3. Locally decomposable Riemannian manifolds

Let <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>g<sup>Þ</sup> be <sup>n</sup>−dimensional Riemannian manifold and <sup>F</sup> be a tensor <sup>ð</sup>1, <sup>1</sup>Þ−type on <sup>M</sup><sup>~</sup> such that <sup>F</sup><sup>2</sup> <sup>¼</sup> <sup>I</sup>, <sup>F</sup>≠∓I.

If the Riemannian metric tensor ~g satisfying

$$
\tilde{\mathcal{g}}(X,Y) = \tilde{\mathcal{g}}(FX,FY) \tag{81}
$$

for any <sup>X</sup>, <sup>Y</sup>∈ΓðTM<sup>~</sup> <sup>Þ</sup> then <sup>M</sup><sup>~</sup> is called almost Riemannian product manifold and <sup>F</sup> is said to be almost Riemannian product structure. If <sup>F</sup> is parallel, that is, <sup>ð</sup>∇<sup>~</sup> XFÞ<sup>Y</sup> <sup>¼</sup> 0, then <sup>M</sup><sup>~</sup> is said to be locally decomposable Riemannian manifold.

Now, let M~ be an almost Riemannian product manifold. We put

$$P = \frac{1}{2}(I + F),\ Q = \frac{1}{2}(I - F). \tag{82}$$

Then, we have

$$P + Q = I, \quad P^2 = P, \quad Q^2 = Q, \quad PQ = QP = 0 \quad \text{and} \quad F = P \text{-} Q. \tag{83}$$

Thus, <sup>P</sup> and <sup>Q</sup> define two complementary distributions <sup>P</sup> and <sup>Q</sup> globally. Since <sup>F</sup><sup>2</sup> <sup>¼</sup> <sup>I</sup>, we easily see that the eigenvalues of F are 1 and −1. An eigenvector corresponding to the eigenvalue 1 is in P and an eigenvector corresponding to −1 is in Q. If F has eigenvalue 1 of multiplicity P and eigenvalue −1 of multiplicity q, then the dimension of P is p and that of Q is q. Conversely, if there exist in M~ two globally complementary distributions P and Q of dimension p and q, respectively. Then, we can define an almost Riemannian product structure <sup>F</sup> on <sup>M</sup><sup>~</sup> by <sup>M</sup><sup>~</sup> by <sup>F</sup> <sup>¼</sup> <sup>P</sup>−<sup>Q</sup> [7].

Let <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>g, <sup>F</sup><sup>Þ</sup> be a locally decomposable Riemannian manifold and we denote the integral manifolds of the distributions P and Q by Mp and Mq , respectively. Then we can write <sup>M</sup><sup>~</sup> <sup>¼</sup> Mp XM<sup>q</sup> , <sup>ð</sup>p, <sup>q</sup> <sup>&</sup>gt; <sup>2</sup>Þ. Also, we denote the components of the Riemannian curvature <sup>R</sup> of <sup>M</sup><sup>~</sup> by Rdcba, 1≤a, b, c, d≤n ¼ p þ q.

Now, we suppose that the two components are both of constant curvature λ and μ. Then, we have

$$R\_{dcbu} = \lambda \{ \mathbf{g}\_{da} \mathbf{g}\_{cb} - \mathbf{g}\_{ca} \mathbf{g}\_{db} \} \tag{84}$$

and

$$R\_{zyw} = \mu \{ \mathbf{g}\_{zw} \mathbf{g}\_{yx} \mathbf{-} \mathbf{g}\_{yw} \mathbf{g}\_{zx} \}. \tag{85}$$

Then, the above equations may also be written in the form

$$\begin{split} R\_{kji l} &= \frac{1}{4} (\lambda + \mu) \{ (\mathcal{g}\_{kli}\mathcal{g}\_{ji} - \mathcal{g}\_{jli}\mathcal{g}\_{ki}) + (F\_{kli}F\_{ji} - F\_{jli}F\_{ki}) \} \\ &+ \frac{1}{4} (\lambda - \mu) \{ (F\_{kli}\mathcal{g}\_{ji} - F\_{jk}\mathcal{g}\_{ki}) + (\mathcal{g}\_{kli}F\_{ji} - \mathcal{g}\_{jli}F\_{ki}) \}. \end{split} \tag{86}$$

Conversely, suppose that the curvature tensor of a locally decomposable Riemannian manifold has the form

$$\begin{split} R\_{kjlh} &= a \{ (\mathcal{g}\_{kh}\mathcal{g}\_{ji} - \mathcal{g}\_{jh}\mathcal{g}\_{ki}) + (F\_{kh}F\_{ji} - F\_{jh}F\_{ki}) \} \\ &+ b \{ (F\_{kh}\mathcal{g}\_{ji} - F\_{jh}\mathcal{g}\_{ki}) + (\mathcal{g}\_{kh}F\_{ji} - \mathcal{g}\_{jh}F\_{ki}) \}. \end{split} \tag{87}$$

Then, we have

$$R\_{clba} = \mathcal{Z}(a+b) \{ \mathbf{g}\_{da} \mathbf{g}\_{cb} \mathbf{-} \mathbf{g}\_{ca} \mathbf{g}\_{db} \} \tag{88}$$

and

$$R\_{zy\alpha w} = \mathcal{Z}(a - b) \{ \mathcal{g}\_{zw} \mathcal{g}\_{yx} - \mathcal{g}\_{yw} \mathcal{g}\_{zx} \}. \tag{89}$$

Let M~ be an m−dimensional almost Riemannian product manifold with the Riemannian structure <sup>ð</sup>F, <sup>~</sup>g<sup>Þ</sup> and <sup>M</sup> be an <sup>n</sup>−dimensional sub-manifold of <sup>M</sup><sup>~</sup> . For any vector field <sup>X</sup> tangent to <sup>M</sup>, we put

$$FX = fX + wX,\tag{90}$$

where f X and wX denote the tangential and normal components of FX, with respect to M, respectively. In the same way, for <sup>V</sup>∈ΓðT<sup>⊥</sup>MÞ, we also put

$$FV = BV + CV,\tag{91}$$

where BV and CV denote the tangential and normal components of FV, respectively. Then, we have

$$\left(f^2 + Bw = I, \mathbb{C}w + wf = 0\right) \tag{92}$$

and

<sup>P</sup> <sup>þ</sup> <sup>Q</sup> <sup>¼</sup> <sup>I</sup>, <sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>P</sup>, <sup>Q</sup><sup>2</sup> <sup>¼</sup> <sup>Q</sup>, PQ <sup>¼</sup> QP <sup>¼</sup> 0 and <sup>F</sup> <sup>¼</sup> <sup>P</sup>−Q: (83)

, respectively. Then we can write

(86)

Thus, <sup>P</sup> and <sup>Q</sup> define two complementary distributions <sup>P</sup> and <sup>Q</sup> globally. Since <sup>F</sup><sup>2</sup> <sup>¼</sup> <sup>I</sup>, we easily see that the eigenvalues of F are 1 and −1. An eigenvector corresponding to the eigenvalue 1 is in P and an eigenvector corresponding to −1 is in Q. If F has eigenvalue 1 of multiplicity P and eigenvalue −1 of multiplicity q, then the dimension of P is p and that of Q is q. Conversely, if there exist in M~ two globally complementary distributions P and Q of dimension p and q, respectively. Then, we can define an almost Riemannian product structure

Let <sup>ð</sup>M<sup>~</sup> , <sup>~</sup>g, <sup>F</sup><sup>Þ</sup> be a locally decomposable Riemannian manifold and we denote the integral

Now, we suppose that the two components are both of constant curvature λ and μ. Then, we have

, <sup>ð</sup>p, <sup>q</sup> <sup>&</sup>gt; <sup>2</sup>Þ. Also, we denote the components of the Riemannian curvature <sup>R</sup> of <sup>M</sup><sup>~</sup>

ðλ þ μÞfðgkhgji−gjhgkiÞþðFkhFji−FjhFkiÞg

ðλ−μÞfðFkhgji−FjhgkiÞþðgkhFji−gjhFkiÞg:

Conversely, suppose that the curvature tensor of a locally decomposable Riemannian manifold

Rkjih ¼ afðgkhgji−gjhgkiÞþðFkhFji−FjhFkiÞg

Let M~ be an m−dimensional almost Riemannian product manifold with the Riemannian structure <sup>ð</sup>F, <sup>~</sup>g<sup>Þ</sup> and <sup>M</sup> be an <sup>n</sup>−dimensional sub-manifold of <sup>M</sup><sup>~</sup> . For any vector field <sup>X</sup> tangent to <sup>M</sup>, we put

Rdcba ¼ λfgdagcb−gcagdbg (84)

Rzyxw ¼ μfgzwgyx−gywgzxg: (85)

<sup>þ</sup>bfðFkhgji−FjhgkiÞþðgkhFji−gjhFkiÞg: (87)

Rcdba ¼ 2ða þ bÞfgdagcb−gcagdbg (88)

Rzyxw ¼ 2ða−bÞfgzwgyx−gywgzxg: (89)

<sup>F</sup> on <sup>M</sup><sup>~</sup> by <sup>M</sup><sup>~</sup> by <sup>F</sup> <sup>¼</sup> <sup>P</sup>−<sup>Q</sup> [7].

60 Manifolds - Current Research Areas

by Rdcba, 1≤a, b, c, d≤n ¼ p þ q.

XM<sup>q</sup>

<sup>M</sup><sup>~</sup> <sup>¼</sup> Mp

and

has the form

Then, we have

and

manifolds of the distributions P and Q by Mp and Mq

Then, the above equations may also be written in the form

þ 1 4

Rkjih <sup>¼</sup> <sup>1</sup> 4

$$fB + BC = 0,\\ wB + C^2 = I. \tag{93}$$

On the other hand, we can easily see that

$$\mathcal{g}(X, fY) = \mathcal{g}(fX, Y) \tag{94}$$

and

$$\mathcal{g}(\mathbf{X}, \mathbf{Y}) = \mathcal{g}(f\mathbf{X}, f\mathbf{Y}) + \mathcal{g}(w\mathbf{X}, w\mathbf{Y}) \tag{95}$$

for any X, Y∈ΓðTMÞ [6].

If wX <sup>¼</sup> 0 for all <sup>X</sup>∈ΓðTMÞ, then <sup>M</sup> is said to be invariant sub-manifold in <sup>M</sup><sup>~</sup> , i:e:, FðTMðpÞÞ⊂TMðpÞ for each p∈M. In this case, f <sup>2</sup> <sup>¼</sup> <sup>I</sup> and <sup>g</sup>ðf X, f YÞ ¼ <sup>g</sup>ðX,YÞ: Thus, <sup>ð</sup><sup>f</sup> , <sup>g</sup><sup>Þ</sup> defines an almost product Riemannian on M.

Conversely, ðf , gÞ is an almost product Riemannian structure on M, the w ¼ 0 and hence M is an invariant sub-manifold in M~ .

Consequently, we can give the following theorem [7].

Theorem 3.1. Let M be a sub-manifold of an almost Riemannian product manifold M~ with almost Riemannian product structure ðF, ~gÞ. The induced structure ðf , gÞ on M is an almost Riemannian product structure if and only if M is an invariant sub-manifold of M~ .

Definition 3.1. Let M be a sub-manifold of an almost Riemannian product M~ with almost product Riemannian structure ðF, ~gÞ. For each non-zero vector Xp∈TMðpÞ at p∈M, we denote the slant angle between FXp and TMðpÞ by θðpÞ. Then M said to be slant sub-manifold if the angle θðpÞ is constant, i:e:, it is independent of the choice of p∈M and Xp∈TMðpÞ [5].

Thus, invariant and anti-invariant immersions are slant immersions with slant angle θ ¼ 0 and <sup>θ</sup> <sup>¼</sup> <sup>π</sup> <sup>2</sup>, respectively. A proper slant immersion is neither invariant nor anti-invariant.

Theorem 3.2. Let M be a sub-manifold of an almost Riemannian product manifold M~ with almost product Riemannian structure ðF, ~gÞ. M is a slant sub-manifold if and only if there exists a constant λ∈ð0, 1Þ, such tha

$$f^2 = \lambda I.\tag{96}$$

Furthermore, if the slant angle is <sup>θ</sup>, then it satisfies <sup>λ</sup> <sup>¼</sup> cos<sup>2</sup><sup>θ</sup> [9].

Definition 3.2. Let M be a sub-manifold of an almost Riemannian product manifold M~ with almost Riemannian product structure ðF, ~gÞ. M is said to be semi-slant sub-manifold if there exist distributions D<sup>θ</sup> and D<sup>T</sup> on M such that

(i) TM has the orthogonal direct decomposition TM <sup>¼</sup> <sup>D</sup>⊕D<sup>T</sup>:

(ii) The distribution D<sup>θ</sup> is a slant distribution with slant angle θ:

(iii) The distribution <sup>D</sup><sup>T</sup> is an invariant distribution, :e:, <sup>F</sup>ðDTÞ⊆D<sup>T</sup>.

In a semi-slant sub-manifold, if <sup>θ</sup> <sup>¼</sup> <sup>π</sup> <sup>2</sup>, then semi-slant sub-manifold is called semi-invariant sub-manifold [8].

Example 3.1. Now, let us consider an immersed sub-manifold M in R<sup>7</sup> given by the equations

$$
\mathbf{x}\_1^2 + \mathbf{x}\_2^2 = \mathbf{x}\_5^2 + \mathbf{x}\_6^2,\\
\mathbf{x}\_3 + \mathbf{x}\_4 = \mathbf{0}.\tag{97}
$$

By direct calculations, it is easy to check that the tangent bundle of M is spanned by the vectors

$$\begin{aligned} z\_1 &= \cos\theta \frac{\partial}{\partial x\_1} + \sin\theta \frac{\partial}{\partial x\_2} + \cos\beta \frac{\partial}{\partial x\_5} + \sin\beta \frac{\partial}{\partial x\_6} \\ z\_2 &= -u\sin\theta \frac{\partial}{\partial x\_1} + u\cos\theta \frac{\partial}{\partial x\_2}, z\_3 = \frac{\partial}{\partial x\_3} - \frac{\partial}{\partial x\_4}, \\ z\_4 &= -u\sin\theta \frac{\partial}{\partial x\_5} + u\cos\beta \frac{\partial}{\partial x\_6}, z\_5 = \frac{\partial}{\partial x\_7}, \end{aligned} \tag{98}$$

where θ, β and u denote arbitrary parameters.

For the coordinate system of <sup>R</sup><sup>7</sup> ¼ fðx1, <sup>x</sup>2, <sup>x</sup>3, <sup>x</sup>4, <sup>x</sup>5, <sup>x</sup>6, <sup>x</sup>7Þjxi∈R, <sup>1</sup> <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>7</sup>g, we define the almost product Riemannian structure F as follows:

$$F\left(\frac{\partial}{\partial \mathbf{x}\_i}\right) = \frac{\partial}{\partial \mathbf{x}\_i}, F\left(\frac{\partial}{\partial \mathbf{x}\_j}\right) = \frac{\partial}{\partial \mathbf{x}\_j}, 1 \le i \le 3 \text{ and } 4 \le j \le 7. \tag{99}$$

Since Fz<sup>1</sup> and Fz<sup>3</sup> are orthogonal to M and Fz2,Fz4,Fz<sup>5</sup> are tangent to M, we can choose a <sup>D</sup> <sup>¼</sup> Spfz2, <sup>z</sup>4, <sup>z</sup>5<sup>g</sup> and <sup>D</sup><sup>⊥</sup> <sup>¼</sup> Spfz1, <sup>z</sup>3g. Thus, <sup>M</sup> is a 5−dimensional semi-invariant sub-manifold of <sup>R</sup><sup>7</sup> with usual almost Riemannian product structure <sup>ð</sup>F, <sup>&</sup>lt; , <sup>&</sup>gt;Þ:

Example 3.2. Let M be sub-manifold of R<sup>8</sup> by given

$$(\mu + \upsilon, \mu - \upsilon, \iota \cos \alpha, \iota \sin \alpha, \iota + \upsilon, \iota - \upsilon, \iota \cos \beta, \iota \sin \beta) \tag{100}$$

where u, v and β are the arbitrary parameters. By direct calculations, we can easily see that the tangent bundle of M is spanned by

$$\begin{array}{c} e\_1 = \frac{\partial}{\partial x\_1} + \frac{\partial}{\partial x\_2} + \cos\alpha \frac{\partial}{\partial x\_3} + \sin\alpha \frac{\partial}{\partial x\_4} + \frac{\partial}{\partial x\_5} - \frac{\partial}{\partial x\_6} + \cos\beta \frac{\partial}{\partial x\_7} + \sin\beta \frac{\partial}{\partial x\_8} \\\ e\_2 = \frac{\partial}{\partial x\_1} - \frac{\partial}{\partial x\_2} + \frac{\partial}{\partial x\_5} + \frac{\partial}{\partial x\_6}, e\_3 = -u\sin\frac{\partial}{\partial x\_3} + u\cos\alpha \frac{\partial}{\partial x\_4}, \\\ e\_4 = -u\sin\beta \frac{\partial}{\partial x\_7} + u\cos\beta \frac{\partial}{\partial x\_8}. \end{array} \tag{101}$$

For the almost Riemannian product structure <sup>F</sup> of <sup>R</sup><sup>8</sup> <sup>¼</sup> <sup>R</sup><sup>4</sup> <sup>x</sup>R4, <sup>F</sup>ðTM<sup>Þ</sup> is spanned by vectors

$$\begin{split} Fe\_1 &= \frac{\partial}{\partial \mathbf{x}\_1} + \frac{\partial}{\partial \mathbf{x}\_2} + \cos \alpha \frac{\partial}{\partial \mathbf{x}\_3} + \sin \alpha \frac{\partial}{\partial \mathbf{x}\_4} - \frac{\partial}{\partial \mathbf{x}\_5} + \frac{\partial}{\partial \mathbf{x}\_6} - \cos \beta \frac{\partial}{\partial \mathbf{x}\_7} - \sin \beta \frac{\partial}{\partial \mathbf{x}\_8}, \\ Fe\_2 &= \frac{\partial}{\partial \mathbf{x}\_1} - \frac{\partial}{\partial \mathbf{x}\_2} - \frac{\partial}{\partial \mathbf{x}\_5} - \frac{\partial}{\partial \mathbf{x}\_6}, \quad Fe\_3 = e\_3 \text{ and } Fe\_4 = -e\_4. \end{split} \tag{102}$$

Since Fe<sup>1</sup> and Fe<sup>2</sup> are orthogonal to M and Fe<sup>3</sup> and Fe<sup>4</sup> are tangent to M, we can choose <sup>D</sup><sup>T</sup> <sup>¼</sup> Spfe3,e4<sup>g</sup> and <sup>D</sup><sup>⊥</sup> <sup>¼</sup> Spfe1,e2g. Thus, <sup>M</sup> is a four-dimensional semi-invariant sub-manifold of <sup>R</sup><sup>8</sup> <sup>¼</sup> <sup>R</sup><sup>4</sup> xR<sup>4</sup> with usual Riemannian product structure F.

Definition 3.3. Let M be a sub-manifold of an almost Riemannian product manifold M~ with almost Riemannian product structure ðF, ~gÞ. M is said to be pseudo-slant sub-manifold if there exist distributions D<sup>θ</sup> and D<sup>⊥</sup> on M such that

i. The tangent bundle TM <sup>¼</sup> <sup>D</sup>θ⊕D<sup>⊥</sup>.

Theorem 3.2. Let M be a sub-manifold of an almost Riemannian product manifold M~ with almost product Riemannian structure ðF, ~gÞ. M is a slant sub-manifold if and only if there exists

Definition 3.2. Let M be a sub-manifold of an almost Riemannian product manifold M~ with almost Riemannian product structure ðF, ~gÞ. M is said to be semi-slant sub-manifold if there

Example 3.1. Now, let us consider an immersed sub-manifold M in R<sup>7</sup> given by the equations

<sup>5</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup>

By direct calculations, it is easy to check that the tangent bundle of M is spanned by the

<sup>þ</sup> <sup>u</sup>cos<sup>θ</sup> <sup>∂</sup>

For the coordinate system of <sup>R</sup><sup>7</sup> ¼ fðx1, <sup>x</sup>2, <sup>x</sup>3, <sup>x</sup>4, <sup>x</sup>5, <sup>x</sup>6, <sup>x</sup>7Þjxi∈R, <sup>1</sup> <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>7</sup>g, we define the

¼ ∂ ∂xj

Since Fz<sup>1</sup> and Fz<sup>3</sup> are orthogonal to M and Fz2,Fz4,Fz<sup>5</sup> are tangent to M, we can choose a <sup>D</sup> <sup>¼</sup> Spfz2, <sup>z</sup>4, <sup>z</sup>5<sup>g</sup> and <sup>D</sup><sup>⊥</sup> <sup>¼</sup> Spfz1, <sup>z</sup>3g. Thus, <sup>M</sup> is a 5−dimensional semi-invariant sub-mani-

<sup>þ</sup> cos<sup>β</sup> <sup>∂</sup> ∂x<sup>5</sup>

∂x<sup>6</sup>

, <sup>z</sup><sup>3</sup> <sup>¼</sup> <sup>∂</sup> ∂x<sup>3</sup> − ∂ ∂x<sup>4</sup> ,

, <sup>z</sup><sup>5</sup> <sup>¼</sup> <sup>∂</sup> ∂x<sup>7</sup> ,

ðu þ v, u−v, ucosα, usinα, u þ v, u−v, ucosβ, usinβÞ (100)

∂x<sup>2</sup>

<sup>þ</sup> <sup>u</sup>cos<sup>β</sup> <sup>∂</sup>

<sup>2</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup>

<sup>þ</sup> sin<sup>θ</sup> <sup>∂</sup> ∂x<sup>2</sup>

<sup>2</sup> <sup>¼</sup> <sup>λ</sup>I: (96)

<sup>2</sup>, then semi-slant sub-manifold is called semi-invariant

<sup>þ</sup> sin<sup>β</sup> <sup>∂</sup> ∂x<sup>6</sup>

<sup>6</sup>, x<sup>3</sup> þ x<sup>4</sup> ¼ 0: (97)

, 1 ≤ i ≤ 3 and 4 ≤ j ≤ 7: (99)

(98)

f

Furthermore, if the slant angle is <sup>θ</sup>, then it satisfies <sup>λ</sup> <sup>¼</sup> cos<sup>2</sup><sup>θ</sup> [9].

(i) TM has the orthogonal direct decomposition TM <sup>¼</sup> <sup>D</sup>⊕D<sup>T</sup>: (ii) The distribution D<sup>θ</sup> is a slant distribution with slant angle θ:

(iii) The distribution <sup>D</sup><sup>T</sup> is an invariant distribution, :e:, <sup>F</sup>ðDTÞ⊆D<sup>T</sup>.

x2 <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup>

<sup>z</sup><sup>1</sup> <sup>¼</sup> cos<sup>θ</sup> <sup>∂</sup>

where θ, β and u denote arbitrary parameters.

almost product Riemannian structure F as follows:

<sup>F</sup> <sup>∂</sup> ∂xi 

Example 3.2. Let M be sub-manifold of R<sup>8</sup> by given

<sup>z</sup><sup>2</sup> <sup>¼</sup> <sup>−</sup>usin<sup>θ</sup> <sup>∂</sup>

¼ ∂ ∂xi

∂x<sup>1</sup>

<sup>z</sup><sup>4</sup> <sup>¼</sup> <sup>−</sup>usin<sup>β</sup> <sup>∂</sup>

∂x<sup>1</sup>

, <sup>F</sup> <sup>∂</sup> ∂xj 

fold of <sup>R</sup><sup>7</sup> with usual almost Riemannian product structure <sup>ð</sup>F, <sup>&</sup>lt; , <sup>&</sup>gt;Þ:

∂x<sup>5</sup>

exist distributions D<sup>θ</sup> and D<sup>T</sup> on M such that

In a semi-slant sub-manifold, if <sup>θ</sup> <sup>¼</sup> <sup>π</sup>

sub-manifold [8].

vectors

a constant λ∈ð0, 1Þ, such tha

62 Manifolds - Current Research Areas


As a special case, if <sup>θ</sup> <sup>¼</sup> 0 and <sup>θ</sup> <sup>¼</sup> <sup>π</sup> <sup>2</sup>, then pseudo-slant sub-manifold becomes semi-invariant and anti-invariant sub-manifolds, respectively.

Example 3.3. Let M be a sub-manifold of R<sup>6</sup> by the given equation

$$(\sqrt{3}u, v, v\sin\theta, v\cos\theta, \text{scost}, \text{-scost})\tag{103}$$

where u, v,s and t arbitrary parameters and θ is a constant.

We can check that the tangent bundle of M is spanned by the tangent vectors

$$\begin{aligned} e\_1 &= \sqrt{3} \frac{\partial}{\partial \mathbf{x}\_1}, e\_2 = \frac{\partial}{\partial y\_1} + \sin \theta \frac{\partial}{\partial \mathbf{x}\_2} + \cos \theta \frac{\partial}{\partial y\_2}, \\ e\_3 &= \cos t \frac{\partial}{\partial \mathbf{x}\_3} - \cos t \frac{\partial}{\partial y\_3}, e\_4 = -\text{ssimt} \frac{\partial}{\partial \mathbf{x}\_3} + \text{ssimt} \frac{\partial}{\partial y\_3}. \end{aligned} \tag{104}$$

For the almost product Riemannian structure F of R<sup>6</sup> whose coordinate systems ðx1, y1, x2, y2, x3, y3Þ choosing

$$\begin{aligned} F\left(\frac{\partial}{\partial \mathbf{x}\_i}\right) &= \frac{\partial}{\partial y\_i}, 1 \le i \le 3, \\ F\left(\frac{\partial}{\partial y\_j}\right) &= \frac{\partial}{\partial \mathbf{x}\_j}, 1 \le j \le 3, \end{aligned} \tag{105}$$

Then, we have

$$\begin{split} Fe\_1 &= \sqrt{3} \frac{\partial}{\partial y\_1}, Fe\_2 = -\frac{\partial}{\partial x\_1} + \sin \theta \frac{\partial}{\partial y\_2} - \cos \theta \frac{\partial}{\partial x\_2} \\ Fe\_3 &= \cos t \frac{\partial}{\partial y\_3} + \cos t \frac{\partial}{\partial x\_3}, Fe\_4 = -\text{ssimt} \frac{\partial}{\partial y\_3} - \text{ssimt} \frac{\partial}{\partial x\_3} .\end{split} \tag{106}$$

Thus, <sup>D</sup><sup>θ</sup> <sup>¼</sup> Spfe1,e2<sup>g</sup> is a slant distribution with slant angle <sup>α</sup> <sup>¼</sup> <sup>π</sup> <sup>4</sup>. Since Fe<sup>3</sup> and Fe<sup>4</sup> are orthogonal to <sup>M</sup>, <sup>D</sup><sup>⊥</sup> <sup>¼</sup> Spfe3,e4<sup>g</sup> is an anti-invariant distribution, that is, <sup>M</sup> is a 4-dimensional proper pseudo-slant sub-manifold of R<sup>6</sup> with its almost Riemannian product structure ðF, < , >Þ:
