Journal of Function Spaces
Volume 2018 (2018), Article ID 2864263, 13 pages
https://doi.org/10.1155/2018/2864263

Research Article

Slant and Semi-Slant Submanifolds in Metallic Riemannian Manifolds

Cristina E. Hretcanu¹ and Adara M. Blaga²

¹Stefan cel Mare University of Suceava, Romania
²West 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: Raúl E. Curto

Copyright © 2018 Cristina E. Hretcanu and Adara M. Blaga. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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. 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. 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. Atçeken et 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. 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 is a polynomial structure, which was generally defined by S.I. Goldberg et al. in ([26, 27]), inspired by the metallic number given by , which is the positive solution of the equation , for positive integer values of and . These numbers are members of the metallic means family or metallic proportions (as generalizations of the Golden number ), 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 of a submanifold in a Riemannian manifold endowed with a metallic (or Golden) structure and the slant angle of the same submanifold of the almost product Riemannian manifold . 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 be an -dimensional manifold endowed with a tensor field of type . We say that the structure is a metallic structure if it verifies

for , , where is the identity operator on the Lie algebra of vector fields on . In this situation, the pair is called metallic manifold.

If one obtains the Golden structure ([17]) determined by a -tensor field which verifies . In this case, is called Golden manifold.

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

for any , then is called a metallic (or a Golden) Riemannian structure and is a metallic (or a Golden) Riemannian manifold.

We can remark that

for any .

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

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

If the almost product structure is a Riemannian one, then and are also metallic Riemannian structures. Also, on a metallic manifold there are two complementary distributions and corresponding to the projection operators and ([21]), given by

and the operators and verify the following relations:

and

In particular, if , we obtain that every Golden structure on induces two almost product structures on this manifold and conversely, an almost product structure on induces two Golden structures on ([17, 19]).

3. Submanifolds of Metallic Riemannian Manifolds

In the next issues we assume that is an -dimensional submanifold, isometrically immersed in the -dimensional metallic (or Golden) Riemannian manifold ( with and . We denote by the tangent space of in a point and by the normal space of in . The tangent space of can be decomposed into the direct sum: , for any . Let be the differential of the immersion . The induced Riemannian metric on is given by , for any , where denotes the set of all vector fields of . For the simplification of the notations, in the rest of the paper we shall note by the vector field , for any .

We consider the decomposition into the tangential and normal parts of and , for any and , are given by

where and , with

We remark that the maps and are -symmetric ([29]):

and

for any and .

For an almost product structure , the decompositions into tangential and normal parts of and , for any and , are given by ([7])

where , , , , with

The maps and are -symmetric ([16]):

for any and .

Remark 1. Let be a Riemannian manifold endowed with an almost product structure and let be the metallic structure induced by on . If is a submanifold in the almost product Riemannian manifold , then

for any and .

Remark 2. Let be a Riemannian manifold endowed with an almost product structure and let be the Golden structure induced by on . If is a submanifold in the almost product Riemannian manifold , then

for any and .

Let be the codimension of in (where ). We fix a local orthonormal basis of the normal space . Hereafter we assume that the indices run over the range .

For any and , the vector fields and can be decomposed into tangential and normal components ([21]):

where , is an -tensor field on , are vector fields on , are -forms on , and is an matrix of smooth real functions on .

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

Theorem 3. The structure induced on the submanifold by the metallic Riemannian structure on satisfies the following equalities ([30]):

for any , where is the Kronecker delta and , are positive integers ([21]).

A structure induced on the submanifold by the metallic Riemannian structure defined on (determined by the -tensor field on , the vector fields on , the -forms on , and the matrix of smooth real functions on ) which verifies the relations (23), (24), (25), and (26) is called - metallic Riemannian structure ([30]).

For , the structure is called -Golden Riemannian structure.

Remark 4. If is the induced structure on the submanifold by the metallic (or Golden) Riemannian structure on , then is an invariant submanifold with respect to if and only if is a metallic (or Golden) Riemannian manifold, whenever is nontrivial ([21]).

Let and be the Levi-Civita connections on and , respectively. The Gauss and Weingarten formulas are given by

for any and , where is the second fundamental form, is the shape operator of . The second fundamental form and the shape operator are related by

Remark 5. Using a local orthonormal basis of the normal space , where is the codimension of in and , for any , we obtain

for any .

Remark 6. For , the normal connection has the decomposition , for any , where is an matrix of -forms on . Moreover, from , we obtain ([30]): , which is equivalent to , for any and .

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

and

for any , and . From , it follows that

for any , , . Moreover, if is an isometrically immersed submanifold of the metallic Riemannian manifold , then ([23])

for any , , .

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 is a metallic (or Golden) Riemannian manifold and is parallel with respect to the Levi-Civita connection on (i.e., ), we say that is a locally metallic (or locally Golden) Riemannian manifold.

Proposition 8. If is a submanifold of a locally metallic (or locally Golden) Riemannian manifold , then

and

for any , where is the Levi-Civita connection on .

Proof. From locally metallic (or locally Golden) Riemannian manifold, we have , for any .
Thus, , which is equivalent to

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

and

Interchanging and and subtracting these equalities, we obtain the tangential and normal components of , which give us (34) and (35).

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

Proposition 9. If is a submanifold of a locally metallic (or Golden) Riemannian manifold , then the covariant derivatives of and verify

and

for any , and .

Proposition 10. If is an -dimensional submanifold of codimension in a locally metallic (or locally Golden) Riemannian manifold , then the structure induced on by the metallic (or Golden) Riemannian structure has the following properties ([30]):

for any .

Proof. From we obtain , for any . Using (27)(i), (29), and (21)(ii), we get

for any . 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 is a submanifold of a locally metallic (or locally Golden) Riemannian manifold , then

for any , where is the Levi-Civita connection on .

4. Slant Submanifolds in Metallic or Golden Riemannian Manifolds

Let be an -dimensional submanifold, isometrically immersed in an -dimensional metallic (or Golden) Riemannian manifold (, where and . Using the Cauchy-Schwartz inequality ([6]), we have

for any . Thus, there exists a function , such that

for any and any nonzero tangent vector . The angle between and is called the Wirtinger angle of and it verifies

Definition 12 (see [29]). A submanifold in a metallic (or Golden) Riemannian manifold ( is called slant submanifold if the angle between and is constant, for any and . In such a case, is called the slant angle of in , and it verifies

The immersion is named slant immersion of in .

Remark 13. The invariant and anti-invariant submanifolds in the metallic (or Golden) Riemannian manifold ( are particular cases of slant submanifolds with the slant angle and , respectively. A slant submanifold in , which is neither invariant nor anti-invariant, is called proper slant submanifold and the immersion is called proper slant immersion.

Proposition 14. ([29]) Let be an isometrically immersed submanifold of the metallic Riemannian manifold . If is a slant submanifold with the slant angle , then, for any , we get

Moreover, we have

where is the identity on and

Remark 15. Let be the identity on . From (23) and (52), we have

Proposition 16. If is an isometrically immersed slant submanifold of the Golden Riemannian manifold with the slant angle , then

for any , . If is the identity on , we have

Definition 17 (see [8]). A submanifold in an almost product Riemannian manifold ( is a slant submanifold if the angle between and is constant, for any and . In such a case, is called the slant angle of the submanifold in and it verifies

Proposition 18 (see [16]). If is a slant submanifold isometrically immersed in an almost product Riemannian manifold with the slant angle then, for any , we get

In the next proposition we find a relation between the slant angles of the submanifold in the metallic Riemannian manifold and the slant angle of the submanifold in the almost product Riemannian manifold .

Theorem 19. Let be a submanifold in the Riemannian manifold ) endowed with an almost product structure on and let be the induced metallic structure by on ). If is a slant submanifold in the almost product Riemannian manifold with the slant angle and ( is the identity on ) and , then is a slant submanifold in the metallic Riemannian manifold with slant angle given by

Proof. From (17)(ii), we obtain , for any . From (51) and (60)(ii) and , we get

for any . Using , we have

for any . Replacing by and using , for any , we obtain

for any . Summing equalities (63) and (64), we obtain

for any . Using , , and in (65), we get (61).

In particular, for , we obtain the relation between slant angle of the immersed submanifold in a Golden Riemannian manifold ) and the slant angle of immersed in the almost product Riemannian manifold .

Proposition 20. Let be a submanifold in the Riemannian manifold ) endowed with an almost product structure on and let be the induced Golden structure by on ). If is a slant submanifold in the almost product Riemannian manifold with the slant angle and ( is the identity on ) and , then is a slant submanifold in the Golden Riemannian manifold with slant angle given by

where 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 be an immersed submanifold of a metallic (or Golden) Riemannian manifold . A differentiable distribution on is called a slant distribution if the angle between and the vector subspace is constant, for any and any nonzero vector field . The constant angle is called the slant angle of the distribution .

Proposition 22. Let be a differentiable distribution on a submanifold of a metallic (or Golden) Riemannian manifold . The distribution is a slant distribution if and only if there exists a constant such that

for any , where is the orthogonal projection on . Moreover, if is the slant angle of , then it satisfies .

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

we get , for any and we obtain (67).
Conversely, if there exists a constant such that (67) holds for any , we obtain and Thus, , and using we get . Thus, is constant and is a slant distribution on .

Definition 23. Let be an immersed submanifold in a metallic (or Golden) Riemannian manifold . We say that is a bi-slant submanifold of if there exist two orthogonal differentiable distributions and on such that and , are slant distributions with the slant angles and , respectively.

For a differentiable distribution on , we denote by the orthogonal distribution of in (i.e., ). Let and be the orthogonal projections on and . Thus, for any , we can consider the decomposition of , where and .

If is a bi-slant submanifold of a metallic Riemannian manifold with the orthogonal distribution and and the slant angles and , respectively, then , for any . In a similar manner as in ([16]), we can prove the following.

Proposition 24. If is a bi-slant submanifold in a metallic (or Golden) Riemannian manifold , with the slant angles and , for any and , then is a slant submanifold in the metallic Riemannian manifold with the slant angle .

Proof. From , for any and , it follows that . Thus, we obtain , for any and , for any . Moreover, using the projections of any on and , respectively, we obtain the decomposition , where and .
From (for ) and using , we obtain , for any . Thus, is a slant submanifold in the metallic (or Golden) Riemannian manifold with the slant angle .

If is a bi-slant submanifold of a manifold , for particular values of the angles and , we obtain the following.

Definition 25. An immersed submanifold in a metallic (or Golden) Riemannian manifold is a semi-slant submanifold if there exist two orthogonal distributions and on such that

(1) admits the orthogonal direct decomposition ;(2)The distribution is invariant distribution (i.e., );(3)The distribution is slant with angle .

Moreover, if , then is a proper semi-slant submanifold.

Rema rk 5.6. If is a semi-slant submanifold of a metallic Riemannian manifold with the slant angle of the distributions , then we get that

(1) is an invariant submanifold if ;(2) is an anti-invariant submanifold if and ;(3) is a semi-invariant submanifold if is anti-invariant (i.e., ).

If is a semi-slant submanifold in a metallic (or Golden) Riemannian manifold then, for any ,

Moreover, we have and the cosine of the slant angle of the distribution is constant, for any nonzero . If , for any nonzero we get

Proposition 27. If is a semi-slant submanifold of the metallic Riemannian manifold with the slant angle of the distribution then, for any , , we get

Proof. Taking in (71) we have , for any , and using (70)(iii) we get (72). From (70)(ii) we get , for any . Thus, we obtain , for any , and it implies (73).

Remark 28. A semi-slant submanifold of a Golden Riemannian manifold with the slant angle of the distribution verifies

for any , .

Proposition 29. Let be a semi-slant submanifold of a metallic Riemannian manifold with the slant angle of the distribution . Then

where is the identity on and

Proof. Using , for any , and (72), we obtain (76). Moreover, we have , for any and . For the identity on we have ; thus, we get (77).

Remark 30. A semi-slant submanifold of a Golden Riemannian manifold with the slant angle of the distribution verifies

where is the identity on and

Proposition 31. Let be an immersed submanifold of a metallic Riemannian manifold . Then is a semi-slant submanifold in if and only if exists a constant such that is a distribution and , for any orthogonal to , where and are given in (1).

Proof. If we consider a semi-slant submanifold of the metallic Riemannian manifold then, in (72) we put . Thus, we obtain and we get . For a nonzero vector field , let , where and . Because is invariant, then and using the property of the metallic structure (1), we obtain , which implies . Because , we obtain and we get ( because and are nonzero natural numbers). Thus, we obtain and , which implies . Therefore, .
Conversely, if there exists a real number such that we have , for any , it follows that which implies that does not depend on . We can consider the orthogonal direct sum . For and (with ), we have From and , we obtain and this implies and . Thus, , for any and we obtain that is an invariant distribution.

Remark 32. An immersed submanifold of the Golden Riemannian manifold is a semi-slant submanifold in if and only if there exists a constant such that

is a distribution and , for any orthogonal to .

Examples 1. Let be the Euclidean space endowed with the usual Euclidean metric . Let be the immersion given by

where .
We can find a local orthonormal frame on given by

We define the metallic structure given by

for and where is the metallic number () and We can verify that and we obtain that is a locally metallic Riemannian manifold.
Moreover, we have , and

We can verify that , ,

On the other hand, we have and , for any , where . We remark that

We define the distributions and . We have (i.e., is an invariant distribution with respect to ). The Riemannian metric tensor of is given by Thus, is a warped product semi-slant submanifold in the locally metallic Riemannian manifold with the slant angle .
If is the Golden structure given by

for and , where is the Golden number and , in the same manner we obtain

We define the distributions and . We obtain that is a warped product semi-slant submanifold in the locally Golden Riemannian manifold , with the slant angle .

Examples 2. Let and is the immersion given by

We can find a local orthonormal frame of the submanifold in , spanned by the vectors:

for any . We remark that , , for any , , for any , and , for , where .
Let be the -tensor field defined by

where is the metallic number and . It is easy to verify that is a metallic structure on (i.e., ). The metric , given by the scalar product on , is compatible and is a metallic Riemannian manifold.
Also, and, for any , we get We can verify that is orthogonal to and
If we consider the distributions and , then is a semi-slant submanifold in the metallic Riemannian manifold , with the Riemannian metric tensor

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 is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold , then
(i) the distribution is integrable if and only if

for any ;
(ii) the distribution is integrable if and only if

for any .

Proof. (i) For , we have and . The distribution is integrable if and only if , which is equivalent to , for any . From we obtain and from (22)(i) we get . Thus, using (45) we have the distribution is integrable if and only if (92) holds.
(ii) For , we have , . The distribution is integrable if and only if , which is equivalent to . Thus, from (44), we obtain that the distribution is integrable if and only if (93) holds.

Remark 34. If is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold , then
(i) the distribution is integrable if and only if

for any ;
(ii) the distribution is integrable if and only if the shape operator of satisfies

for any and ;
(iii) the distribution is integrable if and only if

for any .

Proof. (i) For any , we have if and only if and from (35), we obtain (94).
(ii) For any and any , from (28) and (2) we have

From (35) and (because ) we have

for any and any . Thus, we have .
(iii) For any , we have and . The distribution is integrable if and only if , which is equivalent to or . Thus, from (34), we obtain that is integrable if and only if (96) holds.

Theorem 35. Let be a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold (. If , then the distributions and are integrable.

Proof. First of all, we consider and we prove . For any , we get and using in (39)(i) we obtain , for any , which implies . From

and (69) we get , for any and . Thus, from (1) and (2) we get

for any and . Moreover, for , we obtain

which implies

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

Using (102), (12) and , for any , we obtain

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

From , we obtain .
By using , we have

which implies . Thus, we get and we obtain . In conclusion, for any and this implies . Thus, the distribution is integrable.
Moreover, because is orthogonal to and ( is a Riemannian manifold, we obtain the integrability of the distribution .

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 mixed totally geodesic (i.e., , for any and ), in a similar manner as in the case of semi-slant submanifolds in locally product manifolds ([16]).

Proposition 36. If is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold , then is a mixed totally geodesic submanifold if and only if and , for any , and .

Proof. From (28) we remark that is a mixed totally geodesic submanifolds in the locally metallic (or locally Golden) Riemannian manifolds if and only if , for any and , which is equivalent to and .

Proposition 37. Let be a proper semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold (. If is a mixed totally geodesic submanifold, then , for any , and .

Proof. If is a mixed geodesic submanifold, then and using (39)(ii), we obtain , for any , .

Theorem 38. Let be a proper semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold (. If , for any , , and is not an eigenvector of the tensor field with the eigenvalue , then is a mixed totally geodesic submanifold.

Proof. If is a semi-slant submanifold of a locally metallic (or locally Golden) Riemannian manifold ( and , then from (39)(ii), we get

for any and , where for any . By using , for any and (76), we obtain

where is the slant angle of the distribution . Using , for any , we obtain

Thus, we get

for any and . From (108), (110) and ( is a proper semi-slant distribution), we remark that and this implies , for any and , because is not an eigenvector of with the eigenvalue . Thus, is a mixed totally geodesic submanifold in the locally metallic (or locally Golden) Riemannian manifold (.

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

Proposition 39. Let be a semi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold (). Then is parallel if and only if the shape operator verifies

for any and .

Proof. From (2), we get , for any , . Thus, by using (39)(ii), we obtain

for any , and we have

for any , . Thus, from (113) we obtain (111).

Theorem 40. Let be a proper semi-slant submanifold in a locally metallic (or locally Golden) Riemannian manifold (). If the shape operator verifies , for any , and is not an eigenvector of the tensor field with the eigenvalue then, is a mixed totally geodesic submanifold.

Proof. If , for any , then, from (113) we obtain for any and using the Theorem 38, we obtain that is a mixed totally geodesic submanifold.

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-GRU13-03339, Ref. no. GRU 09-GRAT-20-USV.

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