# Slant and Semi-Slant Submanifolds in Metallic Riemannian Manifolds

Authors: Cristina E. Hretcanu; Adara M. Blaga
Journal: Journal of Function Spaces
 (2018)

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## Abstract

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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## Body

## 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. Cabrerizoet al. ([4, 5])), in para-Hermitian manifold (P. Alegre, A. Carriazo ([6])), and in almost product Riemannian manifolds (B. Sahin ([7]), M. Atçeken ([8, 9])).The notion of slant submanifold was generalized by semi-slant 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. Cariazzoet 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. Atçekenet al. Moreover, many examples of semi-slant, pseudo-slant, 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 product-shaped 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. Etayoet al. in ([24]). Some properties regarding the integrability of the Golden Riemannian structures were investigated by A. Gezeret al. in ([25]).The metallic structureJ is a polynomial structure, which was generally defined by S.I. Goldberget al. in ([26, 27]), inspired by the metallic number given by σp,q=p+p2+4q/2, which is the positive solution of the equation x2-px-q=0, for positive integer values of p and q. These σp,q numbers are members of themetallic means family ormetallic proportions (as generalizations of the Golden number ϕ=1+5/2=1.618…), introduced by Vera W. de Spinadel ([28]). Some examples of the members of the metallic means family arethe 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 anglesθ of a submanifold M in a Riemannian manifold (M¯,g¯) endowed with a metallic (or Golden) structure J and the slant angle ϑ 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.LetM¯ be an m-dimensional manifold endowed with a tensor field J of type (1,1). We say that the structure J is ametallic structure if it verifies(1)J2=pJ+qI,for p, q∈N∗, where I is the identity operator on the Lie algebra Γ(TM¯) of vector fields on M¯. In this situation, the pair (M¯,J) is calledmetallic manifold.Ifp=q=1 one obtains theGolden structure ([17]) determined by a (1,1)-tensor field J which verifies J2=J+I. In this case, (M¯,J) is calledGolden manifold.Moreover, if (M¯,g¯) is a Riemannian manifold endowed with a metallic (or a Golden) structure J, such that the Riemannian metric g¯ is J-compatible, i.e.,(2)g¯JX,Y=g¯X,JY,for any X,Y∈Γ(TM¯), then (g¯,J) is called ametallic (or aGolden)Riemannian structure and (M¯,g¯,J) is ametallic (or aGolden)Riemannian manifold.We can remark that(3)g¯JX,JY=g¯J2X,Y=pg¯JX,Y+qg¯X,Y,for any X,Y∈Γ(TM¯).Any metallic structureJ on M¯ induces two almost product structures on this manifold ([21]):(4)F1=22σp,q-pJ-p2σp,q-pI,F2=-22σp,q-pJ+p2σp,q-pI.Conversely, any almost product structure F on M¯ induces two metallic structures on M¯ ([21]):(5)iJ1=2σp,q-p2F+p2I,iiJ2=-2σp,q-p2F+p2I.If the almost product structureF is a Riemannian one, then J1 and J2 are also metallic Riemannian structures. Also, on a metallic manifold M¯,J there are two complementary distributions D1 and D2 corresponding to the projection operators P and Q ([21]), given by(6)P=-12σp,q-pJ+σp,q2σp,q-pI,Q=12σp,q-pJ+σp,q-p2σp,q-pIand the operators P and Q verify the following relations:(7)P+Q=I,P2=P,Q2=Q,PQ=QP=0and(8)JP=PJ=p-σp,qP,JQ=QJ=σp,qQ.In particular, ifp=q=1, we obtain that every Golden structure J on M¯ induces two almost product structures on this manifold and conversely, an almost product structure F on M¯ induces two Golden structures on M¯ ([17, 19]).


## 3. Submanifolds of Metallic Riemannian Manifolds

In the next issues we assume thatM is an m′-dimensional submanifold, isometrically immersed in the m-dimensional metallic (or Golden) Riemannian manifold (M¯,g¯,J) with m,m′∈N∗ and m>m′. We denote by TxM the tangent space of M in a point x∈M and by Tx⊥M the normal space of M in x. The tangent space TxM¯ of M¯ can be decomposed into the direct sum: TxM¯=TxM⊕Tx⊥M, for any x∈M. Let i∗ be the differential of the immersion i:M→M¯. The induced Riemannian metric g on M is given by g(X,Y)=g¯(i∗X,i∗Y), for any X,Y∈Γ(TM), where Γ(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∗X, for any X∈Γ(TM).We consider the decomposition into the tangential and normal parts ofJX and JV, for any X∈Γ(TM) and V∈Γ(T⊥M), are given by(9)iJX=TX+NX,iiJV=tV+nV,where T:Γ(TM)→Γ(TM),N:Γ(TM)→Γ(T⊥M),t:Γ(T⊥M)→Γ(TM) and n:Γ(T⊥M)→Γ(T⊥M), with(10)TX≔JXT,NX≔JX⊥,tV≔JVT,nV≔JV⊥.We remark that the mapsT and n are g¯-symmetric ([29]):(11)ig¯TX,Y=g¯X,TY,iig¯nU,V=g¯U,nVand(12)g¯NX,U=g¯X,tU,for any X,Y∈Γ(TM) and U,V∈Γ(T⊥M).For an almost product structureF, the decompositions into tangential and normal parts of FX and FV, for any X∈Γ(TM) and V∈Γ(T⊥M), are given by ([7])(13)iFX=fX+ωX,iiFV=BV+CV,where f:Γ(TM)→Γ(TM), ω:Γ(TM)→Γ(T⊥M), B:Γ(T⊥M)→Γ(TM), C:Γ(T⊥M)→Γ(T⊥M), with(14)fX≔FXT,ωX≔FX⊥,BV≔FVT,CV≔FV⊥.The mapsf and C are g¯-symmetric ([16]):(15)g¯fX,Y=g¯X,fY,g¯CU,V=g¯U,CV(16)g¯ωX,V=g¯X,BV,for any X,Y∈Γ(TM) and U,V∈Γ(T⊥M).Remark 1.
Let(M¯,g¯) be a Riemannian manifold endowed with an almost product structure F and let J be the metallic structure induced by F on M¯. If M is a submanifold in the almost product Riemannian manifold (M¯,g¯,F), then(17)iTX=p2X±2σp,q-p2fX,iiNX=±2σp,q-p2ωX(18)itV=±2σp,q-p2BV,iinV=p2V±2σp,q-p2CV,for any X∈Γ(TM) and V∈Γ(T⊥M).Remark 2.
Let(M¯,g¯) be a Riemannian manifold endowed with an almost product structure F and let J be the Golden structure induced by F on M¯. If M is a submanifold in the almost product Riemannian manifold (M¯,g¯,F), then(19)iTX=12X±2ϕ-12fX,iiNX=±2ϕ-12ωX(20)itV=±2ϕ-12BV,iinV=12V±2ϕ-12CV,for any X∈Γ(TM) and V∈Γ(T⊥M).Letr=m-m′ be the codimension of M in M¯ (where r,m,m′∈N∗). We fix a local orthonormal basis {N1,…,Nr} of the normal space Tx⊥M. Hereafter we assume that the indices α,β,γ run over the range {1,…,r}.For anyx∈M and X∈TxM, the vector fields J(i∗X) and JNα can be decomposed into tangential and normal components ([21]):(21)iJX=TX+∑α=1ruαXNα,iiJNα=ξα+∑β=1raαβNβ,where (α∈{1,…,r}), T is an (1,1)-tensor field on M, ξα are vector fields on M, uα are 1-forms on M, and (aαβ)r is an r×r matrix of smooth real functions on M.Using (9) and (21), we remark that(22)iNX=∑α=1ruαXNα,iitNα=ξα,iiinNα=∑β=1raαβNβ.Theorem 3.
The structureΣ=(T,g,uα,ξα,(aαβ)r) induced on the submanifold M by the metallic Riemannian structure (g¯,J) on M¯ satisfies the following equalities ([30]):(23)T2X=pTX+qX-∑α=1ruαXξα,(24)iuαTX=puαX-∑β=1raαβuβX,iiaαβ=aβα,(25)iuβξα=qδαβ+paαβ-∑γ=1raαγaγβ,iiTξα=pξα-∑β=1raαβξβ,(26)uαX=gX,ξαfor any X∈Γ(TM), where δαβ is the Kronecker delta and p, q are positive integers ([21]).A structureΣ=(T,g,uα,ξα,(aαβ)r) induced on the submanifold M by the metallic Riemannian structure (g¯,J) defined on M¯ (determined by the (1,1)-tensor field T on M, the vector fields ξα on M, the 1-forms uα on M, and the r×r matrix (aαβ)r of smooth real functions on M) which verifies the relations (23), (24), (25), and (26) is called Σ- metallic Riemannian structure ([30]).Forp=q=1, the structure Σ=(T,g,uα,ξα,(aαβ)r) is called Σ-Golden Riemannian structure.Remark 4.
IfΣ=(T,g,uα,ξα,(aαβ)r) is the induced structure on the submanifold M by the metallic (or Golden) Riemannian structure (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∇¯ and ∇ be the Levi-Civita connections on (M¯,g¯) and (M,g), respectively. The Gauss and Weingarten formulas are given by(27)i∇¯XY=∇XY+hX,Y,ii∇¯XV=-AVX+∇X⊥V,for any X,Y∈Γ(TM) and V∈Γ(T⊥M), where h is the second fundamental form, AV is the shape operator of M. The second fundamental form h and the shape operator AV are related by(28)g¯hX,Y,V=g¯AVX,Y.Remark 5.
Using a local orthonormal basis{N1,…,Nr} of the normal space Tx⊥M, where r is the codimension of M in M¯ and Aα≔ANα, for any α∈{1,…,r}, we obtain(29)i∇¯XNα=-AαX+∇X⊥Nα,iihαX,Y=gAαX,Y,for any X,Y∈Γ(TM).Remark 6.
Forα∈{1,…,r}, the normal connection ∇X⊥Nα has the decomposition ∇X⊥Nα=∑β=1r‍lαβ(X)Nβ, for any X∈Γ(TM), where (lαβ)r is an r×r matrix of 1-forms on M. Moreover, from g¯(Nα,Nβ)=δαβ, we obtain ([30]): g¯(∇X⊥Nα,Nβ)+g¯(Nα,∇X⊥Nβ)=0, which is equivalent to lαβ=-lβα, for any α,β∈{1,…,r} and X∈Γ(TM).The covariant derivatives of the tangential and normal parts ofJX and JV are given by(30)i∇XTY=∇XTY-T∇XY,ii∇¯XNY=∇X⊥NY-N∇XY,and(31)i∇XtV=∇XtV-t∇X⊥V,ii∇¯XnV=∇X⊥nV-n∇X⊥V,for any X, Y∈Γ(TM) and V∈Γ(T⊥M). From g¯(JX,Y)=g¯(X,JY), it follows that(32)g¯∇¯XJY,Z=g¯Y,∇¯XJZ,for any X, Y, Z∈Γ(TM¯). Moreover, if M is an isometrically immersed submanifold of the metallic Riemannian manifold (M¯,g¯,J), then ([23])(33)g¯∇XTY,Z=g¯Y,∇XTZ,for any X, Y, Z∈Γ(TM).Using an analogy of a locally product manifold ([31]), we can definelocally metallic (or locally Golden) Riemannian manifold as follows ([30]).Definition 7.
If(M¯,g¯,J) is a metallic (or Golden) Riemannian manifold and J is parallel with respect to the Levi-Civita connection ∇¯ on M¯ (i.e., ∇¯J=0), we say that (M¯,g¯,J) is alocally metallic (or locally Golden) Riemannian manifold.Proposition 8.
IfM is a submanifold of a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J), then(34)TX,Y=∇XTY-∇YTX-ANYX+ANXYand(35)NX,Y=hX,TY-hTX,Y+∇X⊥NY-∇Y⊥NX,for any X,Y∈Γ(TM), where ∇ is the Levi-Civita connection on M.Proof.
From(M¯,g¯,J) locally metallic (or locally Golden) Riemannian manifold, we have (∇¯XJ)Y=0, for any X,Y∈Γ(TM).
Thus,∇¯X(TY+NY)=J(∇XY+h(X,Y)), which is equivalent to (36)∇XTY+hX,TY-ANYX+∇X⊥NY=T∇XY+N∇XY+thX,Y+nhX,Y. Taking the normal and the tangential components of this equality, we get(37)N∇XY-∇X⊥NY=hX,TY-nhX,Yand(38)∇XTY-T∇XY=ANYX+thX,Y.Interchanging X and Y and subtracting these equalities, we obtain the tangential and normal components of [X,Y]=∇XY-∇YX, which give us (34) and (35).From (30), (31), (37), and (38) we obtain the following.Proposition 9.
IfM is a submanifold of a locally metallic (or Golden) Riemannian manifold (M¯,g¯,J), then the covariant derivatives of T and N verify(39)i∇XTY=ANYX+thX,Y,ii∇¯XNY=nhX,Y-hX,TY,and(40)i∇XtV=AnVX-TAVX,ii∇¯XnV=-hX,tV-NAVX,for any X, Y∈Γ(TM) and V∈Γ(T⊥M).Proposition 10.
IfM is an n-dimensional submanifold of codimension r in a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J), then the structure Σ=(T,g,uα,ξα,(aαβ)r) induced on M by the metallic (or Golden) Riemannian structure (g¯,J) has the following properties ([30]):(41)∇XTY=∑α=1rhαX,Yξα+∑α=1ruαYAαX,(42)∇XuαY=-hαX,TY+∑β=1ruβYlαβX+hβX,Yaβα,for any X,Y∈Γ(TM).Proof.
From∇¯J=0 we obtain ∇¯XJY=J(∇¯XY), for any X,Y∈Γ(TM¯). Using (27)(i), (29), and (21)(ii), we get(43)∇¯XJY=∇XTY-∑α=1ruαYAαX+∑α=1rhαX,TY+XuαY+∑β=1ruβYlβαXNαJ∇¯XY=T∇XY+∑α=1rhαX,Yξα+∑α=1ruα∇XY+∑β=1rhβX,YaβαNα, for any X,Y∈Γ(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.
IfM is a submanifold of a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J), then(44)TX,Y=∇XTY-∇YTX-∑i=1ruαYAαX-uαXAαY(45)NX,Y=∑α=1n∇YuαX-∇XuαY+uαXlαβY-uαYlαβXNα,for any X,Y∈Γ(TM), where ∇ is the Levi-Civita connection on M.


## 4. Slant Submanifolds in Metallic or Golden Riemannian Manifolds

LetM be an m′-dimensional submanifold, isometrically immersed in an m-dimensional metallic (or Golden) Riemannian manifold (M¯,g¯,J), where m,m′∈N∗ and m>m′. Using the Cauchy-Schwartz inequality ([6]), we have (46)g¯JX,TX≤JX·TX, for any X∈Γ(TM). Thus, there exists a function θ:Γ(TM)→[0,π], such that (47)g¯JXx,TXx=cos⁡θXxTXx·JXx, for any x∈M and any nonzero tangent vector Xx∈TxM. The angle θ(Xx) between JXx and TxM is called theWirtinger angle of X and it verifies (48)cos⁡θXx=g¯JXx,TXxTXx·JXx.Definition 12 (see [29]).
A submanifoldM in a metallic (or Golden) Riemannian manifold (M¯,g¯,J) is called slant submanifold if the angle θ(Xx) between JXx and TxM is constant, for any x∈M and Xx∈TxM. In such a case, θ=:θ(Xx) is called theslant angle of M in M¯, and it verifies(49)cos⁡θ=g¯JX,TXJX·TX=TXJX.The immersion i:M→M¯ is named slant immersion of M in M¯.Remark 13.
The invariant and anti-invariant submanifolds in the metallic (or Golden) Riemannian manifold (M¯,g¯,J) are particular cases of slant submanifolds with the slant angle θ=0 and θ=π/2, respectively. A slant submanifold M in M¯, which is neither invariant nor anti-invariant, is called proper slant submanifold and the immersion i:M→M¯ is called proper slant immersion.Proposition 14.
([29]) Let M be an isometrically immersed submanifold of the metallic Riemannian manifold (M¯,g¯,J). If M is a slant submanifold with the slant angle θ, then, for any X, Y∈Γ(TM) we get(50)g¯TX,TY=cos2⁡θpg¯X,TY+qg¯X,Y(51)g¯NX,NY=sin2⁡θpg¯X,TY+qg¯X,Y.
Moreover, we have(52)T2=cos2⁡θpT+qI,where I is the identity on Γ(TM) and(53)∇T2=pcos2⁡θ∇T.Remark 15.
LetI be the identity on Γ(TM). From (23) and (52), we have(54)sin2⁡θpT+qI=∑α=1ruα⊗ξα.Proposition 16.
IfM is an isometrically immersed slant submanifold of the Golden Riemannian manifold (M¯,g¯,J) with the slant angle θ, then(55)g¯TX,TY=cos2⁡θg¯X,TY+g¯X,Y,(56)g¯NX,NY=sin2⁡θg¯X,TY+g¯X,Y,for any X, Y∈Γ(TM). If I is the identity on Γ(TM), we have(57)T2=cos2⁡θT+I,∇T2=cos2⁡θ∇T,(58)sin2⁡θT+I=∑α=1ruα⊗ξα.Definition 17 (see [8]).
A submanifoldM in an almost product Riemannian manifold (M¯,g¯,F) is a slant submanifold if the angle ϑ(Xx) between JXx and TxM is constant, for any x∈M and Xx∈TxM. In such a case, ϑ=:ϑ(Xx) is called the slant angle of the submanifold M in M¯ and it verifies(59)cos⁡ϑ=g¯FX,fXFX·fX=fXFX.Proposition 18 (see [16]).
IfM is a slant submanifold isometrically immersed in an almost product Riemannian manifold (M¯,g¯,F) with the slant angle ϑ then, for any X,Y∈Γ(TM), we get(60)ig¯fX,fY=cos2⁡ϑg¯X,Y,iig¯ωX,ωY=sin2⁡ϑg¯X,Y.In the next proposition we find a relation between the slant anglesθ of the submanifold M in the metallic Riemannian manifold (M¯,g¯,J) and the slant angle ϑ of the submanifold M in the almost product Riemannian manifold (M¯,g¯,F).Theorem 19.
LetM be a submanifold in the Riemannian manifold (M¯,g¯) endowed with an almost product structure F on M¯ and let J be the induced metallic structure by F on (M¯,g¯). If M is a slant submanifold in the almost product Riemannian manifold (M¯,g¯,F) with the slant angle ϑ and F≠-I (I is the identity on Γ(TM)) and J=2σp,q-p/2F+p/2I, then M is a slant submanifold in the metallic Riemannian manifold (M¯,g¯,J) with slant angle θ given by(61)sin⁡θ=2σp,q-p2σp,qsin⁡ϑ.Proof.
From (17)(ii), we obtain g¯(NX,NY)=(2σp,q-p)2/4g¯(ωX,ωY), for any X,Y∈Γ(TM). From (51) and (60)(ii) and g¯(X,JY)=g¯(X,TY), we get(62)2σp,q-p24g¯X,Ysin2⁡ϑ=pg¯X,JY+qg¯X,Ysin2⁡θ,for any X,Y∈Γ(TM). Using J=(p/2)I+2σp,q-p/2F, we have(63)2σp,q-p2g¯X,Ysin2⁡ϑ=2p2+4qg¯X,Y+2pp2+4qg¯X,FYsin2⁡θ,for any X,Y∈Γ(TM). Replacing Y by FY and using F2Y=Y, for any Y∈Γ(TM), we obtain(64)2σp,q-p2g¯X,FYsin2⁡ϑ=2p2+4qg¯X,FY+2pp2+4qg¯X,Ysin2⁡θ.for any X,Y∈Γ(TM). Summing equalities (63) and (64), we obtain(65)g¯X,FY+Y2σp,q-p2sin2⁡ϑ-4q+pσp,qsin2⁡θ=0,for any X,Y∈Γ(TM). Using q+pσp,q=σp,q2, FY≠-Y, and θ,ϑ∈[0,π) in (65), we get (61).In particular, forp=q=1, we obtain the relation between slant angle θ of the immersed submanifold M in a Golden Riemannian manifold (M¯,g¯,J) and the slant angle ϑ of M immersed in the almost product Riemannian manifold (M¯,g¯,F).Proposition 20.
LetM be a submanifold in the Riemannian manifold (M¯,g¯) endowed with an almost product structure F on M¯ and let J be the induced Golden structure by F on (M¯,g¯). If M is a slant submanifold in the almost product Riemannian manifold (M¯,g¯,F) with the slant angle ϑ and F≠-I (I is the identity on Γ(TM)) and J=2ϕ-1/2F+1/2I, then M is a slant submanifold in the Golden Riemannian manifold (M¯,g¯,J) with slant angle θ given by(66)sin⁡θ=2ϕ-12ϕsin⁡ϑ,where ϕ=1+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.
LetM be an immersed submanifold of a metallic (or Golden) Riemannian manifold (M¯,g¯,J). A differentiable distribution D on M is called a slant distribution if the angle θD between JXx and the vector subspace Dx is constant, for any x∈M and any nonzero vector field Xx∈Γ(Dx). The constant angle θD is called the slant angle of the distribution D.Proposition 22.
LetD be a differentiable distribution on a submanifold M of a metallic (or Golden) Riemannian manifold (M¯,g¯,J). The distribution D is a slant distribution if and only if there exists a constant λ∈[0,1] such that(67)PDT2X=λpPDTX+qX,for any X∈Γ(D), where PD is the orthogonal projection on D. Moreover, if θD is the slant angle of D, then it satisfies λ=cos2θD.Proof.
If the distributionD is a slant distribution on M, by using (68)cos⁡θD=g¯JX,PDTXJX·PDTX=PDTXJX,we get g¯(PDTX,PDTX)=cos2θDg¯(pPDTX+qX,X), for any X∈Γ(D) and we obtain (67).
Conversely, if there exists a constantλ∈[0,1] such that (67) holds for any X∈Γ(D), we obtain g¯(JX,PDTX)=g¯(X,JPDTX)=g¯(X,(PDT)2X) and g¯(JX,PDTX)=λg¯(X,pPDTX+qX)=λg¯(X,pJTX+qX)=λg¯(X,J2X)=λg¯(JX,JX). Thus, cosθD=λ(JX/PDTX), and using cosθ=PDTX/JX we get cos2θD=λ. Thus, cos2θD is constant and D is a slant distribution on M.Definition 23.
LetM be an immersed submanifold in a metallic (or Golden) Riemannian manifold (M¯,g¯,J). We say that M is abi-slant submanifold of M¯ if there exist two orthogonal differentiable distributions D1 and D2 on M such that TM=D1⊕D2 and D1, D2 are slant distributions with the slant angles θ1 and θ2, respectively.For a differentiable distributionD1 on M, we denote by D2≔D1⊥ the orthogonal distribution of D1 in M (i.e., TM=D1⊕D2). Let P1 and P2 be the orthogonal projections on D1 and D2. Thus, for any X∈Γ(TM), we can consider the decomposition of X=P1X+P2X, where P1X∈Γ(D1) and P2X∈Γ(D2).IfM is a bi-slant submanifold of a metallic Riemannian manifold (M¯,g¯,J) with the orthogonal distribution D1 and D2 and the slant angles θ1 and θ2, respectively, then JX=P1TX+P2TX+NX=TP1X+TP2X+NP1X+NP2X, for any X∈Γ(TM). In a similar manner as in ([16]), we can prove the following.Proposition 24.
IfM is a bi-slant submanifold in a metallic (or Golden) Riemannian manifold (M¯,g¯,J), with the slant angles θ1=θ2=θ and g(JX,Y)=0, for any X∈Γ(D1) and Y∈Γ(D2), then M is a slant submanifold in the metallic Riemannian manifold (M¯,g¯,J) with the slant angle θ.Proof.
Fromg¯(JX,Y)=g¯(TX,Y)=0, for any X∈Γ(D1) and Y∈Γ(D2), it follows that g¯(X,JY)=g¯(X,TY)=0. Thus, we obtain TX∈Γ(D1), for any X∈Γ(D1) and TY∈Γ(D2), for any Y∈Γ(D2). Moreover, using the projections of any X∈Γ(TM) on Γ(D1) and Γ(D2), respectively, we obtain the decomposition X=P1X+P2X, where P1X∈Γ(D1) and P2X∈Γ(D2).
Fromg¯(TPiX,TPiX)=cos2θig(JPiX,JPiX) (for i∈{1,2}) and using θ1=θ2=θ, we obtain g¯TX,TX/g¯(JX,JX)=cos2θ, for any X∈Γ(TM). Thus, M is a slant submanifold in the metallic (or Golden) Riemannian manifold (M¯,g¯,J) with the slant angle θ.IfM is abi-slant submanifold of a manifold M¯, for particular values of the angles θ1=0 and θ2≠0, we obtain the following.Definition 25.
An immersed submanifoldM in a metallic (or Golden) Riemannian manifold (M¯,g¯,J) is a semi-slant submanifold if there exist two orthogonal distributions D1 and D2 on M such that (1)
TM admits the orthogonal direct decomposition TM=D1⊕D2;(2)
The distributionD1 is invariant distribution (i.e., J(D1)=D1);(3)
The distributionD2 is slant with angle θ≠0.
Moreover, ifdim(D1)·dim(D2)≠0, then M is a proper semi-slant submanifold.Rema rk 5.6.
IfM is a semi-slant submanifold of a metallic Riemannian manifold (M¯,g¯,J) with the slant angle θ of the distributions D2, then we get that (1)
M is an invariant submanifold if dim(D2)=0;(2)
M is an anti-invariant submanifold if dim(D1)=0 and θ=π/2;(3)
M is a semi-invariant submanifold if D2 is anti-invariant (i.e., θ=π/2).IfM is a semi-slant submanifold in a metallic (or Golden) Riemannian manifold (M¯,g¯,J) then, for any X∈Γ(TM),(69)JX=TP1X+TP2X+NP2X=P1TX+P2TX+NP2X,(70)iJP1X=TP1X,iiNP1X=0,iiiTP2X∈ΓD2.Moreover, we have g¯(JP2X,TP2X)=cosθ(X)TP2X·JP2X and the cosine of the slant angle θ(X) of the distribution D2 is constant, for any nonzero X∈Γ(TM). If θ(X)=:θ, for any nonzero X∈Γ(TM) we get(71)cos⁡θ=g¯JP2X,TP2XTP2X·JP2X=TP2XJP2X.Proposition 27.
IfM is a semi-slant submanifold of the metallic Riemannian manifold (M¯,g¯,J) with the slant angle θ of the distribution D2 then, for any X, Y∈Γ(TM), we get(72)g¯TP2X,TP2Y=cos2⁡θpg¯TP2X,P2Y+qg¯P2X,P2Y,(73)g¯NX,NY=sin2⁡θpg¯TP2X,P2Y+qg¯P2X,P2Y.Proof.
TakingX+Y in (71) we have g¯(TP2X,TP2Y)=cos2θg¯(JP2X,JP2Y)=cos2θ[pg¯(JP2X,P2Y)+qg¯(P2X,P2Y)], for any X, Y∈Γ(TM) and using (70)(iii) we get (72). From (70)(ii) we get TP2X=JP2X-NX, for any X∈Γ(TM). Thus, we obtain g¯(TP2X,TP2Y)=g¯(JP2X,JP2Y)-g¯(NX,NY), for any X, Y∈Γ(TM) and it implies (73).Remark 28.
A semi-slant submanifoldM of a Golden Riemannian manifold (M¯,g¯,J) with the slant angle θ of the distribution D2 verifies(74)g¯TP2X,TP2Y=cos2⁡θg¯TP2X,P2Y+g¯P2X,P2Y,(75)g¯NX,NY=sin2⁡θg¯TP2X,P2Y+g¯P2X,P2Y, for any X, Y∈Γ(TM).Proposition 29.
LetM be a semi-slant submanifold of a metallic Riemannian manifold (M¯,g¯,J) with the slant angle θ of the distribution D2. Then(76)TP22=cos2⁡θpTP2+qI,where I is the identity on Γ(D2) and(77)∇TP22=pcos2⁡θ∇TP2.Proof.
Usingg¯(TP2X,TP2Y)=g¯((TP2)2X,P2Y), for any X, Y∈Γ(TM) and (72), we obtain (76). Moreover, we have (∇X(TP2)2)Y=cos2θ(p(∇XTP2)Y+q(∇XI)Y)=pcos2θ∇X(P2T)Y, for any X∈Γ(D2) and Y∈Γ(TM). For the identity I on Γ(D2) we have (∇XI)P2Y=0; thus, we get (77).Remark 30.
A semi-slant submanifoldM of a Golden Riemannian manifold (M¯,g¯,J) with the slant angle θ of the distribution D2 verifies(78)TP22=cos2⁡θTP2+I,where I is the identity on Γ(D2) and(79)∇TP22=cos2⁡θ∇TP2.Proposition 31.
LetM be an immersed submanifold of a metallic Riemannian manifold (M¯,g¯,J). Then M is a semi-slant submanifold in M¯ if and only if exists a constant λ∈[0,1) such that D={X∈Γ(TM)|T2X=λ(pTX+qX)} is a distribution and NX=0, for any X∈Γ(TM) orthogonal to D, where p and q are given in (1).Proof.
If we considerM a semi-slant submanifold of the metallic Riemannian manifold (M¯,g¯,J) then, in (72) we put λ=cos2θ∈[0,1). Thus, we obtain T2X=λ(pTX+qX) and we get D2⊆D. For a nonzero vector field X∈Γ(D), let X=X1+X2, where X1=P1X∈Γ(D1) and X2=P2X∈Γ(D2). Because D1 is invariant, then JX1=TX1 and using the property of the metallic structure (1), we obtain pTX1+qX1=pJX1+qX1=J2X1=T2X1=λ(pTX1+qX1), which implies (pTX1+qX1)(λ-1)=0. Because λ∈[0,1), we obtain TX1=-q/pX1 and we get X1=0 (q2/p2≠0 because p and q are nonzero natural numbers). Thus, we obtain X∈Γ(D2) and D⊆D2, which implies D=D2. Therefore, D1=D⊥.
Conversely, if there exists a real numberλ∈[0,1) such that we have T2X=λ(pTX+qX), for any X∈Γ(D), it follows that cos2(θ(X))=λ which implies that θ(X)=arccos(λ) does not depend on X. We can consider the orthogonal direct sum TM=D⊕D⊥. For Y∈Γ(D⊥)≔Γ(D1) and X∈Γ(D) (with D≔D2), we have g¯(X,J2Y)=g¯(X,T(JY))=g¯(TX,JY)=g¯(TX,TY)=g¯(T2X,Y)=λ[pg¯(TX,Y)+qg¯(X,Y)]. From g¯(X,J2Y)=pg¯(X,JY)+qg¯(X,Y) and g¯(X,Y)=0, we obtain g¯(X,JY)=λg¯(X,TY) and this implies (1-λ)TY∈Γ(D⊥) and TY∈Γ(D⊥). Thus, JY∈Γ(D⊥), for any X∈Γ(D⊥) and we obtain that D⊥ is an invariant distribution.Remark 32.
An immersed submanifoldM of the Golden Riemannian manifold (M¯,g¯,J) is a semi-slant submanifold in M¯ if and only if there exists a constant λ∈[0,1) such that (80)D=X∈ΓTM∣T2X=λTX+Xis a distribution and NX=0, for any X∈Γ(TM) orthogonal to D.Examples 1.
LetR7 be the Euclidean space endowed with the usual Euclidean metric ·,·. Let f:M→R7 be the immersion given by (81)fu,t1,t2=ucos⁡t1,usin⁡t1,ucos⁡t2,usin⁡t2,u,t1,t2,where M≔{(u,t1,t2)/u>0,t1,t2∈[0,π/2]}.
We can find a local orthonormal frame onTM given by (82)Z1=cos⁡t1∂∂x1+sin⁡t1∂∂x2+cos⁡t2∂∂x3+sin⁡t2∂∂x4+∂∂y1,Z2=-usin⁡t1∂∂x1+ucos⁡t1∂∂x2+∂∂y2,Z3=-usin⁡t2∂∂x3+ucos⁡t2∂∂x4+∂∂y3.
We define the metallic structureJ:R7→R7 given by(83)J∂∂xi,∂∂yj=σ∂∂x1,σ∂∂x2,σ¯∂∂x3,σ¯∂∂x4,σ¯∂∂y1,σ∂∂y2,σ¯∂∂y3,for i∈{1,2,3,4} and j∈{1,2,3} where σ≔σp,q=p+p2+4q/2 is the metallic number (p,q∈N∗) and σ¯=p-σ. We can verify that ∇¯J=0 and we obtain that (R7,·,·,J) is a locally metallic Riemannian manifold.
Moreover, we haveJZ2=σZ2,JZ3=σ¯Z3, and (84)JZ1=σcos⁡t1∂∂x1+σsin⁡t1∂∂x2+σ¯cos⁡t2∂∂x3+σ¯sin⁡t2∂∂x4+σ¯∂∂y1.
We can verify thatJZ22=σ2(u2+1), JZ32=σ¯2(u2+1), (85)JZ12=σ2+2σ¯2,JZ22=σ2u2+1,JZ32=σ¯2u2+1.
On the other hand, we haveJZ1,Z1=σ+2σ¯ and JZi,Zj=0, for any i≠j, where i,j∈{1,2,3}. We remark that (86)cos⁡θ=JZ1,Z1Z1·JZ1=σ+2σ¯3σ2+2σ¯2. We define the distributions D1=span{Z2,Z3} and D2=span{Z1}. We have J(D1)⊂D1 (i.e., D1 is an invariant distribution with respect to J). The Riemannian metric tensor of D1⊕D2 is given by g=3du2+(u2+1)(dt12+dt22). Thus, D1⊕D2 is a warped product semi-slant submanifold in the locally metallic Riemannian manifold (R7,·,·,J) with the slant angle arccosσ+2σ¯/3(σ2+2σ¯2).
IfJ is the Golden structure J:R7→R7 given by(87)J∂∂xi,∂∂yj=ϕ∂∂x1,ϕ∂∂x2,ϕ¯∂∂x3,ϕ¯∂∂x4,ϕ¯∂∂y1,ϕ∂∂y2,ϕ¯∂∂y3,for i∈{1,2,3,4} and j∈{1,2,3}, where ϕ≔1+5/2 is the Golden number and ϕ¯=1-ϕ, in the same manner we obtain (88)cos⁡θ=JZ1,Z1Z1·JZ1=ϕ+2ϕ¯3ϕ2+2ϕ¯2. We define the distributions D1=span{Z2,Z3} and D2=span{Z1}. We obtain that D1⊕D2 is a warped product semi-slant submanifold in the locally Golden Riemannian manifold (R7,·,·,J), with the slant angle arccosϕ+2ϕ¯/3ϕ2+2ϕ¯2.Examples 2.
LetM≔{(u,α1,α2,…,αn)/u>0,αi∈[0,π/2],i∈{1,…,n}} and f:M→R3n+1 is the immersion given by(89)fu,α1,…,αn=ucos⁡α1,…,ucos⁡αn,usin⁡α1,…,usin⁡αn,α1,…,αn,u.We can find a local orthonormal frame of the submanifold TM in R3n+1, spanned by the vectors: (90)Z0=∑j=1ncos⁡αj∂∂xj+sin⁡αj∂∂xn+j+∂∂x3n+1,Zi=-usin⁡αi∂∂xi+ucos⁡αi∂∂xn+i+∂∂x2n+i, for any i∈{1,…,n}. We remark that Z02=n+1, Zi2=u2+1, for any i∈{1,…,n}, Z0⊥Zi, for any i∈{1,…,n}, and Zi⊥Zj, for i≠j, where i,j∈{1,…,n}.
LetJ:R3n+1→R3n+1 be the (1,1)-tensor field defined by(91)JX1,…,X3n,X3n+1=σX1,…,σX3n,σ¯X3n+1,where σ≔σp,q is the metallic number and σ¯=p-σ. It is easy to verify that J is a metallic structure on R3n+1 (i.e., J2=pJ+qI). The metric g¯, given by the scalar product ·,· on R3n+1, is J compatible and (R3n+1,g¯,J) is a metallic Riemannian manifold.
Also,JZ0=σ∑j=1n‍(cos⁡αj(∂/∂xj)+sinαj(∂/∂xn+j))+σ¯(∂/∂x3n+1) and, for any i∈{1,…,n}, we get JZi=σ(-usinαi(∂/∂xi)+ucosαi(∂/∂xn+i)+∂/∂x2n+i)=σZi. We can verify that JZ0 is orthogonal to span{Z1,…,Zn} and cos(JZ0,Z0^)=nσ+σ¯/(n+1)(nσ2+σ¯2).
If we consider the distributionsD1=span{Zi/i∈{1,…,n}} and D2=span{Z0}, then D1⊕D2 is a semi-slant submanifold in the metallic Riemannian manifold (R3n+1,·,·,J), with the Riemannian metric tensor g=(n+1)du2+(u2+1)∑j=1n‍dαj2.


## 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.
IfM is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J), then
(i) the distributionD1 is integrable if and only if(92)∇YuαX=∇XuαY,for any X,Y∈Γ(D1);
(ii) the distributionD2 is integrable if and only if(93)P1∇XTY-∇YTX=∑i=1ruαYP1AαX-uαXP1AαY,for any X,Y∈Γ(D2).Proof.
(i) ForX,Y∈Γ(D1), we have X=P1X and Y=P1Y. The distribution D1 is integrable if and only if [X,Y]∈Γ(D1), which is equivalent to N([X,Y])=0, for any X,Y∈Γ(D1). From J(D1)⊆D1 we obtain NX=NY=0 and from (22)(i) we get uα(X)lαβ(Y)=uα(Y)lαβ(X)=0. Thus, using (45) we have the distribution D1 is integrable if and only if (92) holds.
(ii) ForX,Y∈Γ(D2), we have X=P2X, Y=P2Y. The distribution D2 is integrable if and only if [X,Y]∈Γ(D2), which is equivalent to P1T([X,Y])=0. Thus, from (44), we obtain that the distribution D2 is integrable if and only if (93) holds.Remark 34.
IfM is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J), then
(i) the distributionD1 is integrable if and only if(94)hX,TY=hTX,Y,for any X,Y∈Γ(D1);
(ii) the distributionD1 is integrable if and only if the shape operator of M satisfies(95)JAVX=AVJX,for any X∈Γ(D1) and V∈Γ(T⊥M);
(iii) the distributionD2 is integrable if and only if(96)P1∇XTY-∇YTX=P1ANYX-ANXY,for any X,Y∈Γ(D2).Proof.
(i) For anyX,Y∈Γ(D1), we have [X,Y]∈Γ(D1) if and only if N([X,Y])=0 and from (35), we obtain (94).
(ii) For anyX,Y∈Γ(D1) and any V∈Γ(T⊥M), from (28) and (2) we have (97)gJAVX-AVJX,Y=ghX,JY-hJX,Y,V. From (35) and NX=NY=0 (because J(D1)⊆D1) we have (98)gJAVX-AVJX,Y=gNX,Y,V=0,for any X,Y∈Γ(D1) and any V∈Γ(T⊥M). Thus, we have [X,Y]∈Γ(D1).
(iii) For anyX,Y∈Γ(D2), we have X=P2X and Y=P2Y. The distribution D2 is integrable if and only if [X,Y]∈Γ(D2), which is equivalent to T([X,Y])∈Γ(D2) or P1T([X,Y])=0. Thus, from (34), we obtain that D2 is integrable if and only if (96) holds.Theorem 35.
LetM be a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J). If ∇T=0, then the distributions D1 and D2 are integrable.Proof.
First of all, we considerX,Y∈Γ(D1) and we prove [X,Y]∈Γ(D1). For any Y∈Γ(D1), we get NY=0 and using ∇T=0 in (39)(i) we obtain th(X,Y)=0, for any X,Y∈Γ(D1), which implies Jh(X,Y)=nh(X,Y). From (99)g¯thX,Y,Z=g¯JhX,Y,Z=g¯hX,Y,JZ and (69) we get g¯(h(X,Y),NP2Z)=0, for any X,Y∈Γ(D1) and Z∈Γ(TM). Thus, from (1) and (2) we get (100)g¯JhX,Y,JZ=g¯J2hX,Y,Z=pg¯JhX,Y,Z+qg¯hX,Y,Z=0, for any X,Y∈Γ(D1) and Z∈Γ(TM). Moreover, for Z=∇XY, we obtain (101)0=g¯JhX,Y,NP2∇XY=g¯∇¯XJY,NP2∇XY-g¯J∇XY,NP2∇XY, which implies(102)g¯hX,JY,NP2∇XY=g¯NP2∇XY,NP2∇XY.On the other hand, from (73) and (102) we have(103)g¯hX,JY,NP2∇XY=sin2⁡θpg¯TP2∇XY,P2∇XY+qg¯P2∇XY,P2∇XY.Using (102), (12) and JY=TY, for any Y∈Γ(D1), we obtain(104)g¯hX,JY,NP2∇XY=g¯thX,TY,P2∇XY=0.Thus, from (102) and (104) we have (105)sin2⁡θpg¯TP2∇XY,P2∇XY+qg¯P2∇XY,P2∇XY=0. From θ≠0, we obtain pg¯(TP2∇XY,P2∇XY)+qg¯(P2∇XY,P2∇XY)=0.
By usingg¯(TP2∇XY,P2∇XY)=g¯(JP2∇XY,P2∇XY), we have (106)g¯J2P2∇XY,P2∇XY=pg¯JP2∇XY,P2∇XY+qg¯P2∇XY,P2∇XY=0, which implies g¯(J(P2∇XY),J(P2∇XY))=0. Thus, we get J(P2∇XY)=0 and we obtain P2∇XY=0. In conclusion, ∇XY∈Γ(D1) for any X,Y∈Γ(D1) and this implies [X,Y]∈Γ(D1). Thus, the distribution D1 is integrable.
Moreover, becauseD2 is orthogonal to D1 and (M¯,g¯) is a Riemannian manifold, we obtain the integrability of the distribution D2.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 beD1-D2 mixed totally geodesic (i.e., h(X,Y)=0, for any X∈Γ(D1) and Y∈Γ(D2)), in a similar manner as in the case of semi-slant submanifolds in locally product manifolds ([16]).Proposition 36.
IfM is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J), then M is a D1-D2 mixed totally geodesic submanifold if and only if AVX∈Γ(D1) and AVY∈Γ(D2), for any X∈Γ(D1), Y∈Γ(D2) and V∈Γ(T⊥M).Proof.
From (28) we remark that M is a D1-D2 mixed totally geodesic submanifolds in the locally metallic (or locally Golden) Riemannian manifolds if and only if g(AVX,Y)=g(AVY,X)=0, for any X∈Γ(D1),Y∈Γ(D2) and V∈Γ(T⊥M), which is equivalent to AVX∈Γ(D1) and AVY∈Γ(D2).Proposition 37.
LetM be a proper semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J). If M is a D1-D2 mixed totally geodesic submanifold, then (∇¯XN)Y=0, for any X∈Γ(D1), and Y∈Γ(D2).Proof.
IfM is a D1-D2 mixed geodesic submanifold, then nh(X,Y)=h(X,TY) and using (39)(ii), we obtain (∇¯XN)Y=0, for any X∈Γ(D1), Y∈Γ(D2).Theorem 38.
LetM be a proper semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J). If (∇¯XN)Y=0, for any X∈Γ(D1), Y∈Γ(D2), and h(X,Y) is not an eigenvector of the tensor field n with the eigenvalue -q/p, then M is a D1-D2 mixed totally geodesic submanifold.Proof.
IfM is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J) and (∇¯XN)Y=0, then from (39)(ii), we get(107)n2hX,Y=nhX,TY=hX,T2Y,for any X∈Γ(D1) and Y∈Γ(D2), where nV≔(JV)⊥ for any V∈Γ(T⊥M). By using TY=TP2Y, for any Y∈Γ(D2) and (76), we obtain(108)n2hX,Y=cos2⁡θpnhX,Y+qhX,Y,where θ is the slant angle of the distribution D2. Using TX=TP1X=JX, for any X∈Γ(D1), we obtain (109)n2hX,Y=hT2X,Y=hJ2X,Y=hpJX+qX,Y=phTX,Y+qhX,Y. Thus, we get(110)n2hX,Y=pnhX,Y+qhX,Y,for any X∈Γ(D1) and Y∈Γ(D2). From (108), (110) and cos2θ≠0 (D2 is a proper semi-slant distribution), we remark that nh(X,Y)=-q/ph(X,Y) and this implies h(X,Y)=0, for any X∈Γ(D1) and Y∈Γ(D2), because h(X,Y) is not an eigenvector of n with the eigenvalue -q/p. Thus, M is a D1-D2 mixed totally geodesic submanifold in the locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J).In a similar manner as in ([8], Theorem 4.8), we get the following.Proposition 39.
LetM be a semi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J). Then N is parallel if and only if the shape operator A verifies(111)AnVX=TAVX=AVTX,for any X∈Γ(TM) and V∈Γ(T⊥M).Proof.
From (2), we get g¯(nh(X,Y),V)=g¯(Jh(X,Y),V)=g¯(h(X,Y),nV), for any X,Y∈Γ(TM), V∈Γ(T⊥M). Thus, by using (39)(ii), we obtain (112)g¯∇¯XNY,V=g¯hX,Y,nV-g¯hX,TY,V=g¯AnVX,Y-g¯AVX,TY, for any X,Y∈Γ(TM), V∈Γ(T⊥M) and we have(113)g¯∇¯XNY,V=g¯AnVX-TAVX,Y=g¯AnVY-AVTY,X,for any X,Y∈Γ(TM), V∈Γ(T⊥M). Thus, from (113) we obtain (111).Theorem 40.
LetM be a proper semi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold (M¯,g¯,J). If the shape operator A verifies AnVX=TAVX=AVTX, for any X∈Γ(TM), V∈Γ(T⊥M) and h(X,Y) is not an eigenvector of the tensor field n with the eigenvalue -q/p then, M is a D1-D2 mixed totally geodesic submanifold.Proof.
IfAnVX=TAVX=AVTX, for any X∈Γ(TM), V∈Γ(T⊥M) then, from (113) we obtain (∇¯XN)Y=0 for any X,Y∈Γ(TM) and using the Theorem 38, we obtain that M is a D1-D2 mixed totally geodesic submanifold.



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*Source: 2864263-2018-09-12.xml*