRU1303339, Ref. no. GRU 09-GRAT-20-USV.

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Cristina E. Hretcanu (iD)(1) and Adara M. Blaga (2)

(1) Stefan cel Mare University ofSuceava, Romania

(2) West University of Timisoara, Romania

Correspondence should be addressed to Cristina E. Hretcanu; criselenab@yahoo.com

Received 11 May 2018; Accepted 11 July 2018; Published 12 September 2018

Academic Editor: Raul E. Curto
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Title Annotation:Research Article
Author:Hretcanu, Cristina E.; Blaga, Adara M.
Publication:Journal of Function Spaces
Date:Jan 1, 2018
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