- About this Journal ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Volume 2012 (2012), Article ID 983403, 26 pages
A Review on Metric Symmetries Used in Geometry and Physics
University of Windsor, Windsor, ON, Canada N9B 3P4
Received 8 November 2011; Accepted 19 December 2011
Academic Editor: C. Qu
Copyright © 2012 K. L. Duggal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This is a review paper of the essential research on metric (Killing, homothetic, and conformal) symmetries of Riemannian, semi-Riemannian, and lightlike manifolds. We focus on the main characterization theorems and exhibit the state of art as it now stands. A sketch of the proofs of the most important results is presented together with sufficient references for related results.
The measurement of distances in a Euclidean space is represented by the distance element with respect to a rectangular coordinate system . Back in 1854, Riemann generalized this idea for -dimensional spaces and he defined element of length by means of a quadratic differential form on a differentiable manifold , where the coefficients are functions of the coordinates system , which represent a symmetric tensor field of type . Since then much of the subsequent differential geometry was developed on a real smooth manifold , called a Riemannian manifold, where is a positive definite metric tensor field. Berger’s book  includes the major developments of Riemannian geometry since 1950, citing the works of differential geometers of that time. On the other hand, we refer standard book of O’Neill  on the study of semi-Riemannian geometry where the metric is indefinite and, in particular, Beem and Ehrlich  on the global Lorentzian geometry used in relativity. In general, an inner product on a real vector space is of type where , with nonzero and with nonzero . A metric on a manifold is a symmetric tensor field on of the type on its tangent bundle space . Kupeli  called a manifold of this type a singular semi-Riemannian manifold if admits a Koszul derivative, that is, is Lie parallel along the degenerate vector fields on . Based on this, Kupeli studied the intrinsic geometry of such degenerate manifolds. On the other hand, a degenerate submanifold of a semi-Riemannian manifold may not be studied intrinsically since due to the induced degenerate tensor field on one cannot use, in general, the geometry of . To overcome this difficulty, Kupeli used the quotient space and the canonical projection for the study of intrinsic geometry of . Here denotes the radical distribution of .
In 1991, Bejancu and Duggal  introduced a general geometric technique to study the extrinsic geometry of degenerate submanifolds, popularly known as lightlike submanifolds of a semi-Riemannian manifold. They used the decomposition where is a nondegenerate complementary screen distribution to and is a symbol for orthogonal direct sum. is not unique; however, it is canonically isomorphic to the quotient bundle .
There are three types of metrics, namely, Riemannian, semi-Riemannian, and degenerate (lightlike). The properties of Riemannian metrics which come from their nondegenerate character remain same in the semi-Riemannian case. However, neither "geodesic completeness" nor "sectional curvature" nor "analysis on Lorentzian manifolds" works in the same way as in the Riemannian case. However, the case of degenerate metric is different (see Section 5).
One of the widely used technique is to assume the existence of a metric tensor with a symmetry as follows: consider with the metric of any one of the three types and a vector field (local or global) of such that where is the Lie derivative operator and is a function on . Above equation is known as conformal Killing equation and the symmetry vector is called a conformal Killing vector, briefly denoted by CKV. If is nonconstant, then is called a proper CKV. In particular, is homothetic or Killing according as is a nonzero constant or zero. The set of all proper CKV fields and all Killing vector fields on form a finite-dimensional Lie algebra.
The purpose of this paper is to present a survey of research done on the geometry and physics of Riemannian, semi-Riemannian, in particular, Lorentzian and lightlike manifolds having a metric symmetry defined by (1.3). We collect the results of the two main symmetries, namely, Killing and conformal Killing and their two closely related subsymmetries, called affine Killing and affine conformal Killing symmetries. This approach will help the reader to better understand the differences, similarities, and relations between these two symmetries, with respect to their use in geometry and physics. A sketch of the proof of the most important results is given along with references for their link with several other related results.
The subject matter of metric symmetries is very wide and cannot be covered in one review paper. For this reason we have provided a large number of references for more related results.
2. Riemannian and Semi-Riemannian Metric Symmetries
Given a smooth manifold , the group of all smooth transformations of is a very large group. This leads to the study of those transformations of which leave a certain physical/geometric quantity invariant. Related to the focus of this paper we let be a real -dimensional smooth Riemannian or semi-Riemannian manifold. A diffeomorphism is called an isometry of if it leaves invariant the metric tensor . This means that where is the differential (tangent) map of and denotes the set of all tangent vector fields on . Since each tangent mapping , at , is a linear isomorphism of on , it follows that is an isometry if and only if is a linear isometry for any . The set of all isometries of forms a group under composition of mappings. Myers and Steenrod  proved that the group of all isometries of a Riemannian manifold is a Lie group. For analogous results on semi-Riemannian manifolds see O’Neill [2, chapter 9]. The isometric symmetry is related to a local infinitesimal transformation group as follows.
Let be a smooth vector field on and a neighborhood of each with coordinate system . Let the integral curves of , through any point in , be defined on an open interval for . For each define an isometric map on such that for in , is on the integral curve of through . Then, generates a local 1-parameter group of infinitesimal transformations and we have which, after expanding up to first-order in , yields to Using the Lie derivative operator , the above equation can be rewritten as where is the associated 1-form of and is the Levi-Civita connection on . The above Killing equations were named after a German mathematician Killing  who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. A simple example is a vector field on a circle that points clockwise and has the same length at each point is a Killing vector field since moving each point on the circle along this vector field just rotates the circle. For an -dimensional Euclidean space, there exist independent Killing vector fields. In general, any Riemannian or semi-Riemannian manifold which admits maximum Killing vector fields is called a manifold of constant curvature. This section contains important results on compact Riemannian, Kählerian, contact, and semi-Riemannian manifolds.
2.1. Riemannian Manifolds
The following divergence theorem is used in proofs of some results on the existence or nonexistence of Killing and affine Killing vector fields.
Theorem 2.1. Let be a compact orientable Riemannian manifold with boundary . For a smooth vector field on , one has where and are the unit normal to and its surface element and is the volume element of .
Consider an -dimensional Riemannian manifold without boundary, that is, holds for a smooth vector field on . Let be a Killing vector field of , that is, where denotes a symmetric affine connection on . We start with the following fundamental theorem on the existence of a Killing vector field.
Theorem 2.2 (Bochner ). If Ricci tensor of a compact orientable Riemannian manifold , without boundary, is negative semidefinite, then a Killing vector field on is covariant constant. On the other hand, if the Ricci tensor on is negative definite, then a Killing vector field other than zero does not exist on .
Bochner proved this theorem by assuming that is a gradient of a function and a result of Watanabe  which states “ if is Killing.” Several other results on the geometry of compact Riemannian manifolds, without boundary, presented in Yano [10, 11] are consequences of above result of Bochner.
Remark 2.3. Recall from Berger  that all known examples of compact Riemannian manifolds with positive sectional curvature carry a positively curved metric with a continuous Lie group as its group of isometries. Thus, they carry a nontrivial Killing vector field. Moreover, such a Killing vector is singular at least at one point if the manifold is even dimensional. There are examples of odd-dimensional closed positively curved Riemannian manifolds carrying nonsingular Killing vector fields. A simple case is the 3-sphere which admits 3 pointwise linearly independent Killing vector fields while no two of them commute. Killing symmetry has another closely associated symmetry, with respect to a symmetric affine connection on a nonflat Riemannian or semi-Riemannian manifold , defined as follows.
A vector field on is called affine Killing if . To interpret this relation with respect to the metric we split the tensor (see Killing equation (1.3)) into its symmetric and antisymmetric parts as follows: Then, it follows that is covariant constant, that is, From (2.7) and (2.8) we deduce that is affine Killing if and only if is called a proper tensor if it is different than the metric tensor of and then is called a proper affine Killing vector field. In general, for an -dimensional manifold (), the existence of a proper has its roots back in 1923, when Eisenhart  proved that a Riemannian admits a proper if and only if is reducible. This means that is locally a product manifold of the form () and there exists a local coordinate system in terms of which the distance element of is given by where , , and . Thus, an irreducible Riemannian manifold admits no proper affine Killing vector field.
Observe that the Killing equation (2.6) implies that the condition holds if is Killing. However, not every affine Killing vector field is Killing. For example, it was shown in  that a non-Einstein conformally flat Riemannian manifold can admit an affine vector field for which is a linear combination of the metric tensor and the Ricci tensor. This result also holds for any nonrecurrent, nonconformally flat and non-Einstein manifold which is conformally recurrent with a locally gradient recurrent vector . Thus, affine vector fields in such spaces are proper since they are neither Killing nor homothetic. Also, see Sections 2.3 and 3.1 for some examples of a proper affine Killing vector.
To find a class of Riemannian manifolds for which an affine Killing symmetry is Killing, Yano proved the following result.
Yano . An affine Killing vector field on a compact orientable Riemannian manifold, without boundary, is Killing.
The proof is easy since affine Killing implies is constant on and, in particular, if is without boundary which implies is Killing.
2.2. Kähler Manifolds
A real Riemannian manifold is called a Hermitian manifold if where is a tensor field of type () of the tangent space , at each point of , is the identity morphism of , and denotes the set of all tangent vectors fields on . The fundamental 2-form of is defined by
is called a Kähler manifold if is closed. A vector field on a Kähler manifold is analytic if . It is easy to show that if is analytic (also called holomorphic) on a Kähler manifold, then so is . Using this, one can easily show that if is a Killing vector field on a compact Kähler manifold, then is an analytic gradient vector.
Yano . (a) In a compact Kähler manifold an analytic divergence-free vector field is Killing. (b) A Killing vector field on a compact Kähler manifold is analytic.
(a) follows from and an integral formula (1.14) in [11, p. 41]. Then, (b) follows easily.
Sharm . An affine Killing vector field in a nonflat complex space form is Killing and analytic.
Sharma first proved that the only symmetric (antisymmetric) second-order parallel tensor in a nonflat space form is the Kählerian metric up to a constant multiple. Then, the proof follows from Yano .
Remark 2.4. Comparing Sharma’s result with the part (b) of Yano’s  result, observe that Sharma assumed affine Killing and proved it to be Killing and analytic in (not necessarily compact), whereas Yano required to be compact (not necessarily of constant holomorphic sectional curvature).
In [17, pages 176–178] the reader can find other types of metric and curvature symmetries of Kähler manifolds, as a consequence of above two results.
2.3. Contact Manifolds
A ()-dimensional differentiable manifold is called a contact manifold if it has a global differential 1-form such that everywhere on . For a given contact form , there exists a unique global vector field , called the characteristic vector field, satisfying A Riemannian metric of is called an associated metric of the contact structure if there exists a tensor field , of type () such that These metrics can be constructed by the polarization of evaluated on a local orthonormal basis of the tangent space with respect to an arbitrary metric, on the -dimensional contact subbundle of . The structure () on is called a contact metric structure and its associated manifold is called a contact metric manifold which is orientable and odd dimensional. The contact metric structure is called a -contact structure if its global characteristic vector field is Killing. has a normal contact structure if where is the Nijenhuis tensor field of . A normal contact metric manifold is called a Sasakian manifold which is also -contact but the converse holds only if . The global characteristic Killing vector field of a -contact manifold has played a key role in the contact geometry. For details, see a complete set of Sasaki’s works cited in Blair .
On the existence of a proper affine Killing vector field in contact geometry, we have the following nontrivial example.
Example 2.5. Let be a contact metric manifold such that Blair  proved that is locally the product of a Euclidean manifold and . Using this result, in 1985, Blair and Patnaik  used a tensor field on a contact metric structure of , where is the self-adjoint trace-free operator and proved that They also proved that, if , there exists an affine connection annihilating such that the curvature property is preserved. Thus, there exists a proper affine Killing vector field of the above described locally product contact metric manifold , defined by .
Other than the above isolated example, the present author is not aware of any more cases of proper affine Killing vector field in contact geometry. On the other hand, in  Sharma has proved the following two results.(i)On a -contact manifold a second-order symmetric parallel tensor is a constant multiple of the associated metric tensor.(ii)An affine Killing vector field on a compact -contact manifold without boundary is Killing. Then, in another paper , Sharma generalized the above first result as follows.Let be a contact metric manifold whose -sectional curvature is nowhere vanishing and is independent of the choice of . Then a second-order parallel tensor on is a constant multiple of the associated metric tensor.
Remark 2.6. In [17, pages 182–185] and [21, 22] the reader can find several results on other types of metric and curvature symmetries of contact manifolds. Now we quote the following two results involving manifolds with boundary.
Theorem 2.7 (Yano and Ako ). A vector field on a compact orientable manifold , with compact orientable boundary , is Killing if and only if(1) on , (2) on ,where is the tensor associated to the Ricci tensor of .
For proof of the above result and some side results on with boundary, see Yano [11, pp. 118–120].
Ünal  has proved a similar result for semi-Riemannian manifolds with boundary and subject to the following geometric condition.
For a semi-Riemannian , the validity of divergence theorem is not obvious due to the possible existence of degenerate metric coefficient for some index . Thus the boundary may become degenerate at some of its points or it may be a lightlike hypersurface of . In both these cases, there is no well-defined outward normal. Ünal  studied this problem as follows.
Let be a semi-Riemannian manifold with boundary (possibly ). Its induced tensor on is also symmetric but not necessary a metric tensor as it may be degenerate at some or all points of . Let , , and be the subsets of points where the nonzero vectors orthogonal to are spacelike, timelike, and lightlike respectively. Thus, where the three subsets are pairwise disjoint. Now we quote the following result.
Theorem 2.8 (Ünal ). Let () be a compact orientable semi-Riemannian manifold with boundary such that its lightlike part has measure zero in . Then, a vector field on is Killing if and only if(1) on , (2) on ,where is the tensor associated to the Ricci tensor of , is the unit normal vector field to induced on , and all eigenvalues of are real.
Since the measure of lightlike vanishes in , the proof of above result is exactly as in the case of Theorem 2.7 of Yano and Ako  and the use of following Gauss theorem which is also valid for any semi-Riemannian manifold.
Theorem 2.9 (Gauss). Let be a compact orientable semi-Riemannian manifold with boundary . For a smooth vector field on , one has where is the volume element on and with respect to a suitable local coordinate system (). Here denotes the operator of inner product.
2.4. Conformal Killing and Affine Conformal Symmetries
Recall from (1.3) that a vector field of a Riemannian or semi-Riemannian manifold is a conformal Killing vector field if for some function of . To the best of our recollection, this conformal Killing equation appeared in a 1903 paper of Fubini  who studied the properties of infinitesimal conformal transformations of a metric space. Since then, the subject matter on conformal Killing vector (CKV) fields is indeed very wide both in geometry and physics. Here we present main results on the existence or nonexistence of CKV fields and one of its closely related symmetry.
We first link Bochner’s Theorem 2.2 for Killing vector field with the following general existence theorem for a conformal Killing vector field.
Theorem 2.10 (Yano ). If the Ricci tensor of a compact orientable Riemannian manifold , without boundary, is nonpositive, then a CKV field has a vanishing covariant derivative (hence Killing). If the Ricci tensor is negative-definite, then there does not exist any CKV field on .
Yano proved above theorem by assuming that is a gradient of a function and used an integral formula [11, page 46] which states if is a CKV, where . See in  for several other results coming from Yano’s above theorem, involving conditions on the curvature.
In 1971, Obata proved the following result on "Conformal transformations."
Theorem 2.11 (Obata ). If the group of conformal transformations of a compact Riemannian manifold is noncompact, then this manifold is conformally diffeomorphic to the standard sphere.
This theorem was extended to the noncompact case by Ferrand  in 1994. The following results are direct consequences of the above theorem of Obata.
Yano and Nagano . A complete connected Einstein manifold (dimension ), admitting a proper CKV field, is isometric to a sphere in an ()-dimensional Euclidean space.
Yano . In order that a compact Riemannian manifold (dimension ), with constant scalar curvature constant and admitting proper CKV field (with conformal function ), to be isometric to a sphere, it is necessary and sufficient that .
Lichnerowicz . Let a compact Riemannian manifold admit a proper CKV field, with conformal function , such that one of the following holds:(1)the 1-form associated with is exact,(2) is an eigenvalue of the Ricci tensor with constant eigenvalues,(3), for some smooth function . Then is isometric to a sphere.
Yano . (1) If is complete, of dimension , with constant , and if it admits a proper CKV field, with conformal function , then and equality holds if and only if is isometric to a sphere.
(2) If a complete Riemannian manifold , of dimension , with scalar curvature admits a proper CKV field that leaves the length of the Ricci tensor invariant, that is, , then is isometric to a sphere.
We refer Yano [11, pp 120–124] for results on CKV fields in with boundary.
In the world of mathematical science and engineering, the Stokes and divergence theorems are like founding pillars for a large variety of practical (small and or big) problems. I believe this was the main motivation that Ünal’s  Theorem 2.8 appeared in 1995 to use those founding theorems in semi-Riemannian geometry. However, unfortunately, the idea of this reference has not yet been picked by the research community to show a similar use of Stokes and divergence theorems (even with essential restrictions) for semi-Riemannian manifolds. There is a need to take a step in this direction.
Since there is no generalization to the Hopf-Rinow theorem for the semi-Riemannian case, related to problems with metric symmetry, it remains an open question to verify the above quoted results when the Riemannian metric is replaced by a metric of arbitrary signature.
On the other hand, in recent years a systematic study of timelike Killing and conformal Killing vector fields on Lorentzian manifolds has been developed by using Bochner’s technique for which we refer the works of Romero and Sánchez  and Romero . In case of conformal Killing vector fields in general semi-Riemannian manifolds, we refer to two papers of Kühnel-Rademacher [33, 34].
3. Metric Symmetries in Spacetimes
Let be an -dimensional timeorientable Lorentzian manifold, called a spacetime manifold. This means that is a smooth connected Hausdorff manifold and is a time orientable Lorentz metric of normal hyperbolic signature . For physical reason, we collect main results on Killing symmetry used in a 4-dimensional spacetime of general relativity. Later on we present some general results for -dimensional () compact time orientable Lorentzian manifolds.
Consider the following form of Einstein field equations: where , , and are the stress-energy tensor, the Ricci tensor, and the scalar curvature respectively. is said to obey the mixed energy condition if at any point on any hypersurface, (i) the strong energy condition holds, that is, , and (ii) equality in (i) implies that all components of are zero. is said to obey the dominant energy condition if in any orthonormal basis the energy dominates the other components of , that is, for each . Since the Einstein field equations are a complicated set of nonlinear differential equations, most explicit solutions (see Kramer et al. ) have been found by using Killing or homothetic symmetries. This is due to the fact that these symmetries leave the Levi-Civita connection, all the curvature quantities, and the field equations invariant.
Considerable work is available to show that not any arbitrary timeorientable Lorentzian manifold may be physically important as compared to the choice of a prescribed model of spacetimes. Related to the metric symmetries, following is a widely used model of spacetimes.
A spacetime () is called globally hyperbolic  if there exists an embedded spacelike 3-manifold such that every endless causal curve intersects once and only once. Such a hypersurface , if it exists, is called a Cauchy surface. If is globally hyperbolic, then (a) is homeomorphic to , where is a hypersurface of , and for each , is a Cauchy surface, (b) if is any compact hypersurface without boundary, of , then must be a Cauchy surface. It is obvious from above that Minkowski spacetime is globally hyperbolic. In the following we present a characterization result of Eardley et al.  on the existence of Killing or homothetic vector field in globally hyperbolic spacetimes.
Theorem 3.1 (see 36). Let be a globally hyperbolic spacetime which(1)satisfies the Einstein equations for a stress energy tensor T obeying the mixed energy and the dominant energy conditions,(2)admits a homothetic vector field of g,(3)admits a compact hypersurface of constant mean curvature,Then, either is an expanding hyperbolic model with metric with a 3-dimensional Riemannian metric of constant negative curvature on a compact manifold and vanishing, or is Killing.
Sketch of Proof. According to a result by Geroch  we know that if a globally hyperbolic spacetime satisfies the vacuum Einstein equations, that is, vanishes, then may be completely determined from a set of Cauchy data specified on or if satisfies the Einstein equations coupled to a well-posed hyperbolic systems of matter equations, then the coupled system has the same property, where is the induced 3-metric of . Using this property, above theorem was proved within the environment of 3-dimensional compact spacelike hypersurface of . By hypothesis, if the mean curvature of is zero, then is Killing and so the theorem is obvious. If , then it can be proved that is totally umbilical in and is of negative constant curvature. Then, it follows from a theorem of Bochner  that the standard hyperbolic metric admits no nonzero global Killing vector field. Finally, it is easy to show that (for ) the vacuum spacetime is an expanding hyperbolic model as presented in the form (3.2), which completes the proof.
As an application of the previous theorem, consider the Einstein-Yang-Mills equations , with the gauge group chosen to be a compact Lie group. The Lie-algebra-valued Yang-Mills field has the components where and are the gauge potential and the spacetime covariant derivative operator with respect to , respectively. The Einstein-Yang-Mills equations are The above equations satisfy mixed and dominant energy conditions. It is easy to show that if the condition (1) of Theorem 3.1 is replaced by ((1) satisfies the Einstein-Yang-Mills equations), then one can show that either is expanding hyperbolic model with metric (3.2) and field everywhere or is Killing.
Another application is of a massless scalar field coupled to gravity for which the Einstein-Klein-Gordon equations  are Einstein equations with In this case, since does not satisfy the mixed energy condition, we quote the following theorem (proof is common with the proof of the previous theorem).
Theorem 3.2 (see ). Let () be a globally hyperbolic spacetime which(1)satisfies the Einstein-Klein-Gordon equations,(2)admits a homothetic vector field of , (3)admits a compact hypersurface of constant mean curvature. Then, either is an expanding hyperbolic model with metric (3.2) and is constant everywhere, or is Killing.
3.1. Affine Killing Vector Fields in Spacetimes
We know from Section 2.1 that a vector field of a semi-Riemannian manifold is an affine Killing vector field if where is a covariant constant second-order symmetric tensor. is proper affine if is other than . Eisenhart’s  Riemannian result (see Section 2.1) was generalized by Patterson , in 1951, showing that a semi-Riemannian admitting a proper is reducible if the matrix of has at least two distinct characteristic roots at any point of . Since then, a general characterization of affine Killing symmetry (known as affine collineation symmetry) remains open. However, for a 4-dimensional spacetime , this problem has been completely resolved (see Hall and da Costa ). Global study requires the spacetime to be simply connected (which means that any closed loop through any point can be shrunk continuously to that point), and for local considerations one may restrict to a simply connected region. We now know from  that if a simply connected spacetime admits a global, nowhere zero, covariant constant proper , then one of the following three possibilities exist.
(a) There exists locally a timelike or spacelike, nowhere zero covariant constant vector field such that , and is locally decomposable into spacetime.
(b) There exists locally a null, nowhere zero, covariant constant vector field such that is as in (a) but , in general, is not reducible.
(c) is locally reducible into a spacetime and no covariant constant vector exists unless it decomposes into spacetime (a special case of (b)). For the latter case, there exist two such proper covariant constant tensors of order 2.
In another paper, Hall et al.  have proved that the existence of a proper affine Killing symmetry eliminates all vacuum spacetimes except the plane waves, all perfect fluids when the pressure density and all nonnull Einstein Maxwell fields except the () locally decomposable case. Hence, affine Killing symmetry has very limited use in finding exact solutions. We end this section with two examples of spacetimes admitting proper affine Killing vector fields.
Example 3.3. Consider the Robertson-Walker metric in spherical coordinates () with where . Let be a timelike vector parallel to the fluid flow vector . Using affine Killing equation , we obtain Since is covariant constant, . Calculating this later equation, we get and . Thus, we obtain for some constants , and . Thus, is a timelike vector field parallel to such that a proper is given by (3.8) and and are related by (3.9).
Example 3.4. The Einstein static universe, which is simply connected and complete manifold , with the metric admits  an 8-dimensional transitive Lie group of affine transformations generated by the global proper affine vector field .
3.2. Spacetimes with Conformal Killing Symmetry
Although the use of CKV is not desirable in finding exact solutions (as CKVs do not leave the Einstein tensor invariant), nevertheless, now we know quite a number of physically important results (including exact solutions) using conformal symmetry. To review the main latest results on conformal symmetry, we choose one of the widely used models of -splitting (Arnowitt et al. ) 4-dimensional spacetime (). This assumes a thin sandwich of evolved from a spacelike hypersurface at a coordinate time to another spacelike hypersurface at coordinate time with metric given by where is the lapse function, is the shift vector, are spatial coordinates, and is the 3-metric on spacelike slice . This is known as ADM model which admits a CKV field. In 1986, Maartens and Maharaj  proved that Robertson-Walker spacetimes (which provide a satisfactory cosmological ADM model) admit a of Killing vectors and a of conformal Killing vector fields. By definition, a group of isometric or conformal motions has Killing or conformal Killing vectors as generators, respectively. We need the following constraint and conformal evolution equations for the ADM model.
Denote arbitrary vector fields of by , and the timelike unit vector field normal to by . Then the Gauss and Weingarten formulas are where is the shape operator of defined by ( is the inner product with respect to the metric of and the spacetime metric ), the Levi-Civita connections of , respectively, and is the second fundamental form. The Gauss and Codazzi equations are where and denote curvature tensors of and , respectively. It is straightforward to show that the following relation holds: where and are the Ricci tensors of and , respectively, and times the mean curvature of . Let the Einstein’s field equations be of the form , where and are the scalar curvature and the energy-momentum tensor, respectively. Following are the constraint equations: where is the scalar curvature of , the norm operator with respect to . Assume that () admits a CKV field V, that is, . Decompose along as , where is the tangential component of . A simple calculation using all the above equations provides the following evolution equation: Here is the Ricci operator of , and ( summed over ). The above Evolution equations were first derived in  through a different approach using Berger’s  condition that sets the evolution vector field equal to .
Theorem 3.5 (Sharma ). Let () be an ADM spacetime solution of Einstein’s field equations admitting a field and evolved by a complete spacelike hypersurface such that (a) is totally umbilical in , (b) the normal component of is nonconstant on , and (c) the normal sectional curvature of is independent of the tangential direction at each point of . Then is conformally diffeomorphic to (i) a 3-sphere , (ii) Euclidean space , (iii) hyperbolic space , or (iv) the product of a complete 2-dimensional manifold and an open real interval. If is compact, then only (i) holds.
Sharma’s proof uses the above constraint and evolution equations with the condition that the normal sectional curvature of at a point with respect to a plane section spanned by a unit tangent vector of and the unit normal is independent of the choice of . Note that the normal sectional curvature is defined as (see [3, page 33]). This normal sectional curvature holds when is Minkowski, de Sitter, anti-de Sitter, and Robertson-Walker spacetime.
Example 3.6. Consider the following generalized Robertson-Walker (GRW) spacetime as the warped product defined by where is the time line, is an arbitrary -Riemannian manifold, and is a warping function (see Alías et al. ). They have shown that the normal curvature condition holds for this GRW-spacetime and each slice constant is homothetic to the fiber and totally umbilical in . As a consequence of the Theorem 3.5, the following two results are easy to prove.
Duggal-Sharma . (1) Let be an ADM spacetime evolved out of a complete initial hypersurface that is totally umbilical and has nonzero constant mean curvature. If admits a closed CKV field nonvanishing on , then either is orthogonal to and the lapse function is constant over , or is conformally diffeomorphic to , , , or the product of an open interval and a 2-dimensional Riemannian manifold.
(2) Let a conformally flat perfect fluid solution of the Einstein’s equations be evolved out of an initial spacelike hypersurface that is compact, has constant mean curvature, and is orthogonal to the 4-velocity. If has a nonvanishing non-Killing CKV field which is nowhere tangential to , then is totally umbilical in and is of constant curvature. In the case when is of constant negative curvature, is orthogonal to .
3.3. Spacetimes with Affine Conformal Symmetry
An affine conformal symmetry is defined by a vector field of satisfying where is a second-order symmetric tensor and is called an affine conformal Killing vector , denoted by ACV, which is CKV when vanishes. If is constant, then is affine. Moreover, is an ACV if and only if which is also known as “conformal collineation symmetry” generated by an ACV field . Here are the Christoffel symbols. We state the main results on ACV (proved by Tashiro ) on the local reducibility of a Riemannian manifold . By local reducibility we mean that is locally a product manifold.(1)If has constant scalar curvature and has a flat part, then an ACV on is the sum of an affine and a CKV.(2)If has at least three parts and no part is locally flat, then an ACV on is affine. If is also complete, then the ACV is Killing.(3)Let have constant scalar curvature with no flat part. If is irreducible or is the product of two irreducible parts whose scalar curvatures are signed opposite to each other, then an ACV on is a CKV. Otherwise, it is affine.(4)A globally defined ACV on a Euclidean space is necessarily affine.(5)A Riemannian manifold of constant curvature does not admit an ACV.(6)An irreducible admits no ACV which is not a CKV.(7)If a locally reducible has at least three parts, one of which is flat, then an ACV on is sum of an affine vector and a CKV. If is also complete, then the ACV is affine.
Remark 3.7. For a semi-Riemannian manifold, a general characterization of an ACV still remains open, although limited results are available in [49, 51]. As an attempt to verify some or all results listed above, Mason and Maartens  constructed the following example which supports first part of the result (7).
Example 3.8. Let be a Einstein static fluid spacetime with metric and the velocity vector (). This spacetime admits a CKV and a proper affine vector . Since the metric is reducible, it can be easily verified that a combination is a proper ACV such that Now let be a compact orientable semi-Riemannian manifold with boundary . The divergence theorem is not valid due to the possible degenerate part of . For this reason we call a regular  semi-Riemannian manifold if we exclude the possible degenerate part in . Then, following is a characterization theorem for the existence of a proper ACV.
Theorem 3.9 (Duggal ). A vector field in a compact orientable regular semi-Riemannian manifold (), with boundary , is a proper ACV if and only if(a), (b), ,where , , and are the de-Rham Laplacian, affine conformal function, covariant constant tensor of type (), and its associated () tensor, respectively.
The reader will find several other side results in [17, Chapter 7] on the geometry and physics of affine conformal symmetry.
4. Compact Time Orientable Lorentzian Manifolds
Recall that the famous Hopf-Rinow theorem maintains the equivalence of metric and geodesic completeness and, therefore, guarantees the completeness of all Riemannian metrics, for a compact smooth manifold, with the existence of minimal geodesics. Also, if this theorem holds, then the Riemannian function is finite valued and continuous. Unfortunately, for an indefinite metric, completeness is a more subtle notion than in the Riemannian case, since there is no satisfactory generalization to the Hopf-Rinow theorem for a semi-Riemannian manifold. There are some isolated cases satisfying metric and/geodesic completeness. For example, in 1973, Marsden  proved that “every compact homogeneous semi-Riemannian manifold is geodesically complete.” For the case of Lorentzian manifolds, the singularity theorems (see Hawking and Ellis ) confirm that not all Lorentz manifolds are metric and/geodesic complete. Also, the Lorentz distance function fails to be finite and/or continuous for all arbitrary spacetimes . It has been shown in Beem and Ehrlich’s book  that the globally hyperbolic spacetimes turn out to be the most closely related physical spaces sharing some properties of Hopf-Rinow theorem. Now we know that timelike Cauchy completeness and finite compactness are equivalent and the Lorentz distance function is finite and continuous for this class of spacetimes.
We have seen in previous sections that metric symmetries have a key role in 4-dimensional paracompact globally hyperbolic spacetimes. In this section we let be an -dimensional () compact time orientable Lorentzian manifold. Recall that a compact manifold admits a Lorentzian metric if and only if the Euler number of vanishes. Considerable work has been done on the applications of null geodesics of compact using a conformal Killing symmetry. Since, for Lorentzian metrics, the compactness does not imply geodesic completeness, Romero and Sánchez  have proved that a compact Lorentzian manifold which admits a timelike CKV field yields to its geodesic completeness.
Let be a curve in a Lorentzian manifold , where is a suitable parameter. A vector field on is called a Jacobi vector field if it satisfies the following Jacobi differential equation: where is a metric connection on .
Definition 4.1. We say that a point on a geodesic of is conjugate to a point along if there is a Jacobi field along , not identically zero, which vanishes at and .
From a geometric point of view, a conjugate point of along a geodesic can be interpreted as an “almost-meeting point’’ of a geodesic starting from with initial velocity . In general relativity, since the relative position of neighboring events of a free falling particle is given by the Jacobi field of , the attraction of gravity causes conjugate points, while the nonattraction of gravity will prevent them. Although a physical spacetime is generally assumed to be causal (free of closed causal curves), all compact Lorentzian manifolds are acausal, that is, they admit closed timelike curves. See [3, chapters 10 and 11, Second Edition] in which they have done extensive work on conjugate points along null geodesics of a general Lorentzian manifold which may be causal or acausal. We need the following notion of null sectional curvature .
Let and be a null vector of . A plane of is called a null plane directed by if it contains , for any and there exists such that . Then, the null sectional curvature of , with respect to and , is defined as a real number where is any vector in independent with (and therfore spacelike). It is easy to see that is independent of but depends in a quadratic fashion on . The null congruence associated with a vector field is defined by where is the natural projection. is an oriented embedded submanifold of with dimension and is a fiber bundle with fiber type . Therefore, for a compact , will be compact. If a null congruence is fixed with respect to a timelike vector field , then one can choose, for every null plane , the unique null vector , so that the null sectional curvature can be thought as a function on null planes. This function is called the -normalized null sectional curvature.
Let be an -dimensional () compact Lorentzian manifold that admits a timelike CKV field . If there exists a real number such that every null geodesic , with , has no conjugate points of in , then Equality holds if and only if has -normalized null sectional curvature . Here is the restriction to of the metric on the . denotes the quadratic form associated with the Ricci tensor of and is the canonical measure associated with .
The authors used the previous result in proving several inequalities relating conjugate points along geodesics to global geometric properties . Also, they have shown some classification results on certain compact Lorentzian manifolds without conjugate points along its null geodesics.
Let be a compact Lorentzian manifold admitting a timelike CKV field . If has no conjugate points along its null geodesic, then where so that with . Moreover, equality holds if and only if has constant sectional curvature . If is a timelike Killing vector field, then and equality holds if and only if is isomorphic to a flat Lorentzian -torus up to a (finite) covering. In particular, is parallel, the first Betti number of is nonzero, and the Levi-Civita connection of is Riemannian.
Remark 4.2. Recall the following classical Hopf theorem .
“A Riemannian torus with no conjugate points must be flat.”
As a Lorentzian analogue to Hopf theorem, Palomo and Romero  have recently proved the following result.
“A conformally stationary Lorentzian tori with conjugate points must be flat.”
On the other hand, in another paper Palomo and Romero  have obtained a sequence of integral inequalities for any ()-dimensional compact conformally stationary Lorentzian manifold with no conjugate points along its causal geodesics. The equality for some of them implies that the Lorentzian manifold must be flat.
5. Metric Symmetries in Lightlike Geometry
Let be an -dimensional smooth manifold with a symmetric tensor field . Assume that is degenerate on , that is, there exists a vector field , of , such that . The radical distribution of , with respect to , is defined by such that , where is a nondegenerate complementary screen distribution of in . Suppose . Then, . As in case of semi-Riemannian manifolds, a vector field on a lightlike manifold is said to be a Killing vector field if . A distribution on is called a Killing distribution if each vector field belonging to is a Killing vector field. Due to degenerate on , in general, there does not exist a unique metric (Levi-Civita) connection for which is undesirable. Killing symmetry has the following important role in removing this anomaly.
Theorem 5.1 (see [59, page 49]). There exists a unique Levi-Civita connection on a lightlike manifold with respect to if and only if is Killing.
The above result also holds if is a lightlike submanifold of a semi-Riemannian manifold for which (see [59, page 169]).
Physically useful are the lightlike hypersurfaces of spacetime manifolds which (under some conditions) are models as black hole horizons (see Carter , Galloway , and other, cited therein). To illustrate this use, let be a lightlike hypersurface of a spacetime manifold . We adopt following features of the intrinsic geometry of lightlike hypersurfaces. Assume that the null normal is not entirely in , but is defined in some open subset of around . This well-defines the spacetime covariant derivative , which, in general, is not possible if is restricted to as is the case of extrinsic geometry, where is the Levi-Civita connection on . Following Carter , a simple way is to consider a foliation of (in the vicinity of ) by a family so that is in the part of foliated by this family such that, at each point in this region, is a null normal to for some value of . Although the family is not unique, for our purpose we can manage (with some reasonable condition(s)) to involve only those quantities which are independent of the choice of the foliation once evaluated at, say, constant. For simplicity, we denote by constant. Then the metric is simply the pull-back of the metric , where an under arrow denotes the pullback to . The “bending” of in is described by the Weingarten map: that is, associates each of the variation of along , with respect to the spacetime connection . The second fundamental form, say , of is the symmetric bilinear form and is related with the Weingarten map by Using and symmetric in (5.3), we obtain which is well defined up to conformal rescaling (related to the choice of ). for any null normal and for any implies that has the same degeneracy as that of the induced metric .
Consider a class of lightlike hypersurfaces such that its second fundamental form is conformally equivalent to its degenerate metric . Geometrically, this means that is totally umbilical in if and only if there is a smooth function on such that It is obvious that above definition does not depend on particular choice of . The name “umbilical” means that extrinsic curvature is proportional to the metric . is proper totally umbilical in if and only if is nonzero on . In particular, is totally geodesic if and only if vanishes, that is, if and only if vanishes on . It follows from (5.4) and (5.5) that Thus, is a conformal Killing vector (CKV) field in a totally umbilical , with conformal function , which is Killing if and only if is totally geodesic.
Now we need the following general result on totally umbilical submanifolds.
Proposition 5.2 (Perlick ). Let be a totally umbilical submanifold of a semi-Riemannian manifold . Then, (1)a null geodesic of that starts tangential to remains within (for some parameter interval around the starting point), (2) is totally geodesic if and only if every geodesic of that starts tangential to remains with in (for some parameter interval around the starting point).
Considerable work has been done to show that (under certain conditions) totally geodesic lightlike hypersurfaces are black hole event (e.g., the Kerr family) or isolated horizons (see details with examples in , which include Killing horizons  as a special case). A Killing horizon is defined as the union , where is a connected component of the set of points forming a family of lightlike hypersurfaces whose null geodesic (as per above proposition) generators coincide with the Killing trajectories of nowhere vanishing . The isolated horizon (IH) of a stationary asymptotically flat black hole is represented by the Killing horizon if is analytic and the mixed energy condition holds for the stress-energy tensor of the Einstein field equations (see Section 3). For example, the following physical model of a spacetime can have a Killing horizon.
Consider a 4-dimensional stationary spacetime ) which is chronological, that is, admits no closed timelike curves. It is well known  that a stationary admits a smooth 1-parameter group, say , of isometries whose orbits are timelike curves in . Denote by the Hausdorff and paracompact 3-dimensional Riemannian orbit space of the action . The projection is a principal -bundle, with the timelike fiber . Let be the nonvanishing timelike Killing vector field, where is a global time coordinate function on . Then, the metric induces a Riemannian metric on such that where is a connection 1-form for the -bundle and It is known that a stationary spacetime () uniquely determines the orbit data () as described previously, and conversely. Suppose the orbit space has a nonempty metric boundary . Consider the maximal solution data in the sense that it is not extendible to a larger domain ( with on an extended spacetime . Under these conditions, it is known  that in any neighborhood of a point , either the connection 1-form degenerates, or . The second case implies that the timelike Killing vector becomes null and degenerates into a lightlike hypersurface, say of . Moreover, is a global null Killing vector field of .
In the following we quote a result on physical interpretation of an ADM spacetime (see Section 3.2) which can admit a Killing horizon.
Theorem 5.3 (see ). Let () be an ADM spacetime evolved through a 1-parameter family of spacelike hypersurfaces such that the evolution vector field is a null CKV field on . Then, reduces to a Killing vector field if and only if the part of tangential to is asymptotic everywhere on for all . Moreover, is a geodesic vector field.
There has been extensive study on black hole time-independent Killing horizons for those spacetimes which admit a global Killing vector field. However, in reality, since the black holes are surrounded by a local mass distribution and expand by the inflow of galactic derbies as well as electromagnetic and gravitational radiation, their physical properties can best be represented by time-dependent black hole horizons. Thus, a Killing horizon (and for the same reason an isolated horizon) is not a realistic model. Since the causal structure is invariant under a conformal transformation, there has been interest in the study of the effect of conformal transformations on properties of black holes (see [68–72]). Directly related to the subject matter of this paper, we review the following work of Sultana and Dyer [70, 71].
Consider a spacetime which admits a timelike conformal Killing vector (CKV) field. Let be a lightlike hypersurface of such that its null geodesic trajectories coincide with conformal Killing trajectories of a null CKV field (instead of Killing trajectories of the Killing horizon). This happens when a spacetime becomes null on a boundary as a null geodesic hypersurface. Such a horizon is called conformal Killing horizon (CKH), as defined by Sultana and Dyer [70, 71]. Consider a spacetime related to a black hole spacetime admitting a Killing horizon generated by the null geodesic Killing field, with the conformal factor in , where is a nonvanishing function on . Under this transformation, the Killing vector field is mapped to a conformal Killing field provided . Since the causal structure and null geodesics are invariant under a conformal transformation, still remains a null hypersurface of . Moreover, as per Proposition 5.2, the null geodesic of that starts tangential to will remain within . Also, its null geodesic generators coincide with the conformal Killing trajectories. Thus, is a CKH in .
Theorem 5.4 (Sultana and Dyer ). Let be a spacetime related to an analytic black hole spacetime admitting a Killing horizon , such that the conformal factor in goes to a constant at null infinity. Then the conformal Killing horizon in is globally equivalent to the event horizon, provided that the stress energy tensor satisfies the week energy condition.
The above paper also contains the case as to what happens when the conformal stationary limit hypersurface does not coincide with the CKH. For this case, they have proved a generalization of the weak rigidity theorem which establishes the conformal Killing property of the event horizon and the rigidity of its CKH.
Also, in  they have given an example of a dynamical cosmological black hole spacetime which describes an expanding black hole in the asymptotic background of the Einstein-de Sitter universe. The metric of such a spacetime is obtained by applying a time-dependent conformal transformation on the Schwarzschild metric, such that the result is an exact solution with the matter content described by a perfect fluid and the other a null fluid. They have also studied several physical quantities related to black holes.
- M. Berger, Riemannian Geometry During the Second Half of the Twentieth Century, vol. 17 of Lecture Series, American Mathematical Society, Providence, RI, USA, 2000.
- B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, NY, USA, 1983.
- J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry, vol. 67, Marcel Dekker, New York, NY, USA, 1981, 2nd edition (with Easley, K. L.), 1996.
- D. N. Kupeli, Singular Semi-Riemannian Geometry, vol. 366, Kluwer Academic, Dodrecht, The Netherlands, 1996.
- A. Bejancu and K. L. Duggal, “Degenerated hypersurfaces of semi-Riemannian manifolds,” Buletinul Institutului Politehnic din Iaşi, vol. 37, no. 1–4, pp. 13–22, 1991.
- S. B. Myers and N. E. Steenrod, “The group of isometries of a Riemannian manifold,” Annals of Mathematics, vol. 40, no. 2, pp. 400–416, 1939.
- W. Killing, “Die Zusammensetzung der stetigen endlichen Transformations-gruppen,” Mathematische Annalen, vol. 31, no. 2, pp. 252–290, 1888.
- S. Bochner, “Curvature and Betti numbers,” Annals of Mathematics, vol. 49, pp. 379–390, 1948.
- Y. Watanabe, “Integral inequalities in compact orientable manifolds, Riemannian or Kählerian,” Kōdai Mathematical Seminar Reports, vol. 20, pp. 264–271, 1968.
- K. Yano, “On harmonic and Killing vector fields,” Annals of Mathematics, vol. 55, pp. 38–45, 1952.
- K. Yano, Integral Formulas in Riemannian Geometry, Marcel Dekker, New York, NY, USA, 1970.
- M. Berger, “Trois remarques sur les variétés riemanniennes à courbure positive,” Comptes Rendus de l'Académie des Sciences, vol. 263, pp. A76–A78, 1966.
- L. P. Eisenhart, “Symmetric tensors of the second order whose first covariant derivatives are zero,” Transactions of the American Mathematical Society, vol. 25, no. 2, pp. 297–306, 1923.
- J. Levine and G. H. Katzin, “Conformally flat spaces admitting special quadratic first integrals. I. Symmetric spaces,” Tensor, vol. 19, pp. 317–328, 1968.
- K. Yano, Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, New York, NY, USA, 1965.
- R. Sharma, “Second order parallel tensor in real and complex space forms,” International Journal of Mathematics and Mathematical Sciences, vol. 12, no. 4, pp. 787–790, 1989.
- K. L. Duggal and R. Sharma, Symmetries of Spacetimes and Riemannian Manifolds, vol. 487, Kluwer Academic, Dodrecht, The Netherlands, 1999.
- D. E. Blair, Contact Manifolds in Riemannian Geometry, vol. 509 of Lecture notes in Math, Springer, Berlin, Germany, 1976.
- D. E. Blair, “Two remarks on contact metric structures,” The Tohoku Mathematical Journal, vol. 29, no. 3, pp. 319–324, 1977.
- D. E. Blair and J. N. Patnaik, “Contact manifolds with characteristic vector field annihilated by the curvature,” Bulletin of the Institute of Mathematics, vol. 9, no. 4, pp. 533–545, 1981.
- R. Sharma, “Second order parallel tensors on contact manifolds,” Algebras, Groups and Geometries, vol. 7, no. 2, pp. 145–152, 1990.
- R. Sharma, “Second order parallel tensors on contact manifolds. II,” Comptes Rendus Mathématiques, vol. 13, no. 6, pp. 259–264, 1991.
- K. Yano and M. Ako, “Vector fields in Riemannian and Hermitian manifolds with boundary,” Kōdai Mathematical Seminar Reports, vol. 17, pp. 129–157, 1965.
- B. Ünal, “Divergence theorems in semi-Riemannian geometry,” Acta Applicandae Mathematicae, vol. 40, no. 2, pp. 173–178, 1995.
- G. Fubini, “Sulla teori degli spazii che ammettono un gruppo cinforme, Atti, Torino,” vol. 38, pp. 404–418, 1903.
- M. Obata, “The conjectures on conformal transformations of Riemannian manifolds,” Journal of Differential Geometry, vol. 6, pp. 247–258, 1971.
- J. Ferrand, “The action of conformal transformations on a Riemannian manifold,” Mathematische Annalen, vol. 304, no. 2, pp. 277–291, 1996.
- K. Yano and T. Nagano, “Einstein spaces admitting a one-parameter group of conformal transformations,” Annals of Mathematics, vol. 69, pp. 451–461, 1959.
- K. Yano, “Riemannian manifolds admitting a conformal transformation group,” Proceedings of the National Academy of Sciences of the United States of America, vol. 62, pp. 314–319, 1969.
- A. Lichnerowicz, “Sur les transformations conformes d'une variété riemannienne compacte,” Comptes Rendus de l'Académie des Sciences, vol. 259, pp. 697–700, 1964.
- A. Romero and M. Sánchez, “Completeness of compact Lorentz manifolds admitting a timelike conformal Killing vector field,” Proceedings of the American Mathematical Society, vol. 123, no. 9, pp. 2831–2833, 1995.
- M. Romero, “The introduction of Bochner's technique on Lorentzian manifolds,” Nonlinear Analysis, vol. 47, no. 5, pp. 3047–3059, 2001.
- W. Kühnel and H.-B. Rademacher, “Essential conformal fields in pseudo-Riemannian geometry,” Journal de Mathématiques Pures et Appliquées, vol. 74, no. 5, pp. 453–481, 1995.
- W. Kühnel and H.-B. Rademacher, “Conformal vector fields on pseudo-Riemannian spaces,” Differential Geometry and Its Applications, vol. 7, no. 3, pp. 237–250, 1997.
- D. Kramer, H. Stephani, M. MacCallum, and E. Herlt, Exact Solutions of Einstein's Field Equations, Cambridge University Press, Cambridge, UK, 1980.
- D. Eardley, J. Isenberg, J. Marsden, and V. Moncrief, “Homothetic and conformal symmetries of solutions of Einstein's equations,” Communications in Mathematical Physics, vol. 106, no. 1, pp. 137–158, 1986.
- R. Geroch, “Domain of dependence,” Journal of Mathematical Physics, vol. 11, pp. 437–449, 1970.
- C. N. Yang and R. D. Mills, “Isotropic spin and isotropic gauge invariance,” Physical Review, vol. 96, pp. 191–195, 1954.
- S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge, UK, 1973.
- E. M. Patterson, “On symmetric recurrent tensors of the second order,” The Quarterly Journal of Mathematics, vol. 2, pp. 151–158, 1951.
- G. S. Hall and J. da Costa, “Affine collineations in space-time,” Journal of Mathematical Physics, vol. 29, no. 11, pp. 2465–2472, 1988.
- G. S. Hall, A. D. Hossack, and J. R. Pulham, “Sectional curvature, symmetries, and conformally flat plane waves,” Journal of Mathematical Physics, vol. 33, no. 4, pp. 1408–1414, 1992.
- R. Arnowitt, S. Deser, and C. W. Misner, “The dynamics of general relativity,” in Gravitation: An Introduction to Current Research, L. Witten, Ed., pp. 227–265, John Wiley & Sons, New York, NY, USA, 1962.
- R. Maartens and S. D. Maharaj, “Conformal killing vectors in Robertson-Walker spacetimes,” Classical and Quantum Gravity, vol. 3, no. 5, pp. 1005–1011, 1986.
- B. K. Berger, “Homothetic and conformal motions in spacelike slices of solutions of Einstein's equations,” Journal of Mathematical Physics, vol. 17, no. 7, pp. 1268–1273, 1976.
- R. Sharma, “Conformal symmetries of Einstein's field equations and inital data,” Journal of Mathematical Physics, vol. 46, no. 4, pp. 1–8, 2005.
- L. J. Alías, A. Romero, and M. Sánchez, “Spacelike hypersurfaces of constant mean curvature in certain spacetimes,” Nonlinear Analysis, vol. 30, no. 1, pp. 655–661, 1997.
- K. L. Duggal and R. Sharma, “Conformal killing vector fields on spacetime solutions of Einstein's equations and initial data,” Nonlinear Analysis, vol. 63, no. 5 –7, pp. 447–454, 2005.
- K. L. Duggal, “Affine conformal vector fields in semi-Riemannian manifolds,” Acta Applicandae Mathematicae, vol. 23, no. 3, pp. 275–294, 1991.
- Y. Tashiro, “On conformal collineations,” Mathematical Journal of Okayama University, vol. 10, pp. 75–85, 1960.
- D. P. Mason and R. Maartens, “Kinematics and dynamics of conformal collineations in relativity,” Journal of Mathematical Physics, vol. 28, no. 9, pp. 2182–2186, 1987.
- J. Marsden, “On completeness of homogeneous pseudo-riemannian manifolds,” Indiana University Mathematics, vol. 22, pp. 1065–1066, 1973.
- M. Gutiérrez, F. J. Palomo, and A. Romero, “A Berger-Green type inequality for compact Lorentzian manifolds,” Transactions of the American Mathematical Society, vol. 354, no. 11, pp. 4505–4523, 2002.
- M. Gutiérrez, F. J. Palomo, and A. Romero, “Conjugate points along null geodesics on Lorentzian manifolds with symmetry,” in Proceedings of the Workshop on Geometry and Physics, pp. 169–182, Madrid, Spain, 2001.
- M. Gutiérrez, F. J. Palomo, and A. Romero, “Lorentzian manifolds with no null conjugate points,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 137, no. 2, pp. 363–375, 2004.
- E. Hopf, “Closed surfaces without conjugate points,” Proceedings of the National Academy of Sciences of the United States of America, vol. 34, pp. 47–51, 1948.
- F. J. Palomo and A. Romero, “Conformally stationary Lorentzian tori with no conjugate points are flat,” Proceedings of the American Mathematical Society, vol. 137, no. 7, pp. 2403–2406, 2009.
- F. J. Palomo and A. Romero, “Compact conformally stationary Lorentzian manifolds with no causal conjugate points,” Annals of Global Analysis and Geometry, vol. 38, no. 2, pp. 135–144, 2010.
- K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364, Kluwer Academic, Dodrecht, The Netherlands, 1996.
- K. L. Duggal and D. H. Jin, Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific Publishing, River Edge, NJ, USA, 2007.
- K. L. Duggal and B. Sahin, Differential Geometry of Lightlike Submanifolds, Frontiers in Mathematics, Birkhäuser, Basel, Switzerland, 2010.
- B. Carter, “Killing horizons and orthogonally transitive groups in space-time,” Journal of Mathematical Physics, vol. 10, pp. 70–81, 1969.
- G. J. Galloway, “Maximum principles for null hypersurfaces and null splitting theorems,” Annales Henri Poincaré, vol. 1, no. 3, pp. 543–567, 2000.
- B. Carter, “Extended tensorial curvature analysis for embeddings and foliations,” in Geometry and Nature, vol. 203 of Contemporary Mathematics, pp. 207–219, American Mathematical Society, Providence, RI, USA, 1997.
- V. Perlick, “On totally umbilical submanifolds of semi-Riemannian manifolds,” Nonlinear Analysis, vol. 63, pp. 511–518, 2005.
- E. Gourgoulhon and J. L. Jaramillo, “A (1 + 3)-perspective on null hypersurfaces and isolated horizons,” Physics Reports, vol. 423, no. 4-5, pp. 159–294, 2006.
- K. L. Duggal and R. Sharma, “Conformal evolution of spacetime solutions of Einstein's equations,” Communications in Applied Analysis, vol. 11, no. 1, pp. 15–22, 2007.
- S. Carlip, “Symmetries, horizons, and black hole entropy,” General Relativity and Gravitation, vol. 39, no. 10, pp. 1519–1523, 2007.
- T. Jacobson and G. Kang, “Conformal invariance of black hole temperature,” Classical and Quantum Gravity, vol. 10, no. 11, pp. L201–L206, 1993.
- J. Sultana and C. C. Dyer, “Conformal Killing horizons,” Journal of Mathematical Physics, vol. 45, no. 12, pp. 4764–4776, 2004.
- J. Sultana and C. C. Dyer, “Cosmological black holes: a black hole in the Einstein-de Sitter universe,” General Relativity and Gravitation, vol. 37, no. 8, pp. 1349–1370, 2005.
- K. L. Duggal, “Time-dependent black hole horizons on spacetime solutions of Einstein's equations with initial data,” in Advances in Lorentzian Geometry, M. Plaue and M. Scherfner, Eds., pp. 51–61, Aachen: Shaker, Berlin, Germany, 2008.