Five-Dimensional Contact CR -Submanifolds in S 7 ( 1 ) †

: Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure ( ϕ , ξ , η ) , we study its ﬁve-dimensional contact CR -submanifolds, which are the analogue of CR -submanifolds in (almost) Kählerian manifolds. In the case when the structure vector ﬁeld ξ is tangent to M , the tangent bundle of contact CR -submanifold M can be decomposed as T ( M ) = H ( M ) ⊕ E ( M ) ⊕ R ξ , where H ( M ) is invariant and E ( M ) is anti-invariant with respect to ϕ . On this occasion we obtain a complete classiﬁcation of ﬁve-dimensional proper contact CR -submanifolds in S 7 ( 1 ) whose second fundamental form restricted to H ( M ) and E ( M ) vanishes identically and we prove that they can be decomposed as (multiply) warped products of spheres.


Introduction
Let M be a Riemannian submanifold of the seven-dimensional unit sphere. It is well-known that S 7 (1) possesses the almost contact structure (ϕ, ξ, η), which is also contact and Sasakian. Having in mind the behaviour of the endomorphism ϕ, submanifolds in the Sasakian manifolds carrying a ϕ-invariant distribution such that its orthogonal complement is ϕ-anti-invariant, are called contact CR-submanifolds. This notion is the odd-dimensional analogue of CR-submanifolds in (almost) Kählerian manifolds, introduced by Bejancu in [1], who requested the existence of a differentiable holomorphic distribution such that its orthogonal complement is a totally real distribution. Also, CR-submanifolds of the nearly Kähler six-dimensional unit sphere have also been investigated (see [2,3], for example). As the ϕ-invariant distribution H(M) is always even-dimensional, the lowest possible dimension for a proper contact CR-submanifold (i.e., suchthat the dimensions of both ϕ-invariant and anti-invariant distributions are different from zero) is four. In this paper we continue our study of certain contact CR-submanifolds in seven-dimensional unit sphere, which we started in [4] for the case of four-dimensional submanifolds and continued in [5], where we presented several examples of four and five-dimensional contact CR-submanifolds of S 7 (1), which are of product and warped product type.
One of the natural problems in the theory of submanifolds is the condition of immersibility. For example, it is interesting to investigate totally geodesic submanifolds, that is, those submanifolds for which all geodesics-when the induced Riemannian metric is considered-are also geodesics on the ambient manifold. This property is equivalent to the vanishing of the second fundamental form. It is well-known that any contact CR-submanifold in a Sasakian manifold can never be totally geodesic. respectively, where R is the curvature on M, and S a , S b are the shape operators corresponding to the normal vectors N a , N b , respectively.

Contact CR-Submanifolds
The notion of contact CR-submanifolds in Sasakian manifolds is the odd-dimensional analogue of CR-submanifolds in (almost) Kählerian manifolds. See also [9]. Particularly, a contact CR-submanifold in the Sasakian manifold ( M, ϕ, ξ, η,g) is a submanifold M carrying a ϕ-invariant distribution D, that is, ϕ p D p ⊆ D p , for any p ∈ M, such that the orthogonal complement D ⊥ of D in T(M) is ϕ-anti-invariant, that is, ϕ p D ⊥ p ⊆ T ⊥ p M, for any p ∈ M. This notion was used by Bejancu and Papaghiuc in [10], using the terminology of semi-invariant submanifold. It is standard to require that ξ is tangent to M rather than normal, which is too restrictive (by Prop. 1.1 in [11], p. 43, M must be ϕ-anti-invariant, that is, ϕT p M ⊆ T ⊥ p M, for all p ∈ M), or oblique which leads to highly complicated embedding equations. The contact CR-submanifold is called proper if both distributions D and D ⊥ are non-trivial distributions. is never integrable (see [12]), while H(M) ⊕ Rξ can be, as in the case of contact CR-products (see [13]). On the other hand, the normal bundle of M can be decomposed as

CR Warped Product Submanifolds in Sasakian Manifolds
The notion of warped product is the natural and very fruitful generalization of Riemannian products. It was introduced by Bishop and O'Neill in [14] in order to construct a large class of complete manifolds of negative curvature.
Let B, F be two Riemannian manifolds with Riemannian metrics g B and g F respectively and let f be a smooth positive function on B. Considering the product manifold B × F, let π 1 : B × F → B and π 2 : B × F → F be the canonical projections. The manifold M = B × f F is called the warped product if it is equipped with the Riemannian structure such that ||X|| 2 = ||π 1, * (X)|| 2 + f 2 (π 1 (x))||π 2, * (X)|| 2 for all X ∈ T x (M), x ∈ M, or, equivalently, g = g B + f 2 g F with the usual meaning, while f is called the warping function on the warped product. For more details we refer to [15].
A contact CR-submanifold M in a Sasakian manifold M, tangent to the structure vector field ξ, is called a contact CR warped product, with the warping function f , if it is the warped product N T × f N ⊥ of an invariant submanifold N T , tangent to ξ and a totally real submanifold N ⊥ of M, where f is the warping function (see [13] for more details). It is notable to point out that there is no proper contact CR-submanifolds in Sasakian manifolds in the form N ⊥ × f N T . This fact was proved in [13,16].

Problem
On this occasion, we consider the problem of finding all five-dimensional proper contact CR-submanifolds in S 7 (1) such that 3. Five-Dimensional Nearly Totally Geodesic Contact CR-Submanifolds in S 7 (1) In order to prove our results, we first select an appropriate frame on M 5 in such a way that equations, which are the consequences of (1), become satisfied. Then, we classify all 5-dimensional proper contact CR-submanifolds in the seven-dimensional unit sphere satisfying (1).

Essential Characteristics of Five-Dimensional Contact CR-Submanifolds in S 7 (1)
In this subsection, after choosing the appropriate basis, we introduce some smooth functions to describe the induced connection and we express the shape operators. Using Codazzi and Ricci equations, we obtain relations between these functions and we derive conditions on these functions, namely a system of algebraic and differential equations.
First, it is straightforward, using the formulae of Gauss and Weingarten, as well as the Sasakian structure of the 7-sphere, to prove the following: Lemma 1. If M is a contact CR-submanifold in the Sasakian manifold S 7 (1) we havẽ g(h(X, Z), ϕW) =g(h(X, W), ϕZ), for every X ∈ H(M), Z, W ∈ E(M).
Further, starting with two arbitrary orthonormal bases {e 1 , e 2 = ϕe 1 } in H(M) and {e 3 , e 4 } in E(M), respectively, we will choose a basis in T(M) so that the second fundamental form will depend only on four smooth functions. In that direction, as a consequence of Lemma 1, we define, for each X ∈ H(M), a symmetric operator h(e 1 , e 4 ) = a 2 ϕe 3 + a 3 ϕe 4 , h(e 2 , e 4 ) = b 2 ϕe 3 + b 3 ϕe 4 .
It follows that in Cases (ii)-(iv) we can choose E 1 and E 2 = ϕE 1 in H(M) such that the operator A(E 2 ) is traceless. Additionally, because the operator A(E 1 ) is also symmetric, we will take the basis in E(M) defined by the eigenvectors of this operator. Denote them by E 3 and E 4 . Consequently we have that h(E 1 , E 3 ) is proportional to ϕE 3 and h(E 1 , E 4 ) is proportional to ϕE 4 .
Concerning the Case (i), we (apparently) have the freedom of choosing E 2 . Nevertheless, if we make a rotation about a certain angle s in E(M), we set E 3 = cos s e 3 + sin s e 4 and E 4 = − sin s e 3 + cos s e 4 . If a 2 = 0 take s = 0 and if a 2 = 0 take s such that cot 2s = a 1 a 2 . Consequently, we obtain h(e 1 , E 3 ) =ã 1 ϕE 3 and h(e 1 , E 4 ) = −ã 1 ϕE 4 . Since s depends on a 1 and a 2 and hence on e 1 and e 2 , we set E 1 = e 1 and E 2 = e 2 .
For simplicity of notation, we continue to write E 5 for ϕE 3 where a 1 , a 3 , b 1 , b 2 are smooth functions on M and consequently for certain smooth functions a, l, p, q, r, α, β, c, ω and θ on M.
Proof. Since ∇ E i ξ = ϕE i , we immediately obtain that ∇ E i ξ = 0 for i = 3, 4 and this implies that ∇ E i X has no component along ξ, for every X ∈ H(M) and i = 3, 4. Now let's prove one of the formulae in (4).
and identifying the tangent and the normal parts, respectively, we obtain for a certain function a ∈ C ∞ (M).
In order to have the complete description of the geometry of M, we write the expression of the normal connection, that is ∇ ⊥ Lemma 2. Under the above assumptions, the coefficient b 2 vanishes.
Proof. Using the fact that∇h is totally symmetric, we obtain the equations given in Table 1.
So, from now on we will take b 2 = 0. Let us develop all situations for the Codazzi quation. Due to the totally symmetry of∇h one has 30 non-trivial possibilities. Nevertheless, some of the equations are consequences of the other ones, or they are automatically satisfied. For example we have: Lemma 3. Under the same hypothesis as for Proposition 2, the Codazzi equations are automatically satisfied for the triple (ξ, ξ, Z) for Z ∈ E(M), as well as for the triple (ξ, ξ, X) for X ∈ H(M).
Therefore, we emphasize only the non-trivial conditions we get from the Codazzi equations. We remark, in the Table 2, two types of conditions, namely algebraic equations and differential equations, respectively.  Table 2, we deduce that a 1 + a 3 = 0.
Adding, side by side, the differential equations we have in lines L2 and L3 in Table 2, we get We obtain a contradiction and therefore b 1 vanishes.
From now on we will distinguish two cases: Case 1. a 1 = a 3 and Case 2. a 1 = a 3 .
We will obtain some more equations in each of the two cases and then we completely solve our problem in Sections 3.2 and 3.3.

Case 1.
For the sake of simplicity, we make the following notation a 1 = a 3 := A From line L2 in Table 2 we get which implies that A cannot vanish. Developing all the equations in the Table 2, we obtain: With respect to the orthonormal basis {E 1 , E 2 , E 3 , E 4 , ξ} in T(M) we may express the two shape operators as follows Straightforward computation shows that Ricci Equations (ER) imply new relations between the functions we have considered: Moreover, the normal curvature is completely determined by the following component Table 2. Gauss equations and symmetries of∇h.

The Symmetry We Use The Result We Get
As a 1 = a 3 , we immediately obtain ω = 1, p = a 1 + a 3 and a, c, q, l, r, θ, α and β vanish. Moreover, we should have a 1 a 3 = −1. Again, for the sake of simplicity we denote a 1 = A and hence a 3 = − 1 A and p = A − 1 A . Obviously, A cannot vanish and satisfies the following partial differential equations Similarly to the Case 1, we may express the two shape operators as follows Straightforward computation shows that the normal connection is flat, so the normal bundle is parallel. Additionally, Ricci Equations (ER) imply no new relations.

The Case 1: M is Congruent to
In this subsection we study in detail the case a 1 = a 3 and prove that then the contact nearly totally geodesic CR-submanifold M is congruent to S 3 × f S 2 and we determine the explicit immersion.
As a consequence, we find that the following relations are true: where X 0 = AE 2 . The statements (a) and (c) imply that D and D ⊥ are both involutive. The statements (a) and (b) mean that the maximal integral manifolds of D ⊥ are extrinsic spheres. Finally, the statement (c) says that the integral manifolds of D are totally geodesic. Now, applying a famous result of Hiepko ([18], p. 213): Let (M, g) be a (pseudo-)Riemannian manifold endowed with a pair (L, N) of non-degenerate foliations. This determines a local warped product structure with N as a normal factor, if and only if, the foliations are orthogonal, L is geodesic, and N is spherical, we have: For every point p ∈ M, there exists an isometry Φ from a warped product N 1 × f N 2 to a neighborhood of p in M with the property In order to find the warping function on M, let us consider the following vector field in D: One can immediately prove that Thus, we choose local coordinates x, y, z on M (in fact on Φ(N 1 )) such thatĒ 1 = ∂ ∂x , E 2 = ∂ ∂y and ξ = ∂ ∂z . Using (6), after a possible translation in y-coordinate, we obtain Taking y ∈ (0, π/2), we obtain p = −2 cot 2y and E 1 = 2 sin 2y ∂ ∂x − cot 2y ∂ ∂z . The restriction of the metric g to D can be expressed in terms of the coordinates x, y and z as follows: Moreover, from ( [13], Theorem 3.2), we know that S ϕZ X = η(X) − (ϕX)(log f ) Z, for any X ∈ D and any Z ∈ D ⊥ . Using the expression (7) for the shape operator, we get that which is the warping function on M.
Since Φ(N 1 × {p 2 }) is totally geodesic in M and M is nearly totally geodesic in S 7 , it follows that N 1 × {p 2 } is (isometrically) immersed in S 7 as a totally geodesic submanifold. With a similar argument, {p 1 } × N 2 is immersed in S 7 as a totally umbilical submanifold. Hence Φ(N 1 × {p 2 }) can be considered to be a (portion of) S 3 (1). Additionally, Φ({p 1 } × N 2 ), being totally umbilical in S 7 , can be taken as a (portion of) 2-sphere of a certain radius. Looking back to the expression (20) of the covariant derivative ∇, we conclude that the mean curvature vector field of Φ({p 1 } × N 2 ) in S 7 is AE 2 . Thus, the curvature of the 2-sphere above is 1 + A 2 , and hence its radius is 1 √ 1+A 2 = cos y. Consequently, we will consider M = S 3 × f S 2 with the warping function f = cos y, y ∈ (0, π 2 ).

Proposition 3.
Under the conditions stated for the case 1, it follows that M is locally congruent to a contact CR warped product S 3 × f S 2 .

Remark 2.
Defining the 1-form Ω on M by Ω(U) = −g(AE 2 , U), we conclude it is closed and therefore, locally, there exists a smooth function on M such that its differential is equal to Ω. We may notice that this function is constant on the leaves of D ⊥ , that is, on Φ({p 1 } × N 2 ). This function is nothing but log f . It can be easily checked that AE 2 = −grad log f . Now, let us determine the metric. Choosing isothermal coordinates u, v on Φ(N 2 ), we have where T is a smooth positive function on M, depending on u and v. Here we made the following notations: g uu := g ∂ ∂u , ∂ ∂u and similar for g vv and g uv . Finally, due to the orthogonality of the two distributions D and D ⊥ , we have g xu = g yu = g zu = 0, g xv = g yv = g zv = 0.
We adopted similar notations as before for g xu , g yu and so on. Hence the metric g is completely determined. Now, let us find a more appropriate basis. Taking an arbitrary unit vector Z in E(M), it can be expressed as Z = cos sE 3 + sin sE 4 , where s ∈ C ∞ (M).
Moreover, as the functiong(h(E 1 , Z), h(E 1 , Z)) is equal to A 2 , it is independent of Z. Hence, E 3 is not uniquely defined and it could be replaced by any other unit vector Z in E(M).
So, because of this freedom, we choose Now, we need to do some additional computations. In what follows, by subscripts we mean the partial derivatives; for example, T u = ∂T ∂u . Using (22) and (20), we conclude and on the other hand [E 3 , Thus, α = − T v T 2 cos y and β = T u T 2 cos y . Next we have to calculate the Levi-Civita connection of the metric g in terms of the coordinates x, y, z, u and v and then to compare the results with the relations (20). Being a straightforward computation, we present only one situation, namely we compute ∇ E 2 E 3 : In a similar way we find l = 0 and θ = 0.
Finally, the Ricci Equations (8)- (16) are automatically fulfilled, while the Equation (17) leads to the following partial differential equation for T Further, our aim is to find the isometric immersion F : M 5 −→ S 7 . Let ι : S 7 −→ E 8 = C 4 be the canonical inclusion of the 7-dimensional unit sphere in the 4-dimensional complex space. Denoting by , the scalar product on E 8 and by • ∇ the corresponding flat connection, we have for all X and Y tangent to S 7 (1), where p denotes the position vector of a point of the sphere. Using the Gauss formula (G) we have for all X and Y tangent to M.
For example, if we put X = E 2 = ∂ y and Y = ξ = ∂ z we get We obtain that F also satisfies, simultaneously, the following partial differential equations: Remark 3. Observe that Equation (26) also follows from (27) respectively, that is 2F x − F z depends neither on u, nor on v.
Considering Equations (28), (33) and (37), we deduce F(x, y, z, u, v) = cos z cos y cos x 2 u 1 + cos z cos y sin x 2 v 1 + cos z sin y cos x 2 u 2 + cos z sin y sin x 2 v 2 + sin z cos y cos x 2 u 3 + sin z cos y sin x 2 v 3 + sin z sin y cos x 2 u 4 + sin z sin y sin where u 1 , . . . , u 4 , v 1 , . . . , v 4 are vectors in R 8 which do not depend on x, y and z, but they do depend on u and v. Using (41) we compute Remark 4. Since vectors u 1 , . . . , u 4 , v 1 , . . . , v 4 do not depend on x, y and z and 2F x − F z does not depend on u and v, using Equation (42) we conclude that u 1 + v 3 , u 2 + v 4 , u 3 − v 1 and u 4 − v 2 are constant vectors in R 8 .
Combining now with (41) we obtain 0 = cos z cos x 2 (cos yu 1 − sin yu 2 + cos yv 3 + sin yv 4 ) (45) Since (45) is satisfied for all x, y, z, it follows Replacing (46) in (41) yields F(x, y, z, u, v) = cos y cos z + x 2 u 1 + sin y cos z − x 2 u 2 + cos y sin z + x 2 u 3 + sin y sin z − x 2 u 4 , where u 2 , u 4 are constant vectors in R 8 and u 1 , u 3 may depend on u and v. Now, using (39), we compute and using (40), we get Therefore, we proceed solving the system having in mind P ∈ {u 1 , u 3 }.
Considering the two vector-valued functions U(u, v) = 1 T 2 P u and V(u, v) = 1 T 2 P v and using (48)-(50), we conclude that U(u, v) and V(u, v) satisfy the Cauchy-Riemann equations for each component of the vector valued function U(u, v) + iV (u, v). This means that U(u, v) + iV(u, v) depends only on w = u + iv and not onw = u − iv. Therefore, the function is holomorphic. Denoting it by where the functions a and b satisfy the Cauchy-Riemann equations, we conclude that the function P has to satisfy the equation Up to now, we have kept the conformal factor T in the general form, without thinking at any possible concrete expression. Our motivation has been a possibility to use this technique for solving another problem of the same type.
Recall that N 2 = S 2 (1). There are several ways to consider isothermal coordinates u and v on the 2-sphere, that is, to write the metric as T(u, v) 2 (du 2 + dv 2 ). Recall two of them: • T(u, v) = 2 1+u 2 +v 2 , associated to the parametrization obtained from the stereographic projection; obtained as a surface of revolution.
After setting T = 2 1+u 2 +v 2 and solving the Equation (53), we obtain where a 0 and b 0 also satisfy the Cauchy-Riemann equations. Since P(w,w) ∈ R 8 , we conclude On the other hand, since ∂ w ∂wP = 1 4 (P uu + P vv ) ∈ R 8 , using (53) and the information that a and b satisfy the Cauchy-Riemann equations, we compute Consequently, using (57), we conclude Let us express ub − va from the Equation (57): For simplicity of notation, we write b 0v instead of ∂v , for example. Taking the partial derivatives of (60), with respect to u and with respect to v, multiplying the obtained equations respectively by u and v and adding them, we get Moreover, after computing b 0u and b 0v (using (59)) and replacing it in (61), together with b 0 from (59), we get Further, taking the partial derivatives of (60), with respect to u and with respect to v, multiplying the obtained equations respectively by v and u and subtracting them, we get Using (63) and the partial derivatives of b 0u and b 0v (using (59)), we compute Taking the partial derivative of (62), with respect to v, we compute Having in mind that a is a harmonic function (a uu + a vv = 0) and taking the partial derivative of (62), with respect to u, we conclude Taking the partial derivatives of the equation a uu + a vv = 0 with respect to u and with respect to v, we get a uuu + a uvv = 0, (67) Combining Equations (65) and (68), we obtain Multiplying Equations (66) and (69) respectively by v and u and adding them, we get a uvv = 0, which, together with (67), implies a uuu = 0. From (66) and (65) we deduce a uuv = 0 and a vvv = 0. Since all the third order derivatives of the function a(u, v) vanish, we set where c 0 , c 1 , c 2 , l 1 , l 2 are constant vectors in R 8 . Using (62) and (70), it follows c 2 = 0 and therefore From (59) we compute The Cauchy-Riemann equations for a 0 and b 0 , using (72), are Using (73) and (74), we compute Using (60), (71) and (72), we get Since a(u, v) and b(u, v) satisfy the Cauchy-Riemann equations, from b u = −a v , using (71), Using (76) and (77) we get d 0 = −l 2 and c 0 = −l 1 . Consequently, (71) and (77) become

Remark 7.
Let us see what happens in the case when we work with T(u, v) = 1 cosh v . From the Equation (50) we immediately obtain that P u T does not depend on v. Hence, there exist functions A = A(u) and B = B(v) such that P(u, v) = A(u) cosh v + B(v). Using (48) and (49) we find that A(u) = c + c 1 cos u + c 2 sin u and cosh v , for some constants c, c 1 , c 2 and c 3 . Hence As before, we recall that P ∈ {u 1 , u 3 }; it follows that there exist six constant vectors 10 , 11 , 12 , 30 , 31 , We now replace u 1 and u 3 from (88) in (47) to obtain F(x 1 , y 1 , x 2 , y 2 ; u, v, w) = x 1 u 10 + x 1 v 11 + x 1 w 12 + x 2 u 2 + y 1 u 30 + y 1 v 31 + y 1 w 32 + y 2 u 4 , where x 1 , y 1 , x 2 , y 2 are as in the Remark 6 and u = cos u cosh v , v = sin u cosh v and w = tanh v are obtained using the isothermal coordinates u, v on the 2-sphere. We note that, for an appropriate choice of initial conditions, the immersion is the same as (87).
This confirms that the choice of isothermal coordinates on S 2 is not so important (in our problem) to arrive at the result. However, the most important fact is the ability of the reader in solving (explicitly) the system of PDE equations.

The Case 2: M is Congruent to
In this subsection we continue the study of Case 2, introduced in Section 3.1. Recall that in this case: The only non-zero components of the second fundamental form are In order to obtain the expression of the isometric immersion F : M 5 −→ S 7 in local coordinates, we write the Lie brackets all other being zero. Considering the following vector fields: we can easily prove that the Lie brackets of any two vectors from the set {Ē 1 , E 2 ,Ē 3 ,Ē 4 , ξ} vanish. Therefore, we can set (local) coordinates on M, call them x, y, z, u and v, such that Using (89) we conclude A = tan y, with y ∈ − π 2 , π 2 \ {0} (after a translation in the y-coordinate) and consequently we compute We can write now the expression of the metric g in terms of the (local) coordinates g = 1 4 dx 2 + dy 2 + dz 2 + cos 2y dxdz + cos 2 y du 2 + sin 2 y dv 2 .
Let F be the isometric immersion of M in S 7 . Analogously to Case 1, we obtain the system of partial differential equations satisfied by F: Further, we solve these partial differential equations satisfied by F. Using Equation (94) we conclude where U, V ∈ R 8 and U, V do not depend on y. Equations (97) and (98) imply that ∂U ∂u = 0 and ∂V ∂v = 0, that is U = U(x, z, v) and V = V(x, z, u).
Further, set p 0 to be the initial point on M corresponding to x = 0, y = π 4 , z = 0, u = 0, v = 0 and set the following initial conditions satisfied by F and its first partial derivatives, meaning that we fix the initial point on S 7 and the initial tangent space at p 0 as a subspace in T F(p 0 ) S 7 : (1, 0, 0, 0, 1, 0, 0, 0) Here J is the complex structure on R 8 locally defined by (85). Finally, set also the initial normal space (at p 0 ) as a subspace in T F(p 0 ) S 7 : 2JF v (p 0 ) = (0, 0, 0, 1, 0, 0, 0, 0).
Using (106), (108) and (113), we conclude F(x, y, z, u, v) = cos y cos u cos(z + x 2 ), cos y cos u sin(z + x 2 ), sin y sin v cos(z − x 2 ), sin y sin v sin(z − x 2 ), sin y cos v cos(z − x 2 ), sin y cos v sin(z − x 2 ), cos y sin u cos(z + x 2 ), cos y sin u sin(z + x 2 ) for y ∈ (0, π 2 ), x, z, u, v ∈ R. We will show that M can be expressed in terms of (multiply) warped products. Consider the following mutually orthogonal distributions on M: The key of the proof is to apply a generalization of Hiepko's theorem given by Nölker in 1996 in (Reference [19], Theorem 4). The following conditions are satisfied; they are analogue to the previous conditions (a)-(c): (i) the decomposition T(M) = D 0 ⊥ D 3 ⊥ D 4 is orthogonal; (here ⊥ means the orthogonal decomposition); (ii) the distributions D 3 and D 4 are spherical; (iii) the distributions D ⊥ 3 and D ⊥ 4 are autoparallel, that is ∇ Z W ∈ D ⊥ k , (k = 3, 4), for any Z, W ∈ D ⊥ k .
Let us focus on the second condition: for example, the distributions D 3 and D 4 are spherical since they are totally umbilical and the corresponding mean curvature vector fields, X 0 andX 0 , respectively, are parallel. From the Equation (20) we obtain X 0 = AE 2 andX 0 = − 1 A E 2 , which are parallel with respect to the corresponding normal connections.
Thus, for any point p ∈ M, there exists an isometric immersion Φ of a warped product N 0 × f N 3 ×ˆf N 4 onto a neighborhood of p in M such that Similar computations as in the case a 1 = a 3 imply that the warping functions are given by f = cos y andf = sin y.

Proposition 4.
Under the conditions stated for the case 2, it follows that M is locally congruent to a contact CR multiply warped product S 3 × f 1 S 1 × f 2 S 1 .

Conclusions and Further Research
We proved that a five-dimensional proper nearly totally geodesic contact CR-submanifold of seven-dimensional unit sphere is locally congruent to S 3 × f S 2 or to S 3 × f 1 S 1 × f 2 S 1 , via the immersions (86) and (114). Thus, the list of five-dimensional nearly totally geodesic contact CR-submanifolds in the seven-sphere is now complete. So, to finalize the research in this direction, we have to investigate hypersurfaces in S 7 which are nearly totally geodesic contact CR-submanifolds. This will be done in a future paper.
Author Contributions: Both authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.