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Totally umbilical submanifolds
Abstract
In the first part of this article, we provided an extensive survey on known results on totally umbilical submanifolds. In section 8, we defined and studied twisted product manifolds. In the last section, we presented eight open problems on totally umbilical submanifolds.
Supplementary resource (1)
... 6 Totally umbilical submanifolds of paracontact metric (κ, µ) manifolds Here σ is second fundamental form of the submanifold, g is the induced metric, H is mean curvature vector. X, Y are tangent to M [6]. ...
... Proof. From Codazzi equation [6] we get ...
The object of the present paper is to deduce some necessary and sufficient conditions for invariant Submanifolds of paracontact (κ, µ)-spaces to be totally geodesic. We also establish that a totally umbilical invariant submanifold of a paracontact (κ, µ)-manifold is also totally geodesic. Some more necessary and sufficient conditions for a submanifold of a paracontact (κ, µ)-manifold to be totally geodesic have been deduced using parallelity and pseudo parallelity of the second fundamental form. In the last section we obtain some results on paracontact (κ, µ)-manifold with concircular canonical field.
... where α and β are scalar fields and u i is a time-like vector field [14]. Proof. ...
In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved that spacetimes are Ricci flat in trivial, , and subspaces, whereas perfect fluid in subspaces , , and , and have zero scalar curvature in subspace . Finally, it is proved that pseudo-Ricci symmetric GRW spacetimes are vacuum, and as a consequence of this result, we address several corollaries.
1. Introduction
A pseudo-Ricci symmetric manifold (briefly ) is a nonflat pseudo-Riemannian manifold whose Ricci tensor satisfieswhere is a nonzero -form and indicates the covariant differentiation with respect to the metric [1].
The class of pseudo-Ricci symmetric manifolds is a subclass of weakly Ricci symmetric manifolds which were first introduced and studied by Tamássy and Binh [2]. There has been much focus on the concept of manifolds; for instance, a sufficient condition on manifolds to be quasi-Einstein manifolds was introduced by De and Gazi [3]. manifolds whose scalar curvature satisfies have zero scalar curvature [1]. A concrete example of pseudo-Ricci symmetric manifolds was given in [4]. There are many generalizations of manifolds, for example, see [5, 6].
An invariant orthogonal decomposition of the covariant derivative of the Ricci tensor was coined and studied by Gray in [7] (see also [8–10]). The manifolds in the trivial subspace have parallel Ricci tensor; that is, . The subspace contains manifolds whose Ricci tensor is Killing; that is,
The next subspace is denoted by . The Ricci tensors of manifolds in are Codazzi; that is,
The subspace is characterized by the equation . Manifolds withlie in . In , the tensor is Killing, whereas in , the tensor is a Codazzi tensor. Such manifolds are called Einstein-like manifolds [11]. Recently, there has been growing interest in this decomposition. For example, generalized Robertson–Walker spacetimes are either Einstein or perfect fluid in Gray’s orthogonal subspaces except one in which the Ricci tensor is not restricted [12].
An -dimensional Lorentzian manifold is said to be pseudo-Ricci symmetric spacetime if the Ricci tensor satisfies equation (1). Here, we assume the associated vector is a unit time-like vector ().
In standard theory of gravity, the relation between the matter of spacetimes and the geometry of spacetimes is given by Einstein’s field equation (EFE):where , , , and are the Ricci tensor, scalar curvature tensor, Newtonian constant, and energy-momentum tensor, respectively. EFE implies that the energy-momentum tensor is divergence-free. This requirement is directly satisfied if .
This paper is organized as follows: In Section 2, general properties of spacetimes are considered. In Section 3, spacetimes are investigated in all Gray’s orthogonal subspaces. It is proved that spacetimes in trivial, , and subspaces are Ricci flat, in subspaces , , and are perfect fluid spacetimes, and in have a zero scalar curvature. In Section 4, we prove that pseudo-Ricci symmetric GRW spacetimes are vacuum and as a consequence, we address some corollaries.
2. On Spacetimes
In this section, the main properties of spacetimes are considered. Equation (1) implies
The use of yields
A different contraction of equation (1) with gives
Solving equations (7) and (8) together, one gets
Lemma 1. In spacetimes, the covariant derivative of the scalar curvature is . Moreover, is an eigenvector of the Ricci tensor with zero eigenvalue.
Assume that the scalar curvature is constant. Equation (10) directly leads to .
Lemma 2. In (PRS) spacetimes, the scalar curvature is constant if and only if .
Let us consider ; then, the use of equation (10) in equation (1) implies that
This leads us to the following lemma.
Lemma 3. In spacetimes with nonzero scalar curvature, the covariant derivative of the Ricci tensor takes the formprovided .
The Weyl tensor of type has the form [13]and its divergence is
In virtue of (1) and (10), we have
Assume that the Weyl conformal curvature tensor is divergence-free, that is, ; then,
Contracting with and using equation (9), we obtain
A multiplication with gives , and hence,
Thus, we can conclude the following theorem:
Theorem 1. A spacetime with divergence-free Weyl curvature tensor is Ricci flat.
The use of this result () in the defining property of the conformal curvature tensor entails that
Hence, we have the following corollary.
Corollary 1. Semisymmetric and conformally semisymmetric pseudo-Ricci symmetric spacetimes are equivalent.
The covariant derivative of equation (1) gives
Interchanging the indices and in the last equation, we have
Subtracting the last two equations, we obtain
Making use of equation (1) and simplifying, we get
Now, assume that the is Ricci semisymmetric, that is, ; we have
Contracting with and using equation (9), we infer
Again, contracting with and utilizing equation (9), we get
Thus, we have the following theorem:
Theorem 2. Ricci semisymmetric spacetimes are Ricci flat.
3. Spacetimes in Gray’s Decomposition Subspaces
This section is devoted to study spacetimes in Gray’s seven subspaces. Three main results are obtained in this section. A Lorentzian manifold is said to be perfect fluid if its Ricci tensor satisfieswhere are scalar fields and is a time-like vector field [14].
Theorem 3. spacetimes in trivial, , and subspaces are Ricci flat.
Proof. The trivial subspace of Gray’s decomposition contains spacetimes whose Ricci tensors are parallel and the scalar curvatures are constant. Thus, equation (10) easily gives . And hence, equation (1) becomesA contraction of equation (28) with yieldsAnd consequently,which means that spacetimes with parallel Ricci tensor are Ricci flat.
In subspace spacetimes have a Killing Ricci tensor; that is,It is well known that in this subspace, the scalar curvature is covariantly constant. Equation (10) implies . Using equation (1) in equation (31), we haveContracting equation (32) with and using equation (9), we getwhich means that spacetimes in subspace are Ricci flat.
Next, let us consider the subspace in which has a Codazzi type of Ricci tensor [15]. The Codazzi deviation tensor of is given byA contraction with impliesBut, in this subspace, the spacetimes have Codazzi-type Ricci tensor (that is, ); then,Multiplying with and utilizing equation (9), we getA contraction of equation (36) by giveswhich means that spacetimes in Gray’s subspace are Ricci flat.
Theorem 4. spacetimes in , , and subspaces are perfect fluid spacetimes.
Proof. In subspace , the Ricci tensor of pseudo-Ricci symmetric manifold satisfies the following property:Applying equation (1), we obtainIt follows thatContracting with implieswhich means that spacetimes in subspace are perfect fluid.
In subspace , the Ricci curvature tensor satisfiesUsing equation (1), we inferNow, equation (10) impliesA contraction with yieldswhich means that spacetimes in subspace are perfect fluid.
Assume that are in Gray’s subspace ; that is,Equation (1) impliesThe use of equation (10) givesContracting with , we obtainwhich means that spacetimes in Gray’s subspace are perfect fluid.
Theorem 5. spacetimes in subspace have zero scalar curvature.
Proof. In subspace , the scalar curvature is covariantly constant and hence equation (10) implieswhich means spacetimes in Gray’s subspace have zero scalar curvature.
4. Pseudo-Ricci Symmetric GRW Spacetimes
A generalized Robertson–Walker spacetime (for simplicity, denoted by GRW spacetimes) is the warped product of an open connected interval and a Riemannian manifold , where is a positive smooth function. A Lorentzian manifold is a generalized Robertson–Walker spacetime if and only if possesses a unit time-like vector field with [16, 17]where and are scalar functions. Vector fields satisfying equation (52) are called torse-forming.
Now, assume that is a generalized Robertson–Walker spacetime; that is,
A contraction with yields
Using equation (1), one gets
Therefore,
However,
Thus,
It is well known , where (see [12]); thus,
Since is , equation (9) shows that
Multiplying both the sides by , that is,
Using equation (53), one gets
Now, there are two different possible cases. The first one and consequently does not vanish. Then, equation (59) becomes
A contraction by implies thatwhich is a contradiction. The second case is . Then, equation (60) leads to
Thus, either or .
Theorem 6. A pseudo-Ricci symmetric GRW spacetime is vacuum provided the one form is not codirectional with the torse-forming vector field .
Suppose . Then, the spacetime under consideration is Ricci flat, that is, , which implies . It is known thatwhere is the conformal curvature tensor [13].
Therefore, using and , equation (67) yields , that is, . In [18], Mantica et al. proved that an -dimensional GRW spacetime satisfies if and only if the spacetime is perfect fluid. Therefore, we conclude the following.
Corollary 2. A pseudo-Ricci symmetric GRW spacetime is a perfect fluid spacetime provided .
Since and , from the definition of the conformal curvature tensor, it follows that . Hence, semisymmetric and conformally semisymmetric manifolds are equivalent. Eriksson and Senovilla [19] considered the semisymmetric spacetime and proved that it is of Petrov types , , and . Thus, we have the following.
Corollary 3. A conformally semisymmetric pseudo-Ricci symmetric GRW spacetime is of Petrov types , , and .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the King Saud University, Deanship of Scientific Research, College of Science Research Center.
... In 1979 Bang-Yen Chen introduced twisted spacetimes, eq.(1) with b = 1 [5]. Years later he characterized them through the existence of a timelike torqued vector field [6]: ...
In this note, we characterize [Formula: see text] doubly twisted spacetimes in terms of “doubly torqued” vector fields. They extend Bang–Yen Chen’s characterization of twisted and generalized Robertson–Walker spacetimes with torqued and concircular vector fields. The result is a simple classification of [Formula: see text] doubly-twisted, doubly-warped, twisted and generalized Robertson–Walker spacetimes.
... In 1979 Bang-Yen Chen introduced twisted spacetimes, eq.(1) with b = 1 [5]. Years later he characterized them through the existence of a timelike torqued vector field [6]: ...
In this note we characterize 1+n doubly twisted spacetimes in terms of `doubly torqued' vector fields. They extend Bang-Yen Chen's characterization of twisted and generalized Robertson-Walker spacetimes with torqued and concircular vector fields. The result is a simple classification of 1+n doubly-twisted, doubly-warped, twisted and generalized Robertson-Walker spacetimes.
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This book attempts to present a comprehensive survey of the geometry of CR-submanifolds. The theory of submanifolds is one of the most interesting topics in differential geometry. The topic is
introduced by Aurel Bejancu as a generalization of holomorphic and totally real submanifolds of almost Hermitian manifolds, in 1978. Afterward, the study of CR-submanifolds became a very active research subject.
Organized into 22 chapters, the book starts with basic knowledge of Riemannian manifolds and submanifolds, almost Hermitian manifolds and their subclasses, Hopf fibration, symmetric spaces, and a general inequality for submanifolds in complex space forms (in Chaps. 1 and 2). Later, it presents the main results on CR-submanifolds in Kaehler manifolds, the basic inequalities associated with CR-submanifolds in Kaehler manifolds, and several theories and results related to Kaehler manifolds (in Chaps. 3–11). Further, the book discusses the basics of almost-contact metric manifolds and their subclasses, CR-submanifolds of Sasakian, trans-Sasakian and quasi-Sasakian manifolds, with a particular attention on the normal CR-submanifolds (in Chap. 12). It also investigates the contact CR-submanifolds of S-manifolds, the geometry of submersions of CR-submanifolds, and the results on
contact CR-warped product submanifolds (in Chaps. 16–18, 20). In Chapter 19, we discuss submersions of CR-submanifolds. The book also presents some recent results concerning CR-submanifolds of
holomorphic statistical manifolds. In particular, it gives the classification of totally umbilical CR-statistical submanifolds in holomorphic statistical manifolds, as well as a Chen–Ricci inequality for such submanifolds (Chapter 21). In the last chapter, we present results on CR-submanifolds of indefinite Kaehler manifolds and their applications to physics.
If N is a compact totally umbilical with constant mean curvature immersed in a Hermitian symmetric space M of compact type , then N is either totally geodesic or a real space form immersed in M as a purely real submanifold satisfying 2 dim N < dim M.
We define a notion of contact totally umbilical submanifolds of Sasakian space forms corresponds to those of totally umbilical
submanifolds of complex space forms. We study a contact totally umbilical submanifold M of a Sasakian space form
[`(M)] ( c )\overline M \left( c \right)
(c ≠ −3) and prove that M is an invariant submanifold or an anti-invariant submanifold. Furthermore we study a submanifold
M with parallel second fundamental form of a Sasakian space form
[`(M)] ( c )\overline M \left( c \right)
(c ≠ 1) and prove that M is invariant or anti-invariant.
Many subjects for anti-invariant submanifolds of a Sasakian manifold have been investigated extensively by Yano & Kon (1976) and others. Kenmotsu (1972) introduced a new class of almost contact metric structure known as 'Kenmotsu structure, which is closely related to the warped product of two Riemannian manifolds. The purpose of this note is to study anti-invariant submanifolds of Kenmotsu manifold. We study the theory of such submanifold from two different point of view, namely, one is the case where the anti-invariant submanifolds are tangent to the structure vector field, and the other is the case where these are normal to the structure vector field.