All articles published by MDPI are made immediately available worldwide under an open access license. No special
permission is required to reuse all or part of the article published by MDPI, including figures and tables. For
articles published under an open access Creative Common CC BY license, any part of the article may be reused without
permission provided that the original article is clearly cited. For more information, please refer to
Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature
Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for
future research directions and describes possible research applications.
Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive
positive feedback from the reviewers.
Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world.
Editors select a small number of articles recently published in the journal that they believe will be particularly
interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the
most exciting work published in the various research areas of the journal.
Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure , we study its five-dimensional contact -submanifolds, which are the analogue of -submanifolds in (almost) Kählerian manifolds. In the case when the structure vector field is tangent to M, the tangent bundle of contact -submanifold M can be decomposed as where is invariant and is anti-invariant with respect to . On this occasion we obtain a complete classification of five-dimensional proper contact -submanifolds in whose second fundamental form restricted to and vanishes identically and we prove that they can be decomposed as (multiply) warped products of spheres.
Let M be a Riemannian submanifold of the seven-dimensional unit sphere. It is well-known that possesses the almost contact structure , which is also contact and Sasakian. Having in mind the behaviour of the endomorphism , submanifolds in the Sasakian manifolds carrying a -invariant distribution such that its orthogonal complement is -anti-invariant, are called contact -submanifolds. This notion is the odd-dimensional analogue of -submanifolds in (almost) Kählerian manifolds, introduced by Bejancu in , who requested the existence of a differentiable holomorphic distribution such that its orthogonal complement is a totally real distribution. Also, -submanifolds of the nearly Kähler six-dimensional unit sphere have also been investigated (see [2,3], for example). As the -invariant distribution is always even-dimensional, the lowest possible dimension for a proper contact -submanifold (i.e., suchthat the dimensions of both -invariant and anti-invariant distributions are different from zero) is four. In this paper we continue our study of certain contact -submanifolds in seven-dimensional unit sphere, which we started in  for the case of four-dimensional submanifolds and continued in , where we presented several examples of four and five-dimensional contact -submanifolds of , 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 -submanifold in a Sasakian manifold can never be totally geodesic. Therefore, on this occasion we study those five-dimensional contact -submanifolds in a Sasakian sphere which are close to be totally geodesic, namely those whose second fundamental form restricted to both -invariant and -anti-invariant distributions vanishes identically. Calling them nearly totally geodesic contact -submanifolds, we prove that such submanifolds are (multiply) warped products of spheres and finding the immersions, we obtain their complete classification.
Let M be a five-dimensional proper nearly totally geodesic contact -submanifold of seven-dimensional unit sphere. Then M is locally congruent to (multiply) warped product via the immersions (86) and (114).
In  we presented several examples of four and five-dimensional contact -submanifolds of product and warped product type of seven-dimensional unit sphere, which are nearly totally geodesic, minimal and which satisfy the equality sign in some Chen type inequalities.
2.1. Sasakian Manifolds
Let be a Sasakian manifold with the structure tensors , , and . If denotes the Levi-Civita connection on , then the following relations
hold on for all . For more details we refer to , although we use the convention of [7,8].
Let be a submanifold in the Riemannian manifold , where g is the induced metric and let ∇ and be the Levi Civita connections on M and , respectively. Recall the formulae of Gauss and Weingarten
for and , where and are the second fundamental form and the shape operator corresponding to , respectively, related by . Here is a connection in the normal bundle of and is its curvature tensor. If stands for the van der Waerden–Bortolotti connection, which is defined as
for tangent to M, the Codazzi Theorem can be written as
where is the curvature tensor on defined by , and by we mean the projection on the normal bundle. For submanifolds of real space forms, like spheres, this projection vanishes identically and therefore the covariant derivative of the second fundamental form, defined by , is totally symmetric.
The equations of Gauss and Ricci are given by
respectively, where is the curvature on , and , are the shape operators corresponding to the normal vectors , , respectively.
2.3. Contact -Submanifolds
The notion of contact -submanifolds in Sasakian manifolds is the odd-dimensional analogue of -submanifolds in (almost) Kählerian manifolds. See also . Particularly, a contact -submanifold in the Sasakian manifold is a submanifold M carrying a -invariant distribution , that is, , for any , such that the orthogonal complement of in is -anti-invariant, that is, , for any . This notion was used by Bejancu and Papaghiuc in , using the terminology of semi-invariant submanifold. It is standard to require that is tangent to M rather than normal, which is too restrictive (by Prop. 1.1 in , p. 43, M must be -anti-invariant, that is, , for all ), or oblique which leads to highly complicated embedding equations. The contact -submanifold is called proper if both distributions and are non-trivial distributions.
For a contact -submanifold M of a Sasakian manifold , with tangent to M, the tangent space at each point decomposes orthogonally as
where , along and . It should be remarked that the Levi distribution is never integrable (see ), while can be, as in the case of contact -products (see ). On the other hand, the normal bundle of M can be decomposed as where is the orthogonal complement of in , invariant under the action of .
2.4. 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  in order to construct a large class of complete manifolds of negative curvature.
Let be two Riemannian manifolds with Riemannian metrics and respectively and let f be a smooth positive function on B. Considering the product manifold , let and be the canonical projections. The manifold is called the warped product if it is equipped with the Riemannian structure such that
for all , , or, equivalently, with the usual meaning, while f is called the warping function on the warped product. For more details we refer to .
A contact -submanifold M in a Sasakian manifold , tangent to the structure vector field , is called a contact CR warped product, with the warping function f, if it is the warped product of an invariant submanifold , tangent to and a totally real submanifold of , where f is the warping function (see  for more details). It is notable to point out that there is no proper contact -submanifolds in Sasakian manifolds in the form . This fact was proved in [13,16].
2.5. The Sasakian Structure on
Identifying with , let J denote the multiplication with the imaginary unit , on . The -dimensional unit sphere , where is the usual scalar product in , carries a canonical almost contact metric structure induced from . Strictly speaking, as at any point , the outward unit normal to sphere coincides with the position vector , putting to be the characteristic vector field, for X tangent to , fails, in general, to be tangent and decomposing into the tangent and the normal part we have Moreover, this structure is Sasakian. For more details and proofs we refer to [6,17].
On this occasion, we consider the problem of finding all five-dimensional proper contact -submanifolds in such that
where, as we have already mentioned, and , with , and being the orthogonal complement of in .
3. Five-Dimensional Nearly Totally Geodesic Contact -Submanifolds in
In order to prove our results, we first select an appropriate frame on in such a way that equations, which are the consequences of (1), become satisfied. Then, we classify all 5-dimensional proper contact -submanifolds in the seven-dimensional unit sphere satisfying (1).
3.1. Essential Characteristics of Five-Dimensional Contact
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:
If M is a contact -submanifold in the Sasakian manifold we have
for every , .
As M is a five-dimensional submanifold of , having in mind the condition (1), we conclude that
Further, starting with two arbitrary orthonormal bases in and in , respectively, we will choose a basis in 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 , a symmetric operator
Since , there exist six smooth functions , on M such that
For an arbitrary (for the moment) angle , we consider and we compute
We distinguish the following situations:
if and then for any ;
if and , then take and denote and ;
if and , then take and denote and ;
if both and , then take t such that ; denote the corresponding X by and set .
It follows that in Cases (ii)–(iv) we can choose and in such that the operator is traceless. Additionally, because the operator is also symmetric, we will take the basis in defined by the eigenvectors of this operator. Denote them by and . Consequently we have that is proportional to and is proportional to .
Concerning the Case (i), we (apparently) have the freedom of choosing . Nevertheless, if we make a rotation about a certain angle s in , we set and . If take and if take s such that . Consequently, we obtain and . Since s depends on and and hence on and , we set and .
For simplicity of notation, we continue to write for and for . Moreover, since is Sasakian, using the Gauss formula, we can easily compute , .
Summarizing, we have thus proved
For a proper contact -submanifold in such that the condition (1) is satisfied, we can choose orthonormal differential vector fields defined locally on M, such that , and
where are smooth functions on M and consequently
Further, let us introduce some smooth functions to describe the induced connection on M.
Under the conditions stated above, the Levi-Civita connection ∇ is given by
for certain smooth functions and θ on M.
Since , we immediately obtain that for and this implies that has no component along , for every and .
Considering and identifying the tangent and the normal parts, respectively, we obtain
for a certain function . □
In order to have the complete description of the geometry of M, we write the expression of the normal connection, that is
Under the above assumptions, the coefficient vanishes.
Using the fact that is totally symmetric, we obtain the equations given in Table 1.
If it follows that and . Consequently, we get and . Finally, (on one hand) and (on the other hand). Hence we get a contradiction. □
So, from now on we will take .
Let us develop all situations for the Codazzi quation. Due to the totally symmetry of 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:
Under the same hypothesis as for Proposition 2, the Codazzi equations are automatically satisfied for the triple for , as well as for the triple for .
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.
Under the above conditions, the coefficient vanishes.
Contrary, if in a point, it is different from zero on an open neighborhood. Looking to line L1 in Table 2, we deduce that .
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 vanishes. □
From now on we will distinguish two cases: Case 1. and Case 2. .
We will obtain some more equations in each of the two cases and then we completely solve our problem in Section 3.2 and Section 3.3.
Case 1. For the sake of simplicity, we make the following notation
which implies that A cannot vanish. Developing all the equations in the Table 2, we obtain:
With respect to the orthonormal basis in 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
Case 2. As , we immediately obtain , and a, c, q, l, r, , and vanish. Moreover, we should have . Again, for the sake of simplicity we denote and hence and . 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.
3.2. The Case 1: M Is Congruent to
In this subsection we study in detail the case and prove that then the contact nearly totally geodesic -submanifold M is congruent to and we determine the explicit immersion.
Let us consider the following distributions on M: and . Let and be the orthogonal projections from to , respectively to . From Proposition 2 we write the expression of the Levi-Civita connection on M for the case :
As a consequence, we find that the following relations are true:
, for all in ;
, for all Z in ;
, for all in ;
where . The statements (a) and (c) imply that and are both involutive. The statements (a) and (b) mean that the maximal integral manifolds of are extrinsic spheres. Finally, the statement (c) says that the integral manifolds of are totally geodesic.
Now, applying a famous result of Hiepko (, p. 213): Let be a (pseudo-)Riemannian manifold endowed with a pair 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 , there exists an isometry from a warped product to a neighborhood of p in M with the property
is an integral manifold for for every ;
is an integral manifold for for every ;
where is the warping function on .
In order to find the warping function on M, let us consider the following vector field in : . One can immediately prove that
Thus, we choose local coordinates on M (in fact on ) such that , and . Using (6), after a possible translation in y-coordinate, we obtain
Taking , we obtain and
The restriction of the metric g to can be expressed in terms of the coordinates x, y and z as follows:
Moreover, from (, Theorem 3.2), we know that , for any and any . Using the expression (7) for the shape operator, we get that and combining it with (21) we find
which is the warping function on M.
Since is totally geodesic in M and M is nearly totally geodesic in , it follows that is (isometrically) immersed in as a totally geodesic submanifold. With a similar argument, is immersed in as a totally umbilical submanifold. Hence can be considered to be a (portion of) . Additionally, , being totally umbilical in , 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 in is . Thus, the curvature of the 2-sphere above is , and hence its radius is . Consequently, we will consider with the warping function , .
Under the conditions stated for the case 1, it follows that M is locally congruent to a contact warped product .
Defining the 1-form Ω on M by , 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 , that is, on . This function is nothing but . It can be easily checked that .
Now, let us determine the metric. Choosing isothermal coordinates on , we have
where T is a smooth positive function on M, depending on u and v. Here we made the following notations: and similar for and .
As both and are orthonormal bases in , then one is obtained from the other by a (not necessary positively oriented) rotation.
Finally, due to the orthogonality of the two distributions and , we have
We adopted similar notations as before for 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 , it can be expressed as
It follows that
Moreover, as the function is equal to , it is independent of Z. Hence, is not uniquely defined and it could be replaced by any other unit vector Z in .
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, . Using (22) and (20), we conclude
on one hand ,
and on the other hand .
Thus, and .
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 :
Comparing with , we get
In a similar way we find
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 . Let be the canonical inclusion of the 7-dimensional unit sphere in the 4-dimensional complex space. Denoting by the scalar product on and by the corresponding flat connection, we have
for all X and Y tangent to , where denotes the position vector of a point of the sphere. Using the Gauss formula we have
for all X and Y tangent to M.
For example, if we put and we get
We obtain that F also satisfies, simultaneously, the following partial differential equations:
Observe that Equation (26) also follows from (27) and (28). Moreover, using (37) and (28), we conclude that Equations (27) and (34) are equivalent. So, not all the previous PDEs are independent.
Then, combining (29) with (31) and (35) with (36), we get
respectively, that is depends neither on u, nor on v.
Considering Equations (28), (33) and (37), we deduce
where are vectors in which do not depend on and z, but they do depend on u and v.
Considering the two vector-valued functions and and using (48)–(50), we conclude that and satisfy the Cauchy-Riemann equations for each component of the vector valued function . This means that depends only on and not on . 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 . There are several ways to consider isothermal coordinates u and v on the 2-sphere, that is, to write the metric as . Recall two of them:
, associated to the parametrization
obtained from the stereographic projection;
, associated to the parametrization
obtained as a surface of revolution.
After setting and solving the Equation (53), we obtain
where and also satisfy the Cauchy-Riemann equations. Since , we conclude
On the other hand, since , using (53) and the information that a and b satisfy the Cauchy-Riemann equations, we compute
Multiplying Equations (66) and (69) respectively by v and u and adding them, we get , which, together with (67), implies . From (66) and (65) we deduce and . Since all the third order derivatives of the function vanish, we set
where are constant vectors in . Using (62) and (70), it follows and therefore
Further, set to be the initial point on M corresponding to , , , , . Then , , . Set also the following initial conditions satisfied by F and the first partial derivatives, meaning that we fix the initial point on and the initial tangent space at as a subspace in :
Here J is the complex structure on locally defined by
Using (31) and (36), with and , we compute , , where is the canonical basis in .
Consequently, we conclude
Recall that we used T that corresponds to the stereographic projection. The parametrization (54) can be re-written as .
Besides, we can consider the parametrization , where
With these notations, the immersion given in (86) becomes
Remark that F is nothing but the immersion given in (Reference , Equation (3.8)) up to some permutation of coordinates and orientation of . The warping function is .
Let us see what happens in the case when we work with . From the Equation (50) we immediately obtain that does not depend on v. Hence, there exist functions and such that . Using (48) and (49) we find that and , for some constants c, , and . Hence
As before, we recall that ; it follows that there exist six constant vectors , , , , , such that
where , , , are as in the Remark 6 and , and are obtained using the isothermal coordinates 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 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.
3.3. 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: , , and
The only non-zero components of the second fundamental form are
In order to obtain the expression of the isometric immersion 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 vanish. Therefore, we can set (local) coordinates on M, call them x, y, z, u and v, such that
Using (89) we conclude , with (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
Let F be the isometric immersion of M in . 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.
Finally, using (109), (111) and (112) we obtain the last differential equations , , and . Consequently, we get
for some constant vectors in .
Moreover, since F lies on , we conclude and and, consequently, , , and are unitary and mutually orthogonal.
Further, set to be the initial point on M corresponding to , , , , and set the following initial conditions satisfied by F and its first partial derivatives, meaning that we fix the initial point on and the initial tangent space at as a subspace in :
Here J is the complex structure on locally defined by (85). Finally, set also the initial normal space (at ) as a subspace in :
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 , Theorem 4). The following conditions are satisfied; they are analogue to the previous conditions (a)–(c):
the decomposition is orthogonal;
(here means the orthogonal decomposition);
the distributions and are spherical;
the distributions and are autoparallel, that is , (), for any .
Let us focus on the second condition: for example, the distributions and are spherical since they are totally umbilical and the corresponding mean curvature vector fields, and , respectively, are parallel. From the Equation (20) we obtain and , which are parallel with respect to the corresponding normal connections.
Thus, for any point , there exists an isometric immersion of a warped product onto a neighborhood of p in M such that
is an integral manifold for for every , ;
is an integral manifold for for every , ;
is an integral manifold for for every , .
Similar computations as in the case imply that the warping functions are given by and .
Under the conditions stated for the case 2, it follows that M is locally congruent to a contact multiply warped product .
In accordance to the case 1 (), we consider the same parametrization on . Then, on the two circles we set and , respectively. Thus, the immersion F can be thought (see also Reference ) as the following map
where the warping functions are given by and .
4. Conclusions and Further Research
We proved that a five-dimensional proper nearly totally geodesic contact -submanifold of seven-dimensional unit sphere is locally congruent to or to , via the immersions (86) and (114). Thus, the list of five-dimensional nearly totally geodesic contact -submanifolds in the seven-sphere is now complete. So, to finalize the research in this direction, we have to investigate hypersurfaces in which are nearly totally geodesic contact -submanifolds. This will be done in a future paper.
Both authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.
The first author is partially supported by Ministry of Education, Science and Technological Development, Republic of Serbia, project 174012. The second author is supported by the project funded by the Ministry of Research and Innovation within Program 1—Development of the national RD system, Subprogram 1.2—Institutional Performance—RDI excellence funding projects, Contract no. 34PFE/19.10.2018.
Both authors wish to thank Luc Vrancken (Université de Valenciennes, Université Lille Nord de France) for his useful hints and help given during the preparation of this paper. The first author wishes to express her gratitude to the Technical University “Gheorghe Asachi”’ of Iasi, Romania for the hospitality she received during the research visit in the framework of the project PNII-RU-PD-2012-3-0387, UEFISCDI Romania. The authors gratefully thank to the four referees for the constructive comments and suggestions which definitely helped improve the readability and quality of the paper.
Conflicts of Interest
The authors declare no conflict of interest.
Bejancu, A. CR Submanifolds of a Kähler Manifold I. Proc. Am. Math. Soc.1978, 69, 134–142. [Google Scholar]
Djorić, M.; Vrancken, L. Three-dimensional minimal CR submanifolds in satisfying Chen’s equality. J. Geom. Phys.2006, 56, 2279–2288. [Google Scholar] [CrossRef]
Antić, M.; Djorić, M.; Vrancken, L. 4-dimensional minimal CR submanifolds of the sphere S6 satisfying Chen’s equality. Diff. Geom. Appl.2007, 25, 290–298. [Google Scholar] [CrossRef][Green Version]
Djorić, M.; Munteanu, M.I.; Vrancken, L. Four-dimensional contact CR-submanifolds in S7(1). Math. Nachr.2017, 290, 2585–2596. [Google Scholar] [CrossRef]
Djorić, M.; Munteanu, M.I. On certain contact CR-submanifolds in . In Contemporary Mathematics; American Mathematical Society: Providence, RI, USA, 2020. [Google Scholar]
Blair, D.E. Riemannian geometry of contact and simplectic manifolds. In Progress in Mathematics; Birkäuser: Basel, Switzerland, 2001; Volume 203. [Google Scholar]
Harada, M. On Sasakian submanifolds, (Collection of articles dedicated to Shigeo Sasaki on his sixtieth birthday). Tohoku Math. J.1973, 25, 103–109. [Google Scholar] [CrossRef]
Yano, K.; Kon, M. Generic submanifolds of Sasakian manifolds. Ködai Math. J.1980, 3, 163–196. [Google Scholar] [CrossRef]
Djorić, M.; Okumura, M. CR Submanifolds of Complex Projective Space. In Developments in Mathematics; Springer: Berlin/Heidelberg, Germany, 2010; Volume 19. [Google Scholar]
Bejancu, A.; Papaghiuc, N. Semi-invariant submanifolds of a Sasakian manifold. Analele Ştiinţifice ale Universităţii “Al. I. Cuza” din Iaşi1981, 1, 163–170. [Google Scholar]
Yano, K.; Kon, M. CR submanifolds of Kaehlerian and Sasakian manifolds. In Progress in Mathematics; Birkhäuser: Basel, Switzerland, 1983; Volume 30. [Google Scholar]
Capursi, M.; Dragomir, S. Contact Cauchy-Riemann submanifolds of odd dimensional spheres. Glasnik Matematicki1990, 25, 167–172. [Google Scholar]
Munteanu, M.I. Warped product contact CR-submanifolds of Sasakian space forms. Publ. Math. Debrecen.2005, 66, 75–120. [Google Scholar]
Bishop, R.L.; O’Neill, B. Manifolds of negative curvature. Trans. Am. Math. Soc.1969, 145, 1–49. [Google Scholar] [CrossRef]
Chen, B.-Y. Differential Geometry of Warped Product Manifolds and Submanifolds; World Scientific: Hackensack, NJ, USA, 2017. [Google Scholar]
Hasegawa, I.; Mihai, I. Contact CR-Warped Product Submanifolds in Sasakian Manifolds. Geom. Dedicata2003, 102, 143–150. [Google Scholar] [CrossRef]
Tashiro, Y. On contact structures of hypersurfaces in complex manifolds I. Tohoku Math. J.1963, 15, 62–78. [Google Scholar] [CrossRef]
Hiepko, S. Eine innere Kennzeichnung der verzerrten Produkte. Math. Ann.1979, 241, 209–215. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely
those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or
the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas,
methods, instructions or products referred to in the content.