1. Introduction
A polynomial structure on a differentiable manifold
M, defined in [
1], arises as a
-tensor field
f of type
, which satisfies the algebraic equation
where
is the identity map on the Lie algebra of vector fields on
M and
are linearly independent for every
. The polynomial
is called the structure polynomial.
The almost complex structure and the almost product structure have the structure polynomial and , respectively.
An (
)-golden structure on a differentiable manifold has the structure polynomial of the form
where
p is a nonzero real number and
[
2].
By adding a compatible Riemannian metric, we focus on the study of the structure induced on submanifolds in this setting and on its properties, and we discuss the case of semi-invariant submanifolds in an ()-golden Riemannian manifold.
This structure is a generalization of the golden structure, determined by an endomorphism
which satisfies the polynomial equation
(introduced in [
3]). Also, it is related to the metallic structure, which is a solution of the polynomial equation
, where
p and
q are positive integers (see the references [
4,
5]). These structures can be obtained from the almost product structures on a differentiable manifold.
On the other hand, an almost complex metallic structure is defined as an endomorphism
J which satisfies the relation
(see [
6]). For
, the almost complex metallic structure becomes a complex golden structure (defined in [
3]). These structures are related to the almost complex structures on an even dimensional differentiable manifold.
The almost product and the almost complex structures can be unified under the notion of
-structure, denoted by
(with the property
, where
), defined in [
7]. The structure
studied in this paper is related to the
-structure on an even dimensional differentiable manifold.
Semi-invariant submanifolds in different kinds of ambient manifolds have been defined and studied by many geometers. Firstly, semi-invariant submanifolds in locally Riemannian product manifolds were introduced in 1960 by S. Tachibana [
8] and then studied by A. Bejancu and N. Papaghiuc [
9,
10].
Semi-invariant submanifolds in Riemannian manifolds correspond to CR-submanifolds in complex manifolds. D. Blair and B.Y. Chen studied the properties of CR-submanifolds of Hermitian manifolds in [
11]. Moreover, CR-submanifolds of Kaehler manifolds were studied by B.Y. Chen in [
12,
13] and A. Bejancu in [
14,
15].
In the paper [
16], the authors studied semi-invariant submanifolds of a
-manifold, which are extensions of CR-submanifolds to this general class of manifolds.
The properties of semi-invariant submanifolds in golden (or metallic) Riemannian manifolds were treated in [
17,
18,
19].
In the present paper, we present some properties of submanifolds of the ()-golden Riemannian manifold induced by the ()-golden structure of the ambient manifold, especially in the case of semi-invariant submanifolds.
The paper is organized as follows. In
Section 2, we present some basic facts regarding a Riemannian manifold endowed with an
-golden structure and a compatible Riemannian metric
g, called an almost
-golden Riemannian manifold.
In
Section 3, we deal with the investigation of the structure induced on submanifolds by the almost
-golden structure of a Riemannian manifold. Some properties of the projection operators are given when the condition
is satisfied, where
is the Levi-Civita connection on the ambient almost
-golden Riemannian manifold.
In the last section we focus on the characterization of semi-invariant submanifolds in an -golden Riemannian manifold. We study the integrability conditions of both invariant and anti-invariant distributions. Finally, we obtain a characterization of the mixed totally geodesic semi-invariant submanifolds in an -golden Riemannian manifold.
2. Characterization of the (, )-Golden Riemannian Manifold
In this section, we consider several frameworks useful for our study. Let be an even dimensional manifold and let be the set of smooth sections of .
It is known that a manifold has an (
)-structure if
is endowed with a tensor field
of type
, which, according to [
7], satisfies the equations:
and
for any vector fields
, where
. The tensor field
of type
is an isometry, for
, or an anti-isometry, for
.
In particular, for
= 1 in the equality (
2), one obtains the following:
If
, then
is an almost complex structure and
is an almost Hermitian manifold ([
20] p. 124);
If
, then
is an almost product structure and
is an almost product manifold ([
20] p. 423).
In our paper, we will consider an -structure (having ), which is an isometry ( = 1).
If
is an even dimensional manifold, then the almost golden structure and the almost complex golden structure are related to the
-structure
, which is an endomorphism of the total space
of the tangent bundle that satisfies relation (
1). Moreover, if we fix a Riemannian metric
such that
, we obtain
which is equivalent to
for any vector fields
.
For
, one obtains that
is a pure metric [
21] and the structure
is an almost product structure.
Definition 1 ([
2], Definition 2).
An endomorphism of the total space of the tangent bundle is called an )-golden structure on if it satisfies the equalitywhere p is a nonzero real number and . Definition 2 ([
2]).
An almost -golden Riemannian manifold is a triple , where is a Riemannian metric on an even dimensional manifold , which verifies the equalityfor any . Remark 1 ([
2]).
If is an almost -golden Riemannian manifold, then the Riemannian metric on verifiesfor any . In particular, for
= (1, 1), the almost (
)-golden structure becomes an almost golden structure
and
turns into an almost golden manifold, which was studied in [
3].
On the other hand, if
= (−1, 1), then one obtains an almost complex golden structure determined by the endomorphism
, which satisfies the equation
. In this case,
is called an almost complex golden manifold and it was studied in [
22,
23].
Remark 2. Let be an almost -golden structure. The structure is an almost -golden structure, too ([2]). Proposition 1 ([
2], Proposition 1).
Every α-structure on defines two almost -golden structures, given by the equalityConversely, two α-structures can be associated to a given almost -golden structure as follows: Example 1. Let us assume that is a Riemannian manifold of dimension 2 m. We can define an α-structure , given byfor any and any integer number . The metric g is given byfor any , . We can verify thatfor any . Using the identity (8) in (9), we obtain an -golden structure , given by the equalityMoreover, by using (8) in (10), we can verify that the metric g satisfies the equalityfor any . Thus, is an almost -golden Riemannian manifold.
Now, let
be the Levi-Civita connection on
. The covariant derivative
is a tensor field of the type (1, 2), defined by
for any
.
Using the compatibility formula
, we obtain ([
2])
for any vector fields
.
Let us consider the Nijenhuis tensor field of an
-structure
, defined by the relation
for any
. The Nijenhuis tensor field corresponding to the
-golden structure
is given by the equality ([
2])
and it verifies
for any
.
An -structure on a differentiable manifold is integrable if the Nijenhuis tensor field , corresponding to , vanishes identically (i.e., ).
Remark 3. is integrable if and only if the associated almost α-structure is integrable.
Remark 4. is integrable (i.e., ) if .
From ([
20], Theorem 3.1, p. 125), we remark that if
, then
is an almost complex structure on the manifold
) and
is a complex manifold (i.e.,
) if and only if it admits a linear connection
such that
and
, where
T denotes the torsion of
. Thus, from (
11) and (
12), we obtain the following property:
Proposition 2. The structure on an almost -golden Riemannian manifold is integrable (i.e., ) if and only if it admits a linear connection , having the torsion , such that .
From ([
20], Theorem 2.3, p. 420), it is known that an integrable almost product Riemannian manifold, with structure tensor
J, is a locally product Riemannian manifold. Sufficient conditions for the integrability of almost product structures on Riemannian manifolds were presented in [
21], where it was shown that the condition
is equivalent to decomposability of the pure metric
. In [
24], the authors studied an integrability condition for the locally decomposable metallic Riemannian structures.
Definition 3. The structure on an almost -golden Riemannian manifold is called locally decomposable if , where is the Levi-Civita connection corresponding to the metric .
Taking into account these observations, we may consider an almost -golden Riemannian manifold which is covariant constant (i.e., , where is the Levi-Civita connection corresponding to the metric ) and we introduce the following definition:
Definition 4. A locally -golden Riemannian manifold is an almost -golden Riemannian manifold whose -golden structure is parallel with respect to the Levi-Civita connection (i.e., ).
3. Submanifolds in (, )-Golden Riemannian Manifold
In this section, we assume that
M is an isometrically immersed submanifold in an even dimensional almost
-golden Riemannian manifold
. If
is the Lie algebra of vector fields on
M and
(respectively
) is the tangent space (respectively, the normal space) of
M at a given point
, one obtains the direct sum
Let g be the induced Riemannian metric on M, given by for any , where is the differential of the immersion . We shall assume that all immersions are injective.
In the rest of the paper, one uses the simple notation , for any .
By using (
1), (
3), and (
4), we obtain that the induced metric on the submanifold
M and the
-structure
verify the equalities:
for any
. Moreover, using relations (
6) and (
7), one obtains the equality
which is equivalent to the equality
for any
.
Let be the second fundamental form of M in and let be the shape operator of M with respect to . One denotes by the normal connection on the normal bundle .
The Gauss and Weingarten formulas are
and
respectively, for any tangent vector fields
and for any normal vector field
, where
and ∇ are the Levi-Civita connections on
and on the submanifold
M, respectively. Moreover, the second fundamental form
h and the shape operator
are related by
First of all, we consider the endomorphisms given by the relations
and
for any tangent vector field
and any normal vector field
.
On the other hand, we consider the operators (bundle-valued 2-forms) given by
and
for any tangent vector field
and any normal vector field
.
For any vector field
, we have the decomposition into the tangential and normal parts of
given by the equality
Similarly, for any vector field
, the decomposition into the tangential and normal parts of
is given by the equality
Proposition 3 ([
2], Proposition 8).
Let be a Riemannian manifold endowed with an almost -golden structure . Thus, for any , the maps and satisfy the equalitiesandfor any and any . Moreover, and satisfyfor any . Remark 5. If is an almost -golden Riemannian manifold, then we obtain the equalitiesfor any , andfor any . Proposition 4. The structure induced on a submanifold M by an almost -golden structure on satisfies the following equalities:andfor any . Moreover, for any , one obtains the equalitiesandwhere the operators , and are defined in (16)–(19). Proof. By using (
5) and the decomposition (
20) for
in the equality
, one obtains
for any
. Equalizing the tangential and the normal parts from both members of the last equality, we obtain the relations (
25) and (
26), respectively.
In the same manner, the decomposition Formula (
21) for
applied in
leads to the equality
for any
, which implies the relations (
27) and (
28), respectively. □
We remark that, if we consider
in (
25), then the endomorphism
verifies the equation of the
-golden structure (
5). Since the induced metric
g on the submanifold
M verifies the compatibility relation (
22), we obtain the following property regarding the induced structure on a submanifold:
Proposition 5. If M is a submanifold isometrically immersed in an almost -golden Riemannian manifold () and the operators and , defined in (18) and (19), satisfy , then the submanifold is also an almost -golden Riemannian manifold. Now, we recall a property proved in ([
2], Theorem 2), similar to one in the case of the metallic Riemannian manifolds (see ([
4], Proposition 4.3)).
Proposition 6 ([
2]).
A necessary and sufficient condition for the invariance of a submanifold M in an even dimensional Riemannian manifold , endowed with an almost -golden structure , is that the structure , induced on M by , is also an almost -golden structure. The covariant derivatives of the tangential and normal parts of
are given, for any
, by the equalities
and
Moreover, the covariant derivatives of the tangential and normal parts of
are given, for any
and
, by the equalities
and
Proposition 7. If is a locally -golden Riemannian manifold, then the operators , and , defined in (16)–(19), verify the equalitiesandfor any and . Proof. From
, using the Gauss and Weingarten Equations (
13) and (
14), for any
X,
, one obtains the equality
Equalizing the tangential and the normal components from both members of equality (
37) and using (
29) and (
30), we obtain (
33) and (
34), respectively.
From
, using the Gauss and Weingarten Equations (
13) and (
14), for any
and
, one obtains the equality
Equalizing the tangential and the normal components from both members of equality (
38) and using (
31) and (
32), we obtain (
35) and (
36), respectively. □
Proposition 8. If M is an isometrically immersed submanifold in a locally -golden Riemannian manifold (in the sense of the Definition 4), then the operators , and , defined in (16)–(19), verify the equalitiesandfor any and any . Proof. First of all, using the relations (
15), (
24), and (
33), one obtains
for any
. On the other hand, interchanging
Y and
Z in (
42), we have the equality
Thus, from (
42) and (
43) multiplied by
, we obtain Equation (
39).
Using (
24) and (
36), we obtain the equality
for any
and any
, which implies Equation (
40).
From the relations (
15), (
22)–(
24), (
34) and (
35) we obtain the equalities
for any
and any
, which imply (
41). □
By using Equation (
41), we obtain the following property:
Corollary 1. Let M be an isometrically immersed submanifold in a locally -golden Riemannian manifold . Then, if and only if .
Proposition 9. Let M be an isometrically immersed submanifold in a locally -golden Riemannian manifold . Then, if and only if , for any .
Proof. By using Equation (
33), we have
if and only if
for any
and from relations (
15) and (
24), we get
Thus, we obtain the equality
for any
, which implies the conclusion. □
By using (
35) and Corollary 1, one obtains the following property.
Corollary 2. Let M be an isometrically immersed submanifold in a locally -golden Riemannian manifold . Then, (or equivalently, ) if and only if , for any and any .
Proposition 10. Let M be an isometrically immersed submanifold in a locally -golden Riemannian manifold . Then, if and only if for any .
Proof. From the Equation (
36) we get
if and only if
for any
and
. By using (
15) and (
24) in the last equality, we obtain
for any
and
, which implies the conclusion. □
4. Semi-Invariant Submanifolds in (, )-Golden Riemannian Manifold
4.1. Characterization of Semi-Invariant Submanifolds
Let M be an isometrically immersed submanifold in an even dimensional almost -golden Riemannian manifold (. In the rest of this paper, we suppose that M is a semi-invariant submanifold in an almost -golden Riemannian manifold .
Definition 5. A submanifold M is called a semi-invariant submanifold in if admits two orthogonally complementary distributions D and (i.e., ), where D is an invariant distribution with respect to (i.e., ) and is an anti-invariant distribution with respect to (i.e., ).
The orthogonally complementary distributions D and are called the horizontal and the vertical distribution on M, respectively.
We denote the dimension of the invariant distribution D by and of the anti-invariant distribution by . Thus, we can have the following situations:
For , the semi-invariant submanifold becomes an invariant submanifold;
For , the semi-invariant submanifold becomes an anti-invariant submanifold;
If , the semi-invariant submanifold is called a proper semi-invariant submanifold.
Now, we denote the orthogonal complement of
in
by
. Then, we have the direct sum
Proposition 11. If M is a semi-invariant submanifold isometrically immersed in an almost -golden Riemannian manifold , then the orthogonal complementary distributions of are invariant with respect to the structure and
Proof. If
and
, then
. If
, for
, from (
5) and (
6) we get
Therefore,
.
If
and
, then
and one obtains
and it implies
.
Moreover if
and
, then
and it follows that
which implies
.
Thus, .
On the other hand, if
, then, using (
5), we have
which leads to
. Thus, we have
□
Let us denote by
P and
Q the projection morphism of
to the orthogonally complementary distributions
D and
, respectively. Thus, for any
, one obtains
Proposition 12. If M is a semi-invariant submanifold isometrically immersed in an almost -golden Riemannian manifold and D and are the horizontal and vertical distributions on M, respectively, then we obtainandwhere and were defined in (16) and (20). Proof. From (
20) and (
44), we have
for any
. By using Definition 5, it follows that
and
for any
. Thus, we obtain
and
, for any
. Moreover, for any
, we have
and
, which leads to (
45) and (
46). □
If we replace
X by
in (
25), then we have the following property.
Proposition 13. If M is a semi-invariant submanifold isometrically immersed in an almost -golden Riemannian manifold such that for any , then we obtain the equalitiesandfor any . Proposition 14. Let M be a semi-invariant submanifold isometrically immersed in an almost -golden Riemannian manifold . If D is the horizontal distribution of M and P is the projector operator on D, then one obtainsandfor any . From the relation (
48), we obtain the following property:
Remark 6. If M is a semi-invariant submanifold isometrically immersed in an almost -golden Riemannian manifold, then the structure is an -golden structure on D.
Theorem 1. Let M be a submanifold isometrically immersed in an almost -golden Riemannian manifold , such that , for any . If l and m are two operators on , defined by the equalitiesthen l and m are orthogonal complementary projection operators on . Moreover, we obtain the following equalitiesandfor any . Proof. First of all, applying
in (
49) (i), we have
and using the relation (
47), we obtain (
50) (i).
Similarly, applying
in (
49) (ii), we have
and from (
47), we obtain (
50) (ii).
Now, applying
in (
49) (i), we get
and using
we obtain (
51) (i). Moreover, applying
in (
49) (ii), we have
and using
(
51) (ii) is proved.
On the other hand, we prove that
l and
m are orthogonal complementary projection operators, which means that they satisfy the equalities:
By using (
49) it follows that
. Moreover, applying (
49) (i) to
, it leads to
for any
and from relations (
50) (i) and (
49) (i), we obtain
Similarly, applying (
49) (ii) to
, we obtain
for any
. From the relations (
50) (ii) and (
49) (ii), we have
.
From the relations (
49) (i) and (
50) (ii), we have
for any
. By using (
49) (ii), (
50), and (
49) (i), we get
for any
. □
Theorem 2. Let M be an isometrically immersed submanifold in an almost -golden Riemannian manifold . Then M is a semi-invariant submanifold in if and only if the operators and verify the identity .
Proof. If we suppose that
M is a
semi-invariant submanifold in
and using the decomposition of
X given in (
44), then, from
and
, we obtain
for any
.
Conversely, if we consider that
for any
, then the operators
l and
m defined in (
49) are orthogonal complementary projection operators and they define two complementary distributions
and
on
. Also, from any
, we obtain
, and using (
50) and (
51), we have
By using (
52), we have
for any
, and
for any
. Thus, it follows that the distribution
is an invariant distribution and the distribution
is an anti-invariant distribution with respect to
. Hence,
M is a semi-invariant submanifold in
. □
In particular, for and , we obtain the following property.
Corollary 3 ([
18], Theorem 1).
Let M be any submanifold of a golden Riemannian manifold . Then, a necessary and sufficient condition for the submanifold M to be semi-invariant is that . A similar property to that given in Corollary 3 is obtained in the case of an isometrically immersed submanifold in a metallic Riemannian manifold.
Corollary 4 ([
19], Proposition 3.2).
If M is an isometrically immersed submanifold in a metallic Riemannian manifold , then the submanifold M is semi-invariant if and only if . 4.2. On the Integrability of the Distributions of Semi-Invariant Submanifolds
In order to study the integrability of the invariant distribution D and of the anti-invariant distribution , we calculate the tangential and the normal components of respectively, given in the next lemma.
Lemma 1. If M is an isometrically immersed submanifold in a locally -golden Riemannian manifold , then the normal and the tangential part of are given by the equalitiesandrespectively, for any , where ∇
is the Levi-Civita connection on M. Proof. Using
and (
33), we have the equality
for any
. Interchanging
X by
Y and subtracting these two equalities, it leads to the relation (
53).
By using the relation (
34), it follows that
for any
. Interchanging
X by
Y and subtracting these two equalities, we obtain (
54). □
Theorem 3. Let M be a semi-invariant submanifold isometrically immersed in a locally -golden Riemannian manifold . Thus, we obtain the following:
The invariant distribution D is integrable if and only if we have the equalityfor any ; The anti-invariant distribution is integrable if and only if we have the equalityfor any .
Proof. Let us consider
which implies
. The distribution
D is integrable if and only if
for any
. By using
, for any
and (
54), we obtain that the distribution
D is integrable if and only if (
55) holds.
Now, let us consider
which implies
. The distribution
is integrable if and only if
, for any
. By using
, for any
and (
53), we obtain that the distribution
is integrable if and only if (
56) holds. □
In particular, for , one obtains the following:
Corollary 5 ([
17], Theorem 2.1).
Let M be a semi-invariant submanifold in a golden Riemannian manifold . Then, the distribution D is integrable if and only if , for any . Using (
33) and (
34) in Theorem 3, we obtain the following necessary and sufficient conditions for the integrability of the invariant and anti-invariant distributions, respectively.
Corollary 6. If M is a semi-invariant submanifold isometrically immersed in a locally -golden Riemannian manifold and D and are the invariant and anti-invariant distributions on M, respectively, then the following equivalences hold:
The distribution D is integrable if and only if we have the equalityfor any ; The distribution is integrable if and only if we have the equalityfor any .
Corollary 7. Let M be a semi-invariant submanifold isometrically immersed in a locally -golden Riemannian manifold . The invariant distribution D is integrable if and only if the following equality holdsfor any . Proof. If we consider that the horizontal distribution
D is integrable, then by using (
55) from Theorem 3, we obtain the relation
for any
. Thus, from Equation (
5) of
, we obtain (
57).
Conversely, if (
57) holds, then interchanging
X and
Y and subtracting these two equations, we obtain
. Thus, by using Theorem 3, we have that the horizontal distribution
D of
M is integrable. □
Theorem 4. Let M be a semi-invariant submanifold isometrically immersed in a locally -golden Riemannian manifold . Let D and be the invariant and anti-invariant distributions on M, respectively. The distribution D on M is integrable if and only iffor any and any . Proof. From Theorem 3, the horizontal distribution
D of
M is integrable if and only if relation (
55) holds. This implies the equality
for any
and any
.
From (
15) and (
59), we obtain
for any
and any
. Now, using (
22), we have
for any
and any
. By using (
60) and (
61), it follows that
for any
and any
. Thus, the distribution
D on
M is integrable if and only if relation (
58) holds. □
If we consider
(and
= −1, respectively) in (
58), we obtain the next two corollaries.
Corollary 8. If M is a semi-invariant submanifold isometrically immersed in a locally -golden Riemannian manifold , then the horizontal distribution D on M is integrable if and only iffor any and any . Corollary 9. If M is a semi-invariant submanifold isometrically immersed in a locally -golden Riemannian manifold , then the horizontal distribution D on M is integrable if and only iffor any and any . Theorem 5. Let M be a semi-invariant submanifold isometrically immersed in a locally -golden Riemannian manifold and let D and be the horizontal and vertical distributions on M, respectively.
For , the distribution is integrable if and only iffor any ; For , the distribution is integrable if and only iffor any and any .
Proof. We denote
and we consider
and
. Thus, we have
. By using (
13) and (
15), one obtains the equalities
for any
and any
. Now, using (
6) and (
14) and
(which implies
), we have
Moreover, from
, we get
for any
and any
.
For
, the equality (
64) leads to
for any
; thus,
, for any
. On the other hand, from (
56), the distribution
is integrable if and only
. Thus, we obtain (
62).
For
, from (
64), we have
for any
and any
. Thus, by taking (
56) and (
65) into account, we obtain that the distribution
is integrable if and only
, for any
and any
, which implies (
63). □
4.3. Mixed Totally Geodesic Semi-Invariant Submanifolds
In this subsection we study the conditions which imply that the submanifold M is a mixed totally geodesic submanifold in a locally -golden Riemannian manifold .
Definition 6. Let us consider that M is a semi-invariant submanifold isometrically immersed in a locally -golden Riemannian manifold . If , for any and , where D and are the horizontal and vertical distributions on M, respectively, then M is called a mixed totally geodesic submanifold in .
Theorem 6. If M is a semi-invariant submanifold isometrically immersed in a locally -golden Riemannian manifold , then the submanifold M is a mixed totally geodesic submanifold if and only if or , for any , , and .
Proof. The submanifold
M is a
mixed totally geodesic submanifold if and only if we have
for any
,
, and
, and these imply
and
. □