The de Rham Cohomology Classes of Hemi-Slant Submanifolds in Locally Product Riemannian Manifolds

Abstract

This paper aims to discuss the de Rham cohomology of hemi-slant submanifolds in locally product Riemannian manifolds. The integrability and geodesical invariance conditions of the distributions derived from the definition of a hemi-slant submanifold are given. The existence and non-triviality of de Rham cohomology classes of hemi-slant submanifolds are investigated. Finally, an example is presented.

1. Introduction

Anti-slant submanifolds were introduced and searched by Carriazo [1] as a special case of bi-slant submanifolds in almost Hermitian manifolds. Such a class of submanifolds automatically contains both anti-invariant and slant submanifolds. In addition, anti-slant submanifolds are a particular class of generic submanifolds, which was defined by Ronsse [2]. Also, such a class of submanifolds was investigated by different authors under the names of pseudo-slant submanifolds [3] and hemi-slant submanifolds [4] since the phrase “anti-slant” does not refer to slant factors. Throughout the paper, it is preferred to use the phrase “hemi-slant submanifold” instead of such a type of submanifolds.

In [5], Taştan and Özdemir analyzied hemi-slant submanifolds of almost product Riemannian manifolds regarding their fundamental properties and characterization, the integrability of the related distributions, the hemi-slant product, the total umbilicality, the parallelism of the induced canonical structures and the Chen–Ricci inequality. In this context, a similar study to the previously cited one was performed by Hreţcanu and Blaga [6,7] for hemi-slant submanifolds in the setting of metallic Riemannian manifolds. In [8], Al-Solamy et al. gave classification theorems for totally umbilical hemi-slant submanifolds in Kähler manifolds. In [9], Lone et al. studied the geometry of distributions arising from the definition of hemi-slant submanifolds in cosymplectic manifolds. Moreover, the de Rham cohomology of hemi-slant submanifolds was studied in Kähler manifolds [10], nearly Kähler manifolds [11] and metallic Riemannian manifolds [12]. As is known, semi-invariant and CR-submanifolds are hemi-slant submanifolds. There are also noteworthy works on the Rham cohomology of semi-invariant and CR-submanifolds in other various manifolds, such as Kähler manifolds, nearly Kähler manifolds, quasi-Kähler manifolds, paraquaternionic Kähler manifolds, locally product Riemannian manifolds and metallic Riemannian manifolds (for details, see, e.g., [13,14,15,16,17,18]).

In the light of the above studies, the essential aim of this paper is to examine the de Rham cohomology of hemi-slant submanifolds in locally product Riemannian manifolds. Also, this paper generalizes the results of [18].

This paper is organized as follows: Section 1 gives a brief information about some investigations on the geometry of hemi-slant submanifolds. Section 2 deals with some preliminary definitions and concepts which will be used in the subsequent sections. In Section 3, we mainly focus on finding the de Rham cohomology classes of hemi-slant submanifolds in locally product Riemannian manifolds.

2. Preliminaries

A Riemannian manifold $\left(\overline{M},\overline{g}\right)$ admitting an almost product structure $\overline{\Phi }$ is said to be almost product if the Riemannian metric $\overline{g}$ is $\overline{\Phi }$-compatible, i.e.,

$$\overline{g}\left(\overline{\Phi }X,Y\right)=\overline{g}\left(X,\overline{\Phi }Y\right)$$

(1)

for all $X,Y\in \Gamma \left(T\overline{M}\right)$, where $\Gamma \left(T\overline{M}\right)$ is the Lie algebra of vector fields on $\overline{M}$. Particularly, an almost product Riemannian manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$ is called a locally product Riemannian manifold, or briefly an LPR-manifold, if $\overline{\nabla }$ $\overline{\Phi }=0$, where $\overline{\nabla }$ stands for the Riemannian connection on $\overline{M}$ [19].

We consider any m-dimensional isometrically immersed submanifold M of an $\overline{m}$-dimensional almost product Riemannian manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. We denote by $TM$ and $T{M}^{\perp }$ its tangent and normal bundles, respectively. Let $\Gamma \left(TM\right)$ and $\Gamma \left(T{M}^{\perp }\right)$ be the sets of cross-sections of $TM$ and $T{M}^{\perp }$, respectively. We also use the same notation $\overline{g}$ for the induced metric on M. We denote by ∇ and ${\nabla }^{\perp }$ the induced connection on M and the connection on $T{M}^{\perp }$ induced by $\overline{\nabla }$, respectively. In this case, the formulas of Gauss and Weingarten are given, respectively, by

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y+\sigma \left(X,Y\right)$$

(2)

and

$${\overline{\nabla }}_{X}U=-A_{U}X+{\nabla }_X^{\perp }U$$

(3)

for all $X,Y\in \Gamma \left(TM\right)$ and $U\in \Gamma \left(T{M}^{\perp }\right)$, where $\sigma$ is the second fundamental form of M in $\overline{M}$ and $A_U$ is the Weingarten map in regard to U such that

$$\overline{g}\left(\sigma \left(X,Y\right),U\right)=\overline{g}\left(A_{U}X,Y\right)$$

(4)

for all $X,Y\in \Gamma \left(TM\right)$ and $U\in \Gamma \left(T{M}^{\perp }\right)$.

For any $X\in \Gamma \left(TM\right)$, $\overline{\Phi }X$ is worded as follows:

$$\overline{\Phi }X=fX+hX,$$

(5)

where $fX\in \Gamma \left(TM\right)$ and $hX\in \Gamma \left(T{M}^{\perp }\right)$. Similarly, for any $U\in \Gamma \left(T{M}^{\perp }\right)$, we have

$$\overline{\Phi }U=tU+sU,$$

(6)

where $tU\in \Gamma \left(TM\right)$ and $sU\in \Gamma \left(T{M}^{\perp }\right)$. It follows that the bundle morphisms f and s are $\overline{g}$-symmetric on $TM$ and $T{M}^{\perp }$, respectively. Also, from (1), (5) and (6), there exists a connection between h and t such that

$$\overline{g}\left(hX,U\right)=\overline{g}\left(X,tU\right)$$

(7)

for all $X\in \Gamma \left(TM\right)$ and $U\in \Gamma \left(T{M}^{\perp }\right)$. In addition, we get

$$f^2+th=I,$$

(8)

$$hf+sh=0,$$

(9)

$$ft+ts=0$$

(10)

and

$$s^2+ht=I.$$

(11)

For more details about LPR-manifolds and their submanifolds, we refer the reader to [20].

An isometrically immersed submanifold M of an almost product Riemannian manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$ is called hemi-slant [6] if the tangent bundle $TM$ has the ortogonal direct decomposition $TM=R^{\varphi }\oplus R^{\perp }$, where $R^{\varphi }$ is a slant distribution with the Wirtinger angle $\varphi \in \left[0,\frac{\pi }{2}\right]$ and $R^{\perp }$ is an anti-invariant distribution.

Moreover, a hemi-slant submanifold M of an almost product Riemannian manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$ is said to be proper if $R^{\varphi }\ne \left\{0\right\}$ and $R^{\perp }\ne \left\{0\right\}$, where $\varphi \in \left(0,\frac{\pi }{2}\right)$. Otherwise,

(a)

M is an invariant submanifold if $dim{R}^{\perp }=0$ and $\varphi =0$;

(b)

M is an anti-invariant submanifold if $dim{R}^{\varphi }=0$;

(c)

M is a slant submanifold with the Wirtinger angle $\varphi$ if $dim{R}^{\perp }=0$;

(d)

M is a semi-invariant submanifold if $\varphi =0$.

It can be understood from above that invariant, anti-invariant, slant and semi-invariant submanifolds are non-proper hemi-slant submanifolds.

For a proper hemi-slant submanifold M of an almost product Riemannian manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$, by ([5], Lemmas 3.2 and 3.3), we have

$$f{R}^{\varphi }=R^{\varphi },f{R}^{\perp }=\left\{0\right\},h{R}^{\perp }=\overline{\Phi }{R}^{\perp }\mathrm{and}h{R}^{\varphi }\perp h{R}^{\perp }.$$

In addition, $T{M}^{\perp }$ can be expressed in the following form:

$$T{M}^{\perp }=h{R}^{\varphi }\oplus h{R}^{\perp }\oplus \nu ,$$

where $\nu$ stands for the orthogonal complementary distribution of $h{R}^{\varphi }\oplus h{R}^{\perp }$ in $T{M}^{\perp }$ and is a $\overline{\Phi }$-invariant subbundle of $T{M}^{\perp }$. We also recall the following results:

Theorem 1

([5], Theorem 4.3). Let M be any hemi-slant submanifold of an LPR-manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. Then, the integrability of the slant distribution $R^{\varphi }$ is equivalent to

$$A_{hX}Y-A_{hY}X+{\nabla }_Y^{\perp }hX-{\nabla }_X^{\perp }hY\in \Gamma \left(R^{\varphi }\right)$$

for all $X,Y\in \Gamma \left(R^{\varphi }\right)$.

Corollary 1

([5], Corollary 4.7). Let M be any hemi-slant submanifold of an LPR-manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. Then, the integrability of the anti-invariant distribution $R^{\perp }$ is equivalent to

$$A_{\overline{\Phi }Z}W=0$$

(12)

for all $Z,W\in \Gamma \left(R^{\perp }\right)$.

Let M be a differentiable manifold. We denote by $Z^n\left(M\right)$ and $B^n\left(M\right)$ the vector space of all n-cocycles on M and the vector space of all n-coboundaries on M, respectively, i.e.,

$$Z^n\left(M\right)=ker\left(d:{\Omega }^n\left(M\right)\to {\Omega }^{n+1}\left(M\right)\right)$$

and

$$B^n\left(M\right)=Im\left(d:{\Omega }^{n-1}\left(M\right)\to {\Omega }^n\left(M\right)\right),$$

where ${\Omega }^n$ is the vector space of all differential n-forms on M and d is the exterior derivative. The quotient space

$$H_{dR}^n\left(M\right)=Z^n\left(M\right)/B^n\left(M\right)$$

is said to be the n-th de Rham cohomology group of M. The equivalence class of a differential n-form $\alpha$ in $H_{dR}^n\left(M\right)$ is called the Rham cohomology class of $\alpha$, denoted by $\left[\alpha \right]$. Furthermore, the dimension of $H_{dR}^n\left(M\right)$ is called the n-th Betti number, which is denoted by $b_n$ [21].

In addition, from Hodge theorem ([22], Theorem 8.12), for a closed orientable Riemannian manifold M, there is a canonical isomorphism

$$H_{dR}^n\left(M\right)\cong {\mathcal{H}}_{\Delta }^n\left(M\right),$$

where ${\mathcal{H}}_{\Delta }^n\left(M\right)$ denotes the space of all harmonic n-forms on M, i.e., the kernel space of the Hodge–de Rham Laplacian defined by $\Delta =d\delta +\delta d$ on ${\Omega }^n$, in which $\delta$ is the co-differential operator. It is worth noting that the assumption on a hemi-slant submanifold of an LPR-manifold being closed in this paper is crucial to use Hodge theorem. However, Alexandru Rugina [23] obtained $L^p$ harmonic n-forms, which are neither closed nor co-closed on the hyperbolic space ${\mathbb{H}}_{-1}^m$, $m\ge 3$, where $1<p\ne 2<3$ and $0\le n\le \frac{m}{2}$. On the other hand, by means of condition W, on complete noncompact Riemannian manifolds, Wei [24] showed that a differential n-form is $L^p$ harmonic if and only if it is both closed and co-closed, where $1<p<3$.

A differentiable distribution R on a Riemannian manifold M is called geodesically invariant [25] if

$${\nabla }_{X}X\in \Gamma \left(R\right),$$

or equivalently

$$\left({\nabla }_{X}Y+{\nabla }_{Y}X\right)\in \Gamma \left(R\right)$$

for all $X,Y\in \Gamma \left(R\right)$, where ∇ is the Riemannian connection on M. Moreover, if $\left\{E_1,\dots ,E_r\right\}$ is a local orthonormal frame of R, then the mean curvature vector of R is defined by

$$H_R=\frac{1}{r}\sum _{i=1}^r{\left({\nabla }_{E_i}{E}_i\right)}^{\perp },$$

where ${\left({\nabla }_{E_i}{E}_i\right)}^{\perp }$ stands for the component of ${\nabla }_{E_i}{E}_i$ in the orthogonal complementary of R in M. If $H_R$ vanishes identically, i.e., $H_R=0$, then R is named a minimal distribution [13]. Also, it is not difficult to see that R is minimal if and only if R is geodesically invariant.

3. Main Results

Let M be a hemi-slant submanifold of an almost product Riemannian manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$ with the slant distribution $R^{\varphi }$ and the anti-invariant distribution $R^{\perp }$ such that $dim{R}^{\varphi }=p$ and $dim{R}^{\perp }=q$, where $\varphi$ is its Wirtinger angle.

We consider two globally well-defined differential forms $\alpha$ and $\beta$ on M given by $\alpha ={\alpha }^1\Lambda \cdots \Lambda {\alpha }^p$ and $\beta ={\beta }^{p+1}\Lambda \cdots \Lambda {\beta }^{p+q}$ such that ${\alpha }^1,\dots ,{\alpha }^p,{\beta }^{p+1},\dots ,{\beta }^{p+q}$ are differential one-forms on M with the following properties:

$${\alpha }^k\left(Z\right)=0,{\alpha }^k\left(E_l\right)={\delta }_{kl},1\le k,l\le p$$

(13)

and

$${\beta }^{p+K}\left(X\right)=0,{\beta }^{p+K}\left(E_{p+L}\right)={\delta }_{KL},1\le K,L\le q$$

(14)

for all $X\in \Gamma \left(R^{\varphi }\right)$ and $Z\in \Gamma \left(R^{\perp }\right)$, where ${\mathfrak{B}}_{R^{\varphi }}=\left\{E_1,\dots ,E_p\right\}$ and ${\mathfrak{B}}_{R^{\perp }}=\left\{E_{p+1},\dots ,E_{p+q}\right\}$ are the local orthonormal frames of $R^{\varphi }$ and $R^{\perp }$, respectively. It follows that each of $\alpha$ and $\beta$ is a volume form. Thus, $R^{\varphi }$ and $R^{\perp }$ are orientable in regard to ${\mathfrak{B}}_{R^{\varphi }}$ and ${\mathfrak{B}}_{R^{\perp }}$, respectively. In addition, $\alpha \Lambda \beta$ is a globally well-defined differential m-form and M is orientable in regard to ${\mathfrak{B}}_{R^{\varphi }}\cup {\mathfrak{B}}_{R^{\perp }}$ of $TM$.

Proposition 1.

Let M be any hemi-slant submanifold of an LPR-manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. Then, the geodesical invariance of the slant distribution $R^{\varphi }$ in M is equivalent to

$$\sigma \left(X,fX\right)+{\nabla }_X^{\perp }hX\in \Gamma \left(h{R}^{\varphi }\oplus \nu \right)$$

for all $X\in \Gamma \left(R^{\varphi }\right)$. In particular, $R^{\varphi }$ is geodesically invariant in M if the following expressions hold:

(a)

M is $R^{\varphi }$-geodesic;

(b)

For all $X\in \Gamma \left(R^{\varphi }\right)$,

$${\nabla }_X^{\perp }hX\in \Gamma \left(h{R}^{\varphi }\oplus \nu \right).$$

Proof. 

Taking into account that the fact $\overline{\nabla }$ $\overline{\Phi }=0$, by virtue of (1), (2), (3) and (5), an elementary calculation gives us

$$\overline{g}\left({\nabla }_{X}X,Z\right)=\overline{g}\left(\sigma \left(X,fX\right)+{\nabla }_X^{\perp }hX,\overline{\Phi }Z\right)$$

(15)

for all $X\in \Gamma \left(R^{\varphi }\right)$ and $Z\in \Gamma \left(R^{\perp }\right)$, which is the desired equality. □

Proposition 2.

Let M be any hemi-slant submanifold of an LPR-manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. Then, the geodesical invariance of the anti-invariant distribution $R^{\perp }$ in M is equivalent to

$$f{A}_{\overline{\Phi }Z}Z-t{\nabla }_Z^{\perp }hZ\in \Gamma \left(R^{\perp }\right)$$

for all $Z\in \Gamma \left(R^{\perp }\right)$. Specifically, each leaf of $R^{\perp }$ is geodesically invariant in M if the following expressions are satisfied:

(a)

For all $Z,W\in \Gamma \left(R^{\perp }\right)$,

$$A_{\overline{\Phi }Z}W=0;$$

(b)

For all $Z\in \Gamma \left(R^{\perp }\right)$,

$${\nabla }_Z^{\perp }hZ\in \Gamma \left(h{R}^{\perp }\oplus \nu \right).$$

Proof. 

Because $\overline{M}$ is an LPR-manifold, we deduce from (1), (2), (3) and (5) that

$$\overline{g}\left({\nabla }_{Z}Z,X\right)=-\overline{g}\left(f{A}_{\overline{\Phi }Z}Z-t{\nabla }_Z^{\perp }hZ,X\right)$$

for all $X\in \Gamma \left(R^{\varphi }\right)$ and $Z\in \Gamma \left(R^{\perp }\right)$, which finishes the proof. □

Theorem 2.

Let M be any closed proper hemi-slant submanifold of an LPR-manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. Then, there exists a canonical de Rham cohomology class $\left[\beta \right]$ determined by the differential q-form β in $H_{dR}^q\left(M\right)$ if the following expressions hold:

(a)

For all $X,Y\in \Gamma \left(R^{\varphi }\right)$,

$$A_{hX}Y-A_{hY}X+{\nabla }_Y^{\perp }hX-{\nabla }_X^{\perp }hY\in \Gamma \left(R^{\varphi }\right);$$

(b)

For all $Z\in \Gamma \left(R^{\perp }\right)$,

$$f{A}_{\overline{\Phi }Z}Z-t{\nabla }_Z^{\perp }hZ\in \Gamma \left(R^{\perp }\right).$$

In addition, $\left[\beta \right]\ne 0$ if the following expressions are verified:

(c)

For all $Z,W\in \Gamma \left(R^{\perp }\right)$,

$$A_{\overline{\Phi }Z}W=0;$$

(d)

For all $X\in \Gamma \left(R^{\varphi }\right)$,

$$\sigma \left(X,fX\right)+{\nabla }_X^{\perp }hX\in \Gamma \left(h{R}^{\varphi }\oplus \nu \right).$$

Proof. 

By the definition of $\beta$, we write

$$d\beta =\sum _{K=1}^q{\left(-1\right)}^{p+K}{\beta }^{p+1}\Lambda \cdots \Lambda d{\beta }^{p+K}\Lambda \cdots \Lambda {\beta }^{p+q},$$

(16)

so it is deducible from (14) that $\beta$ is closed if and only if

$$d\beta \left(Z,E_{p+1},\dots ,E_{p+q}\right)=0$$

(17)

and

$$d\beta \left(Z,W,E_{p+1},\dots ,E_{p+q-1}\right)=0$$

(18)

for all $Z,W\in \Gamma \left(R^{\perp }\right)$. On the other hand, taking account of Theorem 1 and Proposition 2, (a) and (b) imply the integrability of $R^{\varphi }$ and the geodesical invariance of $R^{\perp }$, respectively. Thus, by the two expressions mentioned, it is understood that (17) and (18) hold, that is, $d\beta =0$. Therefore, we can talk about the existence of a canonical de Rham cohomology class represented by $\left[\beta \right]$ in $H_{dR}^q\left(M\right)$. Assuming (c) and (d) are true, by a similar argument as above, it can be concluded from Corollary 1 and Proposition 1 that $\alpha$ is closed. Also, as can be clearly seen, $\alpha$ is the Hodge dual of $\beta$, so $\beta$ is a co-closed differential q-form. In addition, since M is a closed submanifold, $\beta$ is an harmonic q-form. As well, M is orientable in regard to ${\mathfrak{B}}_{R^{\varphi }}\cup {\mathfrak{B}}_{R^{\perp }}$. As a result, Hodge theorem says that $\left[\beta \right]\ne 0$ in $H_{dR}^q\left(M\right)$. □

Theorem 3.

Let M be any closed proper hemi-slant submanifold of an LPR-manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. Then, the differential q-form β represents a canonical de Rham cohomology class denoted by $\left[\beta \right]$ in $H_{dR}^q\left(M\right)$ if the following expressions hold:

(a)

For all $X,Y\in \Gamma \left(R^{\varphi }\right)$,

$$A_{hX}Y-A_{hY}X+{\nabla }_Y^{\perp }hX-{\nabla }_X^{\perp }hY\in \Gamma \left(R^{\varphi }\right);$$

(b)

For all $Z,W\in \Gamma \left(R^{\perp }\right)$,

$$A_{\overline{\Phi }Z}W=0;$$

(c)

For all $Z\in \Gamma \left(R^{\perp }\right)$,

$${\nabla }_Z^{\perp }hZ\in \Gamma \left(h{R}^{\perp }\oplus \nu \right).$$

Furthermore, $\left[\beta \right]\ne 0$ if the following expressions are verified:

(d)

M is $R^{\varphi }$-geodesic;

(e)

For all $X\in \Gamma \left(R^{\varphi }\right)$,

$${\nabla }_X^{\perp }hX\in \Gamma \left(h{R}^{\varphi }\oplus \nu \right).$$

Proof. 

By Propositions 1 and 2, the proof follows immediately from Theorem 2. □

Theorem 4.

Let M be any closed totally geodesic proper hemi-slant submanifold of an LPR-manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. Then, the de Rham cohomology class of the differential q-form β in $H_{dR}^q\left(M\right)$ is non-trivial if the following expressions are valid:

(a)

For all $Z,W\in \Gamma \left(R^{\perp }\right)$,

$${\nabla }_Z^{\perp }hW\in \Gamma \left(h{R}^{\perp }\oplus \nu \right);$$

(b)

For all $X,Y\in \Gamma \left(R^{\varphi }\right)$,

$${\nabla }_X^{\perp }hY\in \Gamma \left(h{R}^{\varphi }\oplus \nu \right).$$

Proof. 

If M is a totally geodesic submanifold with (a) and (b), then all the expressions of Theorem 2 are naturally satisfied, so the proof is complete. □

Theorem 5.

Let M be any closed proper hemi-slant submanifold of an LPR-manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. Then, there is a de Rham cohomology class $\left[\alpha \right]$ determined by the differential p-form α in $H_{dR}^p\left(M\right)$ if the following expressions are satisfied:

(a)

For all $Z,W\in \Gamma \left(R^{\perp }\right)$,

$$A_{\overline{\Phi }Z}W=0;$$

(b)

For all $X\in \Gamma \left(R^{\varphi }\right)$,

$$\sigma \left(X,fX\right)+{\nabla }_X^{\perp }hX\in \Gamma \left(h{R}^{\varphi }\oplus \nu \right).$$

Moreover, $\left[\alpha \right]\ne 0$ if the following expressions are verified:

(c)

For all $X,Y\in \Gamma \left(R^{\varphi }\right)$,

$$A_{hX}Y-A_{hY}X+{\nabla }_Y^{\perp }hX-{\nabla }_X^{\perp }hY\in \Gamma \left(R^{\varphi }\right);$$

(d)

For all $Z\in \Gamma \left(R^{\perp }\right)$,

$$f{A}_{\overline{\Phi }Z}Z-t{\nabla }_Z^{\perp }hZ\in \Gamma \left(R^{\perp }\right).$$

Proof. 

The proof is omitted here because it is analogous to that of Theorem 2. □

Theorem 6.

Let M be any closed proper hemi-slant submanifold of an LPR-manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. Then, the differential p-form α induces a de Rham cohomology class represented by $\left[\alpha \right]$ in $H_{dR}^p\left(M\right)$ if the following expressions are verified:

(a)

M is $R^{\varphi }$-geodesic;

(b)

For all $X\in \Gamma \left(R^{\varphi }\right)$,

$${\nabla }_X^{\perp }hX\in \Gamma \left(h{R}^{\varphi }\oplus \nu \right);$$

(c)

For all $Z,W\in \Gamma \left(R^{\perp }\right)$,

$$A_{\overline{\Phi }Z}W=0.$$

In addition, $\left[\alpha \right]\ne 0$ if the following expressions are true:

(d)

For all $X,Y\in \Gamma \left(R^{\varphi }\right)$,

$$A_{hX}Y-A_{hY}X+{\nabla }_Y^{\perp }hX-{\nabla }_X^{\perp }hY\in \Gamma \left(R^{\varphi }\right);$$

(e)

For all $Z\in \Gamma \left(R^{\perp }\right)$,

$${\nabla }_Z^{\perp }hZ\in \Gamma \left(h{R}^{\perp }\oplus \nu \right).$$

Proof. 

Considering Propositions 1 and 2, the proof is easily obtained from Theorem 5. □

Theorem 7.

Let M be any closed totally geodesic proper hemi-slant submanifold of an LPR-manifold $\left(\overline{M},\overline{g},\overline{\Phi }\right)$. Then, the de Rham cohomology class of the differential p-form α in $H_{dR}^p\left(M\right)$ is non-trivial if the following expressions are correct:

(a)

For all $Z,W\in \Gamma \left(R^{\perp }\right)$,

$${\nabla }_Z^{\perp }hW\in \Gamma \left(h{R}^{\perp }\oplus \nu \right);$$

(b)

For all $X,Y\in \Gamma \left(R^{\varphi }\right)$,

$${\nabla }_X^{\perp }hY\in \Gamma \left(h{R}^{\varphi }\oplus \nu \right).$$

Proof. 

The proof can be achieved similarly to that of Theorem 4. □

Lastly, we present an example of the existence of the Rham cohomology class of a hemi-slant submanifold in the eight-dimensional Euclidean space.

Example 1.

Let $\left({\mathbb{R}}^8,⟨,⟩\right)$ be the eight-dimensional Euclidean space. We define an almost product Riemannian structure $\left(⟨,⟩,\overline{\Phi }\right)$ on ${\mathbb{R}}^8$ by

$$\overline{\Phi }\left(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8\right)=\left(x_1,-x_2,x_5,x_6,x_3,x_4,-x_7,x_8\right).$$

Also, it is obvious that $\left({\mathbb{R}}^8,⟨,⟩,\overline{\Phi }\right)$ is an LPR-manifold.

Now, we consider a submanifold M given by the immersion $i:M⟶{\mathbb{R}}^8$ as follows:

$$i\left(u,v,w,z\right)=\left(ucos\frac{\varphi }{2},usin\frac{\varphi }{2},vcos\varphi ,vsin\varphi ,w,0,z,z\right),$$

where $u,v>0$. Hence, the set $\left\{E_1,E_2,E_3,E_4\right\}$ is a local orthonormal frame for $TM$ such that

$$E_1=cos\frac{\varphi }{2}\frac{\partial }{\partial x_1}+sin\frac{\varphi }{2}\frac{\partial }{\partial x_2},E_2=cos\varphi \frac{\partial }{\partial x_3}+sin\varphi \frac{\partial }{\partial x_4},$$

$$E_3=\frac{\partial }{\partial x_5}\mathrm{and}{E}_4=\frac{1}{\sqrt{2}}\left(\frac{\partial }{\partial x_7}+\frac{\partial }{\partial x_8}\right).$$

Putting $R^{\varphi }=Span\left\{E_1,E_2,E_3\right\}$ and $R^{\perp }=Span\left\{E_4\right\}$, a direct calculation gives us that $R^{\varphi }$ is a slant distribution with the Wirtinger angle ϕ and $R^{\perp }$ is the anti-invariant distribution, so M is a four-dimensional hemi-slant submanifold with the Wirtinger angle ϕ of $\left({\mathbb{R}}^8,⟨,⟩,\overline{\Phi }\right)$.

Now, let us consider a differential three-form α and a differential one-form β given by $\alpha ={\alpha }^1\Lambda {\alpha }^2\Lambda {\alpha }^3$ and $\beta ={\beta }^1$, respectively, where

$${\alpha }^1=cos\frac{\varphi }{2}d{x}_1+sin\frac{\varphi }{2}d{x}_2,$$

$${\alpha }^2=cos\varphi d{x}_3+sin\varphi d{x}_4,$$

$${\alpha }^3=d{x}_5$$

and

$${\beta }^1=\frac{1}{\sqrt{2}}\left(d{x}_7+d{x}_8\right).$$

In this case, it follows that $d\beta =\delta \beta =d\alpha =\delta \alpha =0$. Thus, we have

$$\Delta \beta =\Delta \alpha =0,$$

that is, α and β are harmonic forms. For this reason, from Hodge theorem, $H_{dR}^1\left(M\right)$ and $H_{dR}^3\left(M\right)$ are non-trivial, i.e., $b_1\ne 0$ and $b_3\ne 0$.