# A Review on Metric Symmetries Used in Geometry and Physics

Authors: K. L. Duggal
Journal: ISRN Geometry
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

The measurement of distances in a Euclidean spaceR3 is represented by the distance element(1.1)ds2=dx2+dy2+dz2
with respect to a rectangular coordinate system (x,y,z). Back in 1854, Riemann generalized this idea for n-dimensional spaces and he defined element of length by means of a quadratic differential form ds2=gijdxidxj on a differentiable manifold M, where the coefficients gij are functions of the coordinates system (x1,…,xn), which represent a symmetric tensor field g of type (0,2). Since then much of the subsequent differential geometry was developed on a real smooth manifold (M,g), called a Riemannian manifold, where g is a positive definite metric tensor field. Berger’s book [1] 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 [2] on the study of semi-Riemannian geometry where the metric g is indefinite and, in particular, Beem and Ehrlich [3] on the global Lorentzian geometry used in relativity. In general, an inner product g on a real vector space V is of type (r,ℓ,m) where r=dim{u∈V∣g(u,v)=0forallv∈V}, ℓ=sup{dimW∣W⊂V with g(w,w)<0forallnonzero w∈W} and m=sup{dimW∣W⊂V with g(w,w)>0forall nonzero w∈W}. A metric g on a manifold M is a symmetric (0,2) tensor field on M of the type (r,ℓ,m) on its tangent bundle space TM. Kupeli [4] called a manifold (M,g) of this type a singular semi-Riemannian manifold if M admits a Koszul derivative, that is, g is Lie parallel along the degenerate vector fields on M. Based on this, Kupeli studied the intrinsic geometry of such degenerate manifolds. On the other hand, a degenerate submanifold (M,g) of a semi-Riemannian manifold (M̅,g̅) may not be studied intrinsically since due to the induced degenerate tensor field g on M one cannot use, in general, the geometry of M̅. To overcome this difficulty, Kupeli used the quotient space TM*=TM/Rad(TM) and the canonical projection P:TM→TM* for the study of intrinsic geometry of M. Here Rad(TM) denotes the radical distribution of M.In 1991, Bejancu and Duggal [5] 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(1.2)TM=Rad(TM)⊕orthS(TM),
where S(TM) is a nondegenerate complementary screen distribution to Rad(TM) and ⊕orth is a symbol for orthogonal direct sum. S(TM) is not unique; however, it is canonically isomorphic to the quotient bundle TM*=TM/Rad(TM).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 tensorg with a symmetry as follows: consider (M,g,V) with the metric g of any one of the three types and V a vector field (local or global) of M such that(1.3)£Vg=2σg,
where £V is the Lie derivative operator and σ is a function on M. Above equation is known as conformal Killing equation and the symmetry vector V is called a conformal Killing vector, briefly denoted by CKV. If σ is nonconstant, then V is called a proper CKV. In particular, V is homothetic or Killing according as σ is a nonzero constant or zero. The set of all proper CKV fields and all Killing vector fields on M 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(M,g) 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 manifoldM, the group of all smooth transformations of M is a very large group. This leads to the study of those transformations of M which leave a certain physical/geometric quantity invariant. Related to the focus of this paper we let (M,g) be a real n-dimensional smooth Riemannian or semi-Riemannian manifold. A diffeomorphism ϕ:M→M is called an isometry of M if it leaves invariant the metric tensor g. This means that
(2.1)g(ϕ⋆X,ϕ⋆Y)=g(X,Y),∀X,Y∈χ(M),
where ϕ⋆ is the differential (tangent) map of ϕ and χ(M) denotes the set of all tangent vector fields on M. Since each tangent mapping (ϕ⋆)p, at p∈M, is a linear isomorphism of Tp(M) on Tϕ(p)(M), it follows that ϕ is an isometry if and only if (ϕ⋆)p is a linear isometry for any p∈M. The set of all isometries of M forms a group under composition of mappings. Myers and Steenrod [6] 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.LetV be a smooth vector field on M and 𝒰 a neighborhood of each p∈M with coordinate system (xi). Let the integral curves of V, through any point q in 𝒰, be defined on an open interval (-ϵ,ϵ) for ϵ>0. For each t∈(-ϵ,ϵ) define an isometric map ϕt on 𝒰 such that for q in 𝒰, ϕt(q) is on the integral curve of V through q. Then, V generates a local 1-parameter group of infinitesimal transformations ϕt(xi)=xi+tVi and we have(2.2)∂k(xi+tVi)∂m(xj+tVj)gij(x+tV)=gkm,
which, after expanding gij(x+tV) up to first-order in t, yields to(2.3)Vi∂igjk+∂j(Vi)gik+∂k(Vi)gji=0.
Using the Lie derivative operator £V, the above equation can be rewritten as(2.4)£Vgij=∇iVj+∇jVi=0,
where Vi=gijVj is the associated 1-form of V and ∇ is the Levi-Civita connection on M. The above Killing equations were named after a German mathematician Killing [7] 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 n-dimensional Euclidean space, there exist n(n+1)/2 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(M,g) be a compact orientable Riemannian manifold with boundary ∂M. For a smooth vector field V on M, one has
(2.5)∫Mdiv⁡Vdv=∫∂Mg(N,V)dS,
where N and dS are the unit normal to ∂M and its surface element and dv is the volume element of M.Consider ann-dimensional Riemannian manifold (M,g) without boundary, that is, ∫MdivV=0 holds for a smooth vector field V on M. Let V be a Killing vector field of (M,g), that is,(2.6)£Vg=0or£Vgij=∇jVi+∇iVj=0,(1≤i,j≥n),
where ∇ denotes a symmetric affine connection on M. We start with the following fundamental theorem on the existence of a Killing vector field.Theorem 2.2 (Bochner [8]).
If Ricci tensor of a compact orientable Riemannian manifold(M,g), without boundary, is negative semidefinite, then a Killing vector field V on M is covariant constant. On the other hand, if the Ricci tensor on M is negative definite, then a Killing vector field other than zero does not exist on M.Bochner proved this theorem by assuming thatV is a gradient of a function and a result of Watanabe [9] which states “ ∫M[Ric(V,V)-|∇V|2]=0ifVis 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 [12] 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 S3 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 (M,g), defined as follows.
A vector fieldV on (M,g) is called affine Killing if £V∇=0. To interpret this relation with respect to the metric g we split the tensor ∇iVj (see Killing equation (1.3)) into its symmetric and antisymmetric parts as follows:(2.7)∇iVj=Kij+Fij,(Kij=Kji,Fij=-Fji).
Then, it follows that Kij is covariant constant, that is,
(2.8)∇kKij=0.
From (2.7) and (2.8) we deduce that V is affine Killing if and only if
(2.9)£Vgij=2Kij.Kij is called a proper tensor if it is different than the metric tensor gij of M and then V is called a proper affine Killing vector field. In general, for an n-dimensional manifold (M,g), the existence of a proper Kij has its roots back in 1923, when Eisenhart [13] proved that a Riemannian M admits a proper Kij if and only if M is reducible. This means that M is locally a product manifold of the form (M=M1×M2,g=g1⊕g2) and there exists a local coordinate system in terms of which the distance element of g is given by
(2.10)ds2=gab(xc)dxadxb+gAB(xC)dxAdxB,
where a,b,c=1,…,r, A,B,C=r+1,…,n, and 1≤r≤n. Thus, an irreducible Riemannian manifold admits no proper affine Killing vector field.
Observe that the Killing equation (2.6) implies that the condition £V∇=0 holds if V is Killing. However, not every affine Killing vector field is Killing. For example, it was shown in [14] that a non-Einstein conformally flat Riemannian manifold can admit an affine vector field for which Kij 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 [14]. 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 [11]. An affine Killing vector field on a compact orientable Riemannian manifold, without boundary, is Killing.The proof is easy sinceV affine Killing implies divV is constant on M and, in particular, divV=0 if M is without boundary which implies V is Killing.


### 2.2. Kähler Manifolds

AC∞ real Riemannian manifold (M2n,g) is called a Hermitian manifold if(2.11)J2=-I,g(JX,JY)=g(X,Y),∀X,Y∈χ(M),
where J is a tensor field of type (1,1) of the tangent space Tp(M), at each point p of M, I is the identity morphism of T(M), and χ(M) denotes the set of all tangent vectors fields on M. The fundamental 2-form Ω of M is defined by(2.12)Ω(X,Y)=g(X,JY),∀X,Y∈χ(M).(M,g,J) is called a Kähler manifold if Ω is closed. A vector field V on a Kähler manifold (M,g,J) is analytic if £VJ=0. It is easy to show that if V is analytic (also called holomorphic) on a Kähler manifold, then so is JV. Using this, one can easily show that if V is a Killing vector field on a compact Kähler manifold, then JV is an analytic gradient vector.Yano [15]. (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 fromdivV=0 and an integral formula (1.14) in [11, p. 41]. Then, (b) follows easily.Sharm [16]. An affine Killing vector field V in a nonflat complex space form M(c) is Killing and analytic.Sharma first proved that the only symmetric (antisymmetric) second-order parallel tensor in a nonflat space formM(c) is the Kählerian metric up to a constant multiple. Then, the proof follows from Yano [15].Remark 2.4.
Comparing Sharma’s result with the part (b) of Yano’s [15] result, observe that Sharma assumed V affine Killing and proved it to be Killing and analytic in M(c) (not necessarily compact), whereas Yano required M(c) 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 (2n+1)-dimensional differentiable manifold M is called a contact manifold if it has a global differential 1-form η such that η∧(dη)n≠0 everywhere on M. For a given contact form η, there exists a unique global vector field ξ, called the characteristic vector field, satisfying(2.13)η(ξ)=1,(dη)(ξ,X)=0,∀X∈χ(M).
A Riemannian metric g of M is called an associated metric of the contact structure if there exists a tensor field ϕ, of type (1,1) such that(2.14)dη(X,Y)=g(X,ϕY),g(X,ξ)=η(X),ϕ2(X)=-X+η(X)ξ,∀X,Y∈χ(M).
These metrics can be constructed by the polarization of dη evaluated on a local orthonormal basis of the tangent space with respect to an arbitrary metric, on the 2n-dimensional contact subbundle D of M. The structure (ϕ,η,ξ,g) on M 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 K-contact structure if its global characteristic vector field ξ is Killing. M has a normal contact structure if(2.15)Nϕ+2dη⊗ξ=0,
where Nϕ is the Nijenhuis tensor field of ϕ. A normal contact metric manifold is called a Sasakian manifold which is also K-contact but the converse holds only if dim(M)=3. The global characteristic Killing vector field ξ of a K-contact manifold has played a key role in the contact geometry. For details, see a complete set of Sasaki’s works cited in Blair [18].On the existence of a proper affine Killing vector field in contact geometry, we have the following nontrivial example.Example 2.5.
Let(M2n+1,g) be a contact metric manifold such that
(2.16)R(X,Y)ξ=0,∀X,Y∈χ(M2n+1).
Blair [19] proved that (M2n+1,g) is locally the product of a Euclidean manifold En+1 and Sn. Using this result, in 1985, Blair and Patnaik [20] used a tensor field K=h-η⊕ξ on a contact metric structure (ϕ,ξ,η,g) of M, where h=(1/2)£ξϕ is the self-adjoint trace-free operator and proved that
(2.17)R(X,Y)ξ=0isequivalentto∇K(X,Y)=0.
They also proved that, if R(ξ,X)ξ=0, there exists an affine connection annihilating K such that the curvature property is preserved. Thus, there exists a proper affine Killing vector field V of the above described locally product contact metric manifold (M2n+1,g), defined by £Vgij=2Kij.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 [21] Sharma has proved the following two results.(i)
On aK-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 compactK-contact manifold without boundary is Killing. Then, in another paper [22], Sharma generalized the above first result as follows.LetMbe a contact metric manifold whoseξ-sectional curvatureK(ξ,X)is nowhere vanishing and is independent of the choice ofX. Then a second-order parallel tensor onMis 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 [23]).
A vector fieldV on a compact orientable manifold (M,g), with compact orientable boundary ∂M, is Killing if and only if(1)
ΔV+QV=0,divV=0 on M,(2)
(£Vg)(V,N)=0 on ∂M,where Q is the (1,1) tensor associated to the Ricci tensor of M.For proof of the above result and some side results onM with boundary, see Yano [11, pp. 118–120].Ünal [24] has proved a similar result for semi-Riemannian manifolds with boundary and subject to the following geometric condition.For a semi-RiemannianM, the validity of divergence theorem is not obvious due to the possible existence of degenerate metric coefficient gii=0 for some index i. Thus the boundary ∂M may become degenerate at some of its points or it may be a lightlike hypersurface of M. In both these cases, there is no well-defined outward normal. Ünal [24] studied this problem as follows.LetM be a semi-Riemannian manifold with boundary ∂M (possibly ∂M=ϕ). Its induced tensor g∂M on ∂M is also symmetric but not necessary a metric tensor as it may be degenerate at some or all points of ∂M. Let ∂M+, ∂M-, and ∂M0 be the subsets of points where the nonzero vectors orthogonal to ∂M are spacelike, timelike, and lightlike respectively. Thus,
(2.18)∂M=∂M+∪∂M-∪∂M0,
where the three subsets are pairwise disjoint. Now we quote the following result.Theorem 2.8 (Ünal [24]).
Let (M,g) be a compact orientable semi-Riemannian manifold with boundary ∂M such that its lightlike part ∂M0 has measure zero in ∂M. Then, a vector field V on M is Killing if and only if(1)
ΔV+QV=0,divV=0 on M,(2)
(£Vg)(V,N)=0 on ∂M′=∂M+∪∂M-,where Q is the (1,1) tensor associated to the Ricci tensor of M, N is the unit normal vector field to ∂M induced on ∂M′, and all eigenvalues of £Vg are real.Since the measure of lightlike∂M0 vanishes in ∂M, the proof of above result is exactly as in the case of Theorem 2.7 of Yano and Ako [23] and the use of following Gauss theorem which is also valid for any semi-Riemannian manifold.Theorem 2.9 (Gauss).
LetM be a compact orientable semi-Riemannian manifold with boundary ∂M. For a smooth vector field V on M, one has
(2.19)∫M(div⁡V)ϵ=∫∂MiVϵ,
where ϵ=|g|dx1∧…∧dxn is the volume element on M and g=det(gij) with respect to a suitable local coordinate system (x1,…,xn). Here i denotes the operator of inner product.


### 2.4. Conformal Killing and Affine Conformal Symmetries

Recall from (1.3) that a vector field V of a Riemannian or semi-Riemannian manifold (Mn,g) is a conformal Killing vector field if £Vg=2ρg for some function ρ of M. To the best of our recollection, this conformal Killing equation appeared in a 1903 paper of Fubini [25] 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 Theorem2.2 for Killing vector field with the following general existence theorem for a conformal Killing vector field.Theorem 2.10 (Yano [10]).
If the Ricci tensor of a compact orientable Riemannian manifold(M,g), without boundary, is nonpositive, then a CKV field V has a vanishing covariant derivative (hence Killing). If the Ricci tensor is negative-definite, then there does not exist any CKV field on M.Yano proved above theorem by assuming thatV is a gradient of a function and used an integral formula [11, page 46] which states(2.20)∫M[Ric(V,V)-|∇V|2-n-2n(δV)2]dV=0
if V is a CKV, where ρ=(1/n)δV. See in [11] 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 [26]).
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 [27] in 1994. The following results are direct consequences of the above theorem of Obata.Yano and Nagano [28]. A complete connected Einstein manifold M (dimension n≥2), admitting a proper CKV field, is isometric to a sphere in an (n+1)-dimensional Euclidean space.Yano [29]. In order that a compact Riemannian manifold M (dimension n>2), with constant scalar curvature r= constant and admitting proper CKV field (with conformal function σ), to be isometric to a sphere, it is necessary and sufficient that ∫M(Ric-r/ng)(gradσ,gradσ)dv=0.Lichnerowicz [30]. Let a compact Riemannian manifold (M,g) admit a proper CKV field, with conformal function σ, such that one of the following holds:(1)
the 1-form associated withV is exact,(2)
gradσ is an eigenvalue of the Ricci tensor with constant eigenvalues,(3)
LVRic=fg, for some smooth function f. Then M is isometric to a sphere.Yano [11]. (1) If (M,g) is complete, of dimension n>2, with r= constant >0, and if it admits a proper CKV field, with conformal function σ, then(2.21)σ2r2≤n(n-1)2|∇∇σ|2,
and equality holds if and only if M is isometric to a sphere.(2) If a complete Riemannian manifold(M,g), of dimension n>2, with scalar curvature r admits a proper CKV field V that leaves the length of the Ricci tensor Ric invariant, that is, V(|Ric|)=0, then M is isometric to a sphere.We refer Yano [11, pp 120–124] for results on CKV fields in M with boundary.Critical Remark
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 [24] 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 [31] and Romero [32]. In case of conformal Killing vector fields in general semi-Riemannian manifolds, we refer to two papers of Kühnel-Rademacher [33, 34].


## 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(M,g) be a compact orientable Riemannian manifold with boundary ∂M. For a smooth vector field V on M, one has
(2.5)∫Mdiv⁡Vdv=∫∂Mg(N,V)dS,
where N and dS are the unit normal to ∂M and its surface element and dv is the volume element of M.Consider ann-dimensional Riemannian manifold (M,g) without boundary, that is, ∫MdivV=0 holds for a smooth vector field V on M. Let V be a Killing vector field of (M,g), that is,(2.6)£Vg=0or£Vgij=∇jVi+∇iVj=0,(1≤i,j≥n),
where ∇ denotes a symmetric affine connection on M. We start with the following fundamental theorem on the existence of a Killing vector field.Theorem 2.2 (Bochner [8]).
If Ricci tensor of a compact orientable Riemannian manifold(M,g), without boundary, is negative semidefinite, then a Killing vector field V on M is covariant constant. On the other hand, if the Ricci tensor on M is negative definite, then a Killing vector field other than zero does not exist on M.Bochner proved this theorem by assuming thatV is a gradient of a function and a result of Watanabe [9] which states “ ∫M[Ric(V,V)-|∇V|2]=0ifVis 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 [12] 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 S3 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 (M,g), defined as follows.
A vector fieldV on (M,g) is called affine Killing if £V∇=0. To interpret this relation with respect to the metric g we split the tensor ∇iVj (see Killing equation (1.3)) into its symmetric and antisymmetric parts as follows:(2.7)∇iVj=Kij+Fij,(Kij=Kji,Fij=-Fji).
Then, it follows that Kij is covariant constant, that is,
(2.8)∇kKij=0.
From (2.7) and (2.8) we deduce that V is affine Killing if and only if
(2.9)£Vgij=2Kij.Kij is called a proper tensor if it is different than the metric tensor gij of M and then V is called a proper affine Killing vector field. In general, for an n-dimensional manifold (M,g), the existence of a proper Kij has its roots back in 1923, when Eisenhart [13] proved that a Riemannian M admits a proper Kij if and only if M is reducible. This means that M is locally a product manifold of the form (M=M1×M2,g=g1⊕g2) and there exists a local coordinate system in terms of which the distance element of g is given by
(2.10)ds2=gab(xc)dxadxb+gAB(xC)dxAdxB,
where a,b,c=1,…,r, A,B,C=r+1,…,n, and 1≤r≤n. Thus, an irreducible Riemannian manifold admits no proper affine Killing vector field.
Observe that the Killing equation (2.6) implies that the condition £V∇=0 holds if V is Killing. However, not every affine Killing vector field is Killing. For example, it was shown in [14] that a non-Einstein conformally flat Riemannian manifold can admit an affine vector field for which Kij 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 [14]. 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 [11]. An affine Killing vector field on a compact orientable Riemannian manifold, without boundary, is Killing.The proof is easy sinceV affine Killing implies divV is constant on M and, in particular, divV=0 if M is without boundary which implies V is Killing.


## 2.2. Kähler Manifolds

AC∞ real Riemannian manifold (M2n,g) is called a Hermitian manifold if(2.11)J2=-I,g(JX,JY)=g(X,Y),∀X,Y∈χ(M),
where J is a tensor field of type (1,1) of the tangent space Tp(M), at each point p of M, I is the identity morphism of T(M), and χ(M) denotes the set of all tangent vectors fields on M. The fundamental 2-form Ω of M is defined by(2.12)Ω(X,Y)=g(X,JY),∀X,Y∈χ(M).(M,g,J) is called a Kähler manifold if Ω is closed. A vector field V on a Kähler manifold (M,g,J) is analytic if £VJ=0. It is easy to show that if V is analytic (also called holomorphic) on a Kähler manifold, then so is JV. Using this, one can easily show that if V is a Killing vector field on a compact Kähler manifold, then JV is an analytic gradient vector.Yano [15]. (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 fromdivV=0 and an integral formula (1.14) in [11, p. 41]. Then, (b) follows easily.Sharm [16]. An affine Killing vector field V in a nonflat complex space form M(c) is Killing and analytic.Sharma first proved that the only symmetric (antisymmetric) second-order parallel tensor in a nonflat space formM(c) is the Kählerian metric up to a constant multiple. Then, the proof follows from Yano [15].Remark 2.4.
Comparing Sharma’s result with the part (b) of Yano’s [15] result, observe that Sharma assumed V affine Killing and proved it to be Killing and analytic in M(c) (not necessarily compact), whereas Yano required M(c) 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 (2n+1)-dimensional differentiable manifold M is called a contact manifold if it has a global differential 1-form η such that η∧(dη)n≠0 everywhere on M. For a given contact form η, there exists a unique global vector field ξ, called the characteristic vector field, satisfying(2.13)η(ξ)=1,(dη)(ξ,X)=0,∀X∈χ(M).
A Riemannian metric g of M is called an associated metric of the contact structure if there exists a tensor field ϕ, of type (1,1) such that(2.14)dη(X,Y)=g(X,ϕY),g(X,ξ)=η(X),ϕ2(X)=-X+η(X)ξ,∀X,Y∈χ(M).
These metrics can be constructed by the polarization of dη evaluated on a local orthonormal basis of the tangent space with respect to an arbitrary metric, on the 2n-dimensional contact subbundle D of M. The structure (ϕ,η,ξ,g) on M 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 K-contact structure if its global characteristic vector field ξ is Killing. M has a normal contact structure if(2.15)Nϕ+2dη⊗ξ=0,
where Nϕ is the Nijenhuis tensor field of ϕ. A normal contact metric manifold is called a Sasakian manifold which is also K-contact but the converse holds only if dim(M)=3. The global characteristic Killing vector field ξ of a K-contact manifold has played a key role in the contact geometry. For details, see a complete set of Sasaki’s works cited in Blair [18].On the existence of a proper affine Killing vector field in contact geometry, we have the following nontrivial example.Example 2.5.
Let(M2n+1,g) be a contact metric manifold such that
(2.16)R(X,Y)ξ=0,∀X,Y∈χ(M2n+1).
Blair [19] proved that (M2n+1,g) is locally the product of a Euclidean manifold En+1 and Sn. Using this result, in 1985, Blair and Patnaik [20] used a tensor field K=h-η⊕ξ on a contact metric structure (ϕ,ξ,η,g) of M, where h=(1/2)£ξϕ is the self-adjoint trace-free operator and proved that
(2.17)R(X,Y)ξ=0isequivalentto∇K(X,Y)=0.
They also proved that, if R(ξ,X)ξ=0, there exists an affine connection annihilating K such that the curvature property is preserved. Thus, there exists a proper affine Killing vector field V of the above described locally product contact metric manifold (M2n+1,g), defined by £Vgij=2Kij.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 [21] Sharma has proved the following two results.(i)
On aK-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 compactK-contact manifold without boundary is Killing. Then, in another paper [22], Sharma generalized the above first result as follows.LetMbe a contact metric manifold whoseξ-sectional curvatureK(ξ,X)is nowhere vanishing and is independent of the choice ofX. Then a second-order parallel tensor onMis 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 [23]).
A vector fieldV on a compact orientable manifold (M,g), with compact orientable boundary ∂M, is Killing if and only if(1)
ΔV+QV=0,divV=0 on M,(2)
(£Vg)(V,N)=0 on ∂M,where Q is the (1,1) tensor associated to the Ricci tensor of M.For proof of the above result and some side results onM with boundary, see Yano [11, pp. 118–120].Ünal [24] has proved a similar result for semi-Riemannian manifolds with boundary and subject to the following geometric condition.For a semi-RiemannianM, the validity of divergence theorem is not obvious due to the possible existence of degenerate metric coefficient gii=0 for some index i. Thus the boundary ∂M may become degenerate at some of its points or it may be a lightlike hypersurface of M. In both these cases, there is no well-defined outward normal. Ünal [24] studied this problem as follows.LetM be a semi-Riemannian manifold with boundary ∂M (possibly ∂M=ϕ). Its induced tensor g∂M on ∂M is also symmetric but not necessary a metric tensor as it may be degenerate at some or all points of ∂M. Let ∂M+, ∂M-, and ∂M0 be the subsets of points where the nonzero vectors orthogonal to ∂M are spacelike, timelike, and lightlike respectively. Thus,
(2.18)∂M=∂M+∪∂M-∪∂M0,
where the three subsets are pairwise disjoint. Now we quote the following result.Theorem 2.8 (Ünal [24]).
Let (M,g) be a compact orientable semi-Riemannian manifold with boundary ∂M such that its lightlike part ∂M0 has measure zero in ∂M. Then, a vector field V on M is Killing if and only if(1)
ΔV+QV=0,divV=0 on M,(2)
(£Vg)(V,N)=0 on ∂M′=∂M+∪∂M-,where Q is the (1,1) tensor associated to the Ricci tensor of M, N is the unit normal vector field to ∂M induced on ∂M′, and all eigenvalues of £Vg are real.Since the measure of lightlike∂M0 vanishes in ∂M, the proof of above result is exactly as in the case of Theorem 2.7 of Yano and Ako [23] and the use of following Gauss theorem which is also valid for any semi-Riemannian manifold.Theorem 2.9 (Gauss).
LetM be a compact orientable semi-Riemannian manifold with boundary ∂M. For a smooth vector field V on M, one has
(2.19)∫M(div⁡V)ϵ=∫∂MiVϵ,
where ϵ=|g|dx1∧…∧dxn is the volume element on M and g=det(gij) with respect to a suitable local coordinate system (x1,…,xn). Here i denotes the operator of inner product.


## 2.4. Conformal Killing and Affine Conformal Symmetries

Recall from (1.3) that a vector field V of a Riemannian or semi-Riemannian manifold (Mn,g) is a conformal Killing vector field if £Vg=2ρg for some function ρ of M. To the best of our recollection, this conformal Killing equation appeared in a 1903 paper of Fubini [25] 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 Theorem2.2 for Killing vector field with the following general existence theorem for a conformal Killing vector field.Theorem 2.10 (Yano [10]).
If the Ricci tensor of a compact orientable Riemannian manifold(M,g), without boundary, is nonpositive, then a CKV field V has a vanishing covariant derivative (hence Killing). If the Ricci tensor is negative-definite, then there does not exist any CKV field on M.Yano proved above theorem by assuming thatV is a gradient of a function and used an integral formula [11, page 46] which states(2.20)∫M[Ric(V,V)-|∇V|2-n-2n(δV)2]dV=0
if V is a CKV, where ρ=(1/n)δV. See in [11] 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 [26]).
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 [27] in 1994. The following results are direct consequences of the above theorem of Obata.Yano and Nagano [28]. A complete connected Einstein manifold M (dimension n≥2), admitting a proper CKV field, is isometric to a sphere in an (n+1)-dimensional Euclidean space.Yano [29]. In order that a compact Riemannian manifold M (dimension n>2), with constant scalar curvature r= constant and admitting proper CKV field (with conformal function σ), to be isometric to a sphere, it is necessary and sufficient that ∫M(Ric-r/ng)(gradσ,gradσ)dv=0.Lichnerowicz [30]. Let a compact Riemannian manifold (M,g) admit a proper CKV field, with conformal function σ, such that one of the following holds:(1)
the 1-form associated withV is exact,(2)
gradσ is an eigenvalue of the Ricci tensor with constant eigenvalues,(3)
LVRic=fg, for some smooth function f. Then M is isometric to a sphere.Yano [11]. (1) If (M,g) is complete, of dimension n>2, with r= constant >0, and if it admits a proper CKV field, with conformal function σ, then(2.21)σ2r2≤n(n-1)2|∇∇σ|2,
and equality holds if and only if M is isometric to a sphere.(2) If a complete Riemannian manifold(M,g), of dimension n>2, with scalar curvature r admits a proper CKV field V that leaves the length of the Ricci tensor Ric invariant, that is, V(|Ric|)=0, then M is isometric to a sphere.We refer Yano [11, pp 120–124] for results on CKV fields in M with boundary.Critical Remark
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 [24] 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 [31] and Romero [32]. 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(M,g) be an n-dimensional timeorientable Lorentzian manifold, called a spacetime manifold. This means that M is a smooth connected Hausdorff manifold and g 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 n-dimensional (n≥3) compact time orientable Lorentzian manifolds.Consider the following form of Einstein field equations:(3.1)Rij-12rgij=Tij,(i,j=1,…4),
where Tij, Rij, and r are the stress-energy tensor, the Ricci tensor, and the scalar curvature respectively. Tij is said to obey the mixed energy condition if at any point x on any hypersurface, (i) the strong energy condition holds, that is, T11+Tii|x≥0, and (ii) equality in (i) implies that all components of T are zero. T is said to obey the dominant energy condition if in any orthonormal basis the energy dominates the other components of Tij, that is, T11≥|Tij| for each i,j. Since the Einstein field equations are a complicated set of nonlinear differential equations, most explicit solutions (see Kramer et al. [35]) 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 (M,g) is called globally hyperbolic [3] if there exists an embedded spacelike 3-manifold Σ such that every endless causal curve intersects Σ once and only once. Such a hypersurface Σ, if it exists, is called a Cauchy surface. If M is globally hyperbolic, then (a) M is homeomorphic to R×S, where S is a hypersurface of M, and for each t, {t}×S is a Cauchy surface, (b) if S′ is any compact hypersurface without boundary, of M, then S′ 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. [36] on the existence of Killing or homothetic vector field in globally hyperbolic spacetimes.Theorem 3.1 (see 36).
Let(M,g) 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 fieldV of g,(3)
admits a compact hypersurfaceΣ of constant mean curvature,Then, either (M,g) is an expanding hyperbolic model with metric
(3.2)ds2=eλt(-dt2+habdxadxb),
with habdxadxb a 3-dimensional Riemannian metric of constant negative curvature on a compact manifold and T vanishing, or V is Killing.Sketch of Proof. According to a result by Geroch [37] we know that if a globally hyperbolic spacetime (M,g) satisfies the vacuum Einstein equations, that is, T vanishes, then g may be completely determined from a set of Cauchy data specified on (Σ,γ) or if M satisfies the Einstein equations coupled to a well-posed hyperbolic systems of matter equations, then the coupled system has the same property, where γ is the induced 3-metric of Σ. Using this property, above theorem was proved within the environment of 3-dimensional compact spacelike hypersurface Σ of (M,g). By hypothesis, if the mean curvature c of Σ is zero, then V is Killing and so the theorem is obvious. If c≠0, then it can be proved that Σ is totally umbilical in M and is of negative constant curvature. Then, it follows from a theorem of Bochner [8] that the standard hyperbolic metric admits no nonzero global Killing vector field. Finally, it is easy to show that (for c≠0) the vacuum spacetime M 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 [38], with the gauge group chosen to be a compact Lie group. The Lie-algebra-valued Yang-Mills field F has the components
(3.3)Fij=DiAj-DjAi+[Ai,Aj],
where Ai and Di are the gauge potential and the spacetime covariant derivative operator with respect to g, respectively. The Einstein-Yang-Mills equations are(3.4)Rij-12rgij=8πTij,Tij=14FkmFkmgij-FikFjk,DiFij+[Ai,Fij]=0.
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 M is expanding hyperbolic model with metric (3.2) and field F≡0 everywhere or V is Killing.Another application is of a massless scalar fieldψ coupled to gravity for which the Einstein-Klein-Gordon equations [39] are Einstein equations with(3.5)Tij=(Diψ)(Djψ)-12gij(Dkψ)(Dkψ),DiDiψ=0.
In this case, since Tij 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 [36]).
Let (M,g) be a globally hyperbolic spacetime which(1)
satisfies the Einstein-Klein-Gordon equations,(2)
admits a homothetic vector fieldV of g,(3)
admits a compact hypersurface of constant mean curvature. Then, either M is an expanding hyperbolic model with metric (3.2) and ψ is constant everywhere, or V is Killing.


### 3.1. Affine Killing Vector Fields in Spacetimes

We know from Section2.1 that a vector field V of a semi-Riemannian manifold (M,g) is an affine Killing vector field if
(3.6)£Vgij=2Kij,Kij;k=0,
where Kij is a covariant constant second-order symmetric tensor. V is proper affine if Kij is other than gij. Eisenhart’s [13] Riemannian result (see Section 2.1) was generalized by Patterson [40], in 1951, showing that a semi-Riemannian (M,g) admitting a proper Kij is reducible if the matrix of Kij has at least two distinct characteristic roots at any point of M. Since then, a general characterization of affine Killing symmetry (known as affine collineation symmetry) remains open. However, for a 4-dimensional spacetime M, this problem has been completely resolved (see Hall and da Costa [41]). 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 [41] that if a simply connected spacetime (M,g) admits a global, nowhere zero, covariant constant proper Kij, then one of the following three possibilities exist.(a) There exists locally a timelike or spacelike, nowhere zero covariant constant vector fieldξ such that Kij=ηiηj,ηi=gijξj, and M is locally decomposable into (1+3) spacetime.(b) There exists locally a null, nowhere zero, covariant constant vector fieldξ such that Kij is as in (a) but (M,g), in general, is not reducible.(c)M is locally reducible into a (2+2) spacetime and no covariant constant vector exists unless it decomposes into (1+1+2) 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. [42] 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 (2+2) 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 (t,r,θ,ϕ) with
(3.7)ds2=dt2-S2(t)((1-Kr2)-1dr2+r2dθ2+r2sin2θdϕ2),
where 𝒦=0,±1. Let Vi=λ(t)δti be a timelike vector parallel to the fluid flow vector ui=δti. Using affine Killing equation Vi;j+Vj;i=2Kij, we obtain
(3.8)Vi;j=Kij=δitδjtλ̇-λSṠ[δirδjr(1-Kr2)-1+δiθδjθr2+δiϕδjϕr2sin⁡2θ].
Since Kij is covariant constant, V(i;j);k=0. Calculating this later equation, we get λṠ-Sλ̇=0 and λ̈=0. Thus, we obtain
(3.9)λ=aS(t),S=bt+c,
for some constants a,b, and c. Thus, V is a timelike vector field parallel to u such that a proper Kij is given by (3.8) and λ and S are related by (3.9).Example 3.4.
The Einstein static universe, which is simply connected and complete manifoldM=R1×S3, with the metric
(3.10)ds2=-dt2+dr2+sin⁡2r(dθ2+sin⁡2θdϕ2)
admits [41] an 8-dimensional transitive Lie group of affine transformations generated by the global proper affine vector field V=t∂t.


### 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(1+3)-splitting (Arnowitt et al. [43]) 4-dimensional spacetime (M,g). This assumes a thin sandwich of M evolved from a spacelike hypersurface Σt at a coordinate time t to another spacelike hypersurface Σt+dt at coordinate time t+dt with metric g given by
(3.11)gαβdxαdxβ=(-λ2+SaSa)dt2+2γabSadxbdt+γabdxadxb,
where λ is the lapse function, S is the shift vector, x0=t,xa(a=1,2,3) are spatial coordinates, and γab is the 3-metric on spacelike slice Σ. This is known as ADM model which admits a CKV field. In 1986, Maartens and Maharaj [44] proved that Robertson-Walker spacetimes (which provide a satisfactory cosmological ADM model) admit a G6 of Killing vectors and a G9 of conformal Killing vector fields. By definition, a group Gr of isometric or conformal motions has r 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 ofΣ by X,Y,Z,W, and the timelike unit vector field normal to Σ by N. Then the Gauss and Weingarten formulas are(3.12)∇̅XY=∇XY+B(X,Y)N,∇̅XN=ANX,
where AN is the shape operator of Σ defined by B(X,Y)=<ANX,Y> (<,> is the inner product with respect to the metric γ of Σ and the spacetime metric g), ∇̅,∇ the Levi-Civita connections of g,γ, respectively, and B is the second fundamental form. The Gauss and Codazzi equations are(3.13)<R̅(X,Y)Z,W>=<R(X,Y)Z,W>+B(Y,Z)B(X,W)=-B(X,Z)B(Y,W)<R̅(X,Y)N,Z>=(∇XB)(Y,Z)-(∇YB)(X,Z),
where R̅ and R denote curvature tensors of g and γ, respectively. It is straightforward to show that the following relation holds:(3.14)R̅ic(X,Y)+〈R̅(N,X)Y,N〉=Ric(X,Y)+τ〈ANX,Y〉=-〈ANX,ANY〉R̅ic(X,N)=(div⁡AN)X-Xτ,
where R̅ic and Ric are the Ricci tensors of g and γ, respectively, and τ=Tr·AN=3 times the mean curvature of Σ. Let the Einstein’s field equations be of the form R̅ic-(r̅/2)g=T, where r̅ and T are the scalar curvature and the energy-momentum tensor, respectively. Following are the constraint equations:(3.15)r+2τ23-|L|2=2T(N,N),L=AN-τ3I,(div⁡L)X-23Xτ=T(X,N),
where r is the scalar curvature of γ, || the norm operator with respect to γ. Assume that (M,g) admits a CKV field V, that is, £Vg=2σg. Decompose V along Σ as V=ξ+ρN, where ξ is the tangential component of V. A simple calculation using all the above equations provides the following evolution equation:(3.16)(£ξγ)(X,Y)=2σγ(X,Y)-2ρ〈LX,Y〉-2ρτ3γ(X,Y).(£ξL)X=-(∇XDρ-∇2ρ3X)-ρ(TX-Tii3X)+(ρτ-σ2)LX+ρ(QX-r3X)£ξτ=σ(3N-τ)-∇2ρ+ρ[τ23+|L|2+12(Tii+T(N,N))].
Here Q is the Ricci operator of γ, and ∇2=∇a∇a (a summed over 1,2,3). The above Evolution equations were first derived in [36] through a different approach using Berger’s [45] condition that sets the evolution vector field equal to V.Theorem 3.5 (Sharma [46]).
Let (M,g) be an ADM spacetime solution of Einstein’s field equations admitting a CKV field V and evolved by a complete spacelike hypersurface Σ such that (a) Σ is totally umbilical in M, (b) the normal component ρ of V is nonconstant on Σ, and (c) the normal sectional curvature of M is independent of the tangential direction at each point of Σ. Then Σ is conformally diffeomorphic to (i) a 3-sphere S3, (ii) Euclidean space E3, (iii) hyperbolic space H3, or (iv) the product of a complete 2-dimensional manifold and an open real interval. If Σ is compact, then only (i) holds.Sharma’s proof uses the above constraint and evolution equations with the condition that the normal sectional curvatureS̅(N,X) of M at a point p with respect to a plane section spanned by a unit tangent vector X of Σ and the unit normal N is independent of the choice of X. Note that the normal sectional curvature is defined as 〈R̅(N,X)N,X〉 (see [3, page 33]). This normal sectional curvature holds when M 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(M=I×fΣ,g) defined by
(3.17)ds2=-dt2+(f(t))2γabdxadxb,
where I is the time line, (Σ,γ) is an arbitrary 3D-Riemannian manifold, and f>0 is a warping function (see Alías et al. [47]). They have shown that the normal curvature condition holds for this GRW-spacetime and each slice t= constant is homothetic to the fiber Σ and totally umbilical in (M,g). As a consequence of the Theorem 3.5, the following two results are easy to prove.Duggal-Sharma [48]. (1) Let (M,g) be an ADM spacetime evolved out of a complete initial hypersurface Σ that is totally umbilical and has nonzero constant mean curvature. If (M,g) admits a closed CKV field V nonvanishing on Σ, then either V is orthogonal to Σ and the lapse function is constant over Σ, or Σ is conformally diffeomorphic to E3,  S3, H3, or the product of an open interval and a 2-dimensional Riemannian manifold.(2) Let a conformally flat perfect fluid solution(M,g) of the Einstein’s equations be evolved out of an initial spacelike hypersurface Σ that is compact, has constant mean curvature, and is orthogonal to the 4-velocity. If (M,g) has a nonvanishing non-Killing CKV field V which is nowhere tangential to Σ, then Σ is totally umbilical in M and is of constant curvature. In the case when M is of constant negative curvature, V is orthogonal to Σ.


### 3.3. Spacetimes with Affine Conformal Symmetry

An affine conformal symmetry is defined by a vector fieldV of (M,g) satisfying(3.18)£Vg=σg+K,∇K=0,
where K≠g is a second-order symmetric tensor and V is called an affine conformal Killing vector [49], denoted by ACV, which is CKV when K vanishes. If σ is constant, then V is affine. Moreover, V is an ACV if and only if(3.19)£VΓijk=δik∂j(σ)+δjk∂i(σ)-gijσk,
which is also known as “conformal collineation symmetry” generated by an ACV field V. Here Γijk are the Christoffel symbols. We state the main results on ACV (proved by Tashiro [50]) on the local reducibility of a Riemannian manifold (M,g). By local reducibility we mean that M is locally a product manifold.(1)
IfM has constant scalar curvature and has a flat part, then an ACV on M is the sum of an affine and a CKV.(2)
IfM has at least three parts and no part is locally flat, then an ACV on M is affine. If M is also complete, then the ACV is Killing.(3)
LetM have constant scalar curvature with no flat part. If M is irreducible or is the product of two irreducible parts whose scalar curvatures are signed opposite to each other, then an ACV on M 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 irreducibleM admits no ACV which is not a CKV.(7)
If a locally reducibleM has at least three parts, one of which is flat, then an ACV on M is sum of an affine vector and a CKV. If M 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 [51] constructed the following example which supports first part of the result (7).Example 3.8.
Let(M4,g) be a Einstein static fluid spacetime with metric
(3.20)ds2=-dt2+(1-r2)-1dr2+r2(dθ2+sin⁡2θdϕ2)
and the velocity vector ua=δ0i (i=0,1,2,3). This spacetime admits a CKV
(3.21)V1i=(1-r2)1/2{cos⁡⁡tui-rsin⁡⁡tδ1i}
and a proper affine vector V2i=tui. Since the metric is reducible, it can be easily verified that a combination V=V1+V2 is a proper ACV such that
(3.22)Vi=[t+(1-r2)1/2cos⁡⁡t]ui-r(1-r2)1/2sin⁡⁡tδ1i,σ=-(1-r2)1/2sin⁡⁡t,Kij=-2t,it,b.
Now let (Mn,g) be a compact orientable semi-Riemannian manifold with boundary ∂M. The divergence theorem is not valid due to the possible degenerate part of ∂M. For this reason we call (M,g) a regular [49] semi-Riemannian manifold if we exclude the possible degenerate part in ∂M. Then, following is a characterization theorem for the existence of a proper ACV.Theorem 3.9 (Duggal [49]).
A vector fieldV in a compact orientable regular semi-Riemannian manifold (M,g), with boundary ∂M, is a proper ACV if and only if(a)
∫∂M(K-(tr·H/n)g)(V,N)ds≠0,(b)
𝒟V=-(n-2)gradσ∈M, 𝒟V=QV+ΔV,where σ, K, and H are the de-Rham Laplacian, affine conformal function, covariant constant tensor of type (0,2), and its associated (1,1) tensor, respectively.The reader will find several other side results in [17, Chapter 7] on the geometry and physics of affine conformal symmetry.


## 3.1. Affine Killing Vector Fields in Spacetimes

We know from Section2.1 that a vector field V of a semi-Riemannian manifold (M,g) is an affine Killing vector field if
(3.6)£Vgij=2Kij,Kij;k=0,
where Kij is a covariant constant second-order symmetric tensor. V is proper affine if Kij is other than gij. Eisenhart’s [13] Riemannian result (see Section 2.1) was generalized by Patterson [40], in 1951, showing that a semi-Riemannian (M,g) admitting a proper Kij is reducible if the matrix of Kij has at least two distinct characteristic roots at any point of M. Since then, a general characterization of affine Killing symmetry (known as affine collineation symmetry) remains open. However, for a 4-dimensional spacetime M, this problem has been completely resolved (see Hall and da Costa [41]). 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 [41] that if a simply connected spacetime (M,g) admits a global, nowhere zero, covariant constant proper Kij, then one of the following three possibilities exist.(a) There exists locally a timelike or spacelike, nowhere zero covariant constant vector fieldξ such that Kij=ηiηj,ηi=gijξj, and M is locally decomposable into (1+3) spacetime.(b) There exists locally a null, nowhere zero, covariant constant vector fieldξ such that Kij is as in (a) but (M,g), in general, is not reducible.(c)M is locally reducible into a (2+2) spacetime and no covariant constant vector exists unless it decomposes into (1+1+2) 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. [42] 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 (2+2) 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 (t,r,θ,ϕ) with
(3.7)ds2=dt2-S2(t)((1-Kr2)-1dr2+r2dθ2+r2sin2θdϕ2),
where 𝒦=0,±1. Let Vi=λ(t)δti be a timelike vector parallel to the fluid flow vector ui=δti. Using affine Killing equation Vi;j+Vj;i=2Kij, we obtain
(3.8)Vi;j=Kij=δitδjtλ̇-λSṠ[δirδjr(1-Kr2)-1+δiθδjθr2+δiϕδjϕr2sin⁡2θ].
Since Kij is covariant constant, V(i;j);k=0. Calculating this later equation, we get λṠ-Sλ̇=0 and λ̈=0. Thus, we obtain
(3.9)λ=aS(t),S=bt+c,
for some constants a,b, and c. Thus, V is a timelike vector field parallel to u such that a proper Kij is given by (3.8) and λ and S are related by (3.9).Example 3.4.
The Einstein static universe, which is simply connected and complete manifoldM=R1×S3, with the metric
(3.10)ds2=-dt2+dr2+sin⁡2r(dθ2+sin⁡2θdϕ2)
admits [41] an 8-dimensional transitive Lie group of affine transformations generated by the global proper affine vector field V=t∂t.


## 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(1+3)-splitting (Arnowitt et al. [43]) 4-dimensional spacetime (M,g). This assumes a thin sandwich of M evolved from a spacelike hypersurface Σt at a coordinate time t to another spacelike hypersurface Σt+dt at coordinate time t+dt with metric g given by
(3.11)gαβdxαdxβ=(-λ2+SaSa)dt2+2γabSadxbdt+γabdxadxb,
where λ is the lapse function, S is the shift vector, x0=t,xa(a=1,2,3) are spatial coordinates, and γab is the 3-metric on spacelike slice Σ. This is known as ADM model which admits a CKV field. In 1986, Maartens and Maharaj [44] proved that Robertson-Walker spacetimes (which provide a satisfactory cosmological ADM model) admit a G6 of Killing vectors and a G9 of conformal Killing vector fields. By definition, a group Gr of isometric or conformal motions has r 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 ofΣ by X,Y,Z,W, and the timelike unit vector field normal to Σ by N. Then the Gauss and Weingarten formulas are(3.12)∇̅XY=∇XY+B(X,Y)N,∇̅XN=ANX,
where AN is the shape operator of Σ defined by B(X,Y)=<ANX,Y> (<,> is the inner product with respect to the metric γ of Σ and the spacetime metric g), ∇̅,∇ the Levi-Civita connections of g,γ, respectively, and B is the second fundamental form. The Gauss and Codazzi equations are(3.13)<R̅(X,Y)Z,W>=<R(X,Y)Z,W>+B(Y,Z)B(X,W)=-B(X,Z)B(Y,W)<R̅(X,Y)N,Z>=(∇XB)(Y,Z)-(∇YB)(X,Z),
where R̅ and R denote curvature tensors of g and γ, respectively. It is straightforward to show that the following relation holds:(3.14)R̅ic(X,Y)+〈R̅(N,X)Y,N〉=Ric(X,Y)+τ〈ANX,Y〉=-〈ANX,ANY〉R̅ic(X,N)=(div⁡AN)X-Xτ,
where R̅ic and Ric are the Ricci tensors of g and γ, respectively, and τ=Tr·AN=3 times the mean curvature of Σ. Let the Einstein’s field equations be of the form R̅ic-(r̅/2)g=T, where r̅ and T are the scalar curvature and the energy-momentum tensor, respectively. Following are the constraint equations:(3.15)r+2τ23-|L|2=2T(N,N),L=AN-τ3I,(div⁡L)X-23Xτ=T(X,N),
where r is the scalar curvature of γ, || the norm operator with respect to γ. Assume that (M,g) admits a CKV field V, that is, £Vg=2σg. Decompose V along Σ as V=ξ+ρN, where ξ is the tangential component of V. A simple calculation using all the above equations provides the following evolution equation:(3.16)(£ξγ)(X,Y)=2σγ(X,Y)-2ρ〈LX,Y〉-2ρτ3γ(X,Y).(£ξL)X=-(∇XDρ-∇2ρ3X)-ρ(TX-Tii3X)+(ρτ-σ2)LX+ρ(QX-r3X)£ξτ=σ(3N-τ)-∇2ρ+ρ[τ23+|L|2+12(Tii+T(N,N))].
Here Q is the Ricci operator of γ, and ∇2=∇a∇a (a summed over 1,2,3). The above Evolution equations were first derived in [36] through a different approach using Berger’s [45] condition that sets the evolution vector field equal to V.Theorem 3.5 (Sharma [46]).
Let (M,g) be an ADM spacetime solution of Einstein’s field equations admitting a CKV field V and evolved by a complete spacelike hypersurface Σ such that (a) Σ is totally umbilical in M, (b) the normal component ρ of V is nonconstant on Σ, and (c) the normal sectional curvature of M is independent of the tangential direction at each point of Σ. Then Σ is conformally diffeomorphic to (i) a 3-sphere S3, (ii) Euclidean space E3, (iii) hyperbolic space H3, or (iv) the product of a complete 2-dimensional manifold and an open real interval. If Σ is compact, then only (i) holds.Sharma’s proof uses the above constraint and evolution equations with the condition that the normal sectional curvatureS̅(N,X) of M at a point p with respect to a plane section spanned by a unit tangent vector X of Σ and the unit normal N is independent of the choice of X. Note that the normal sectional curvature is defined as 〈R̅(N,X)N,X〉 (see [3, page 33]). This normal sectional curvature holds when M 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(M=I×fΣ,g) defined by
(3.17)ds2=-dt2+(f(t))2γabdxadxb,
where I is the time line, (Σ,γ) is an arbitrary 3D-Riemannian manifold, and f>0 is a warping function (see Alías et al. [47]). They have shown that the normal curvature condition holds for this GRW-spacetime and each slice t= constant is homothetic to the fiber Σ and totally umbilical in (M,g). As a consequence of the Theorem 3.5, the following two results are easy to prove.Duggal-Sharma [48]. (1) Let (M,g) be an ADM spacetime evolved out of a complete initial hypersurface Σ that is totally umbilical and has nonzero constant mean curvature. If (M,g) admits a closed CKV field V nonvanishing on Σ, then either V is orthogonal to Σ and the lapse function is constant over Σ, or Σ is conformally diffeomorphic to E3,  S3, H3, or the product of an open interval and a 2-dimensional Riemannian manifold.(2) Let a conformally flat perfect fluid solution(M,g) of the Einstein’s equations be evolved out of an initial spacelike hypersurface Σ that is compact, has constant mean curvature, and is orthogonal to the 4-velocity. If (M,g) has a nonvanishing non-Killing CKV field V which is nowhere tangential to Σ, then Σ is totally umbilical in M and is of constant curvature. In the case when M is of constant negative curvature, V is orthogonal to Σ.


## 3.3. Spacetimes with Affine Conformal Symmetry

An affine conformal symmetry is defined by a vector fieldV of (M,g) satisfying(3.18)£Vg=σg+K,∇K=0,
where K≠g is a second-order symmetric tensor and V is called an affine conformal Killing vector [49], denoted by ACV, which is CKV when K vanishes. If σ is constant, then V is affine. Moreover, V is an ACV if and only if(3.19)£VΓijk=δik∂j(σ)+δjk∂i(σ)-gijσk,
which is also known as “conformal collineation symmetry” generated by an ACV field V. Here Γijk are the Christoffel symbols. We state the main results on ACV (proved by Tashiro [50]) on the local reducibility of a Riemannian manifold (M,g). By local reducibility we mean that M is locally a product manifold.(1)
IfM has constant scalar curvature and has a flat part, then an ACV on M is the sum of an affine and a CKV.(2)
IfM has at least three parts and no part is locally flat, then an ACV on M is affine. If M is also complete, then the ACV is Killing.(3)
LetM have constant scalar curvature with no flat part. If M is irreducible or is the product of two irreducible parts whose scalar curvatures are signed opposite to each other, then an ACV on M 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 irreducibleM admits no ACV which is not a CKV.(7)
If a locally reducibleM has at least three parts, one of which is flat, then an ACV on M is sum of an affine vector and a CKV. If M 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 [51] constructed the following example which supports first part of the result (7).Example 3.8.
Let(M4,g) be a Einstein static fluid spacetime with metric
(3.20)ds2=-dt2+(1-r2)-1dr2+r2(dθ2+sin⁡2θdϕ2)
and the velocity vector ua=δ0i (i=0,1,2,3). This spacetime admits a CKV
(3.21)V1i=(1-r2)1/2{cos⁡⁡tui-rsin⁡⁡tδ1i}
and a proper affine vector V2i=tui. Since the metric is reducible, it can be easily verified that a combination V=V1+V2 is a proper ACV such that
(3.22)Vi=[t+(1-r2)1/2cos⁡⁡t]ui-r(1-r2)1/2sin⁡⁡tδ1i,σ=-(1-r2)1/2sin⁡⁡t,Kij=-2t,it,b.
Now let (Mn,g) be a compact orientable semi-Riemannian manifold with boundary ∂M. The divergence theorem is not valid due to the possible degenerate part of ∂M. For this reason we call (M,g) a regular [49] semi-Riemannian manifold if we exclude the possible degenerate part in ∂M. Then, following is a characterization theorem for the existence of a proper ACV.Theorem 3.9 (Duggal [49]).
A vector fieldV in a compact orientable regular semi-Riemannian manifold (M,g), with boundary ∂M, is a proper ACV if and only if(a)
∫∂M(K-(tr·H/n)g)(V,N)ds≠0,(b)
𝒟V=-(n-2)gradσ∈M, 𝒟V=QV+ΔV,where σ, K, and H are the de-Rham Laplacian, affine conformal function, covariant constant tensor of type (0,2), and its associated (1,1) 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 [52] proved that “every compact homogeneous semi-Riemannian manifold is geodesically complete.” For the case of Lorentzian manifolds, the singularity theorems (see Hawking and Ellis [39]) 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 [3]. It has been shown in Beem and Ehrlich’s book [3] 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(M,g) be an n-dimensional (n≥3) compact time orientable Lorentzian manifold. Recall that a compact manifold M admits a Lorentzian metric if and only if the Euler number of M vanishes. Considerable work has been done on the applications of null geodesics of compact (M,g) using a conformal Killing symmetry. Since, for Lorentzian metrics, the compactness does not imply geodesic completeness, Romero and Sánchez [31] have proved that a compact Lorentzian manifold which admits a timelike CKV field yields to its geodesic completeness.LetC(s) be a curve in a Lorentzian manifold (M,g), where s is a suitable parameter. A vector field V on C is called a Jacobi vector field if it satisfies the following Jacobi differential equation:(4.1)∇C′∇C′V=R(C′,V)C′,
where ∇ is a metric connection on M.Definition 4.1.
We say that a pointp on a geodesic C(s) of M is conjugate to a point q along C(s) if there is a Jacobi field along C(s), not identically zero, which vanishes at q and p.From a geometric point of view, a conjugate pointC(a) of p=C(0) along a geodesic C can be interpreted as an “almost-meeting point’’ of a geodesic starting from p with initial velocity C′(0). In general relativity, since the relative position of neighboring events of a free falling particle C is given by the Jacobi field of C, 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 [3].Letx∈(M,g) and ξ be a null vector of TxM. A plane H of TxM is called a null plane directed by ξ if it contains ξ, gx(ξ,W)=0 for any W∈H and there exists Wo∈H such that gx(Wo,Wo)≠0. Then, the null sectional curvature of H, with respect to ξ and ∇, is defined as a real number(4.2)Kξ(H)=gx(R(W,ξ)ξ,W)gx(W,W),
where W≠0 is any vector in H independent with ξ (and therfore spacelike). It is easy to see that Kξ(H) is independent of W but depends in a quadratic fashion on ξ. The null congruence associated with a vector field V is defined by(4.3)CVM={ξ∈TM:g(ξ,ξ)=0,g(ξ,Vπ(ξ))=1},
where π:TM→M is the natural projection. CKM is an oriented embedded submanifold of TM with dimension 2(n-1) and (CVM,π,M) is a fiber bundle with fiber type Sn-2. Therefore, for a compact M,CVM will be compact. If a null congruence CVM is fixed with respect to a timelike vector field V, then one can choose, for every null plane H, the unique null vector ξ∈CVM∩H, so that the null sectional curvature can be thought as a function on null planes. This function is called the V-normalized null sectional curvature.Gutiérrez et al. [53–55] have done following work on conjugate points along null geodesics of compact Lorentzian manifolds.Let(M,g) be an n-dimensional (n≥3) compact Lorentzian manifold that admits a timelike CKV field V [53]. If there exists a real number a∈(0,+∞) such that every null geodesic Cξ:[0,a]→M, with ξ∈CVM, has no conjugate points of Cξ(0) in [0,a), then (4.4)Vol(CVM,ĝ)≥a2π2n(n-1)∫CVMR̅icdμĝ.
Equality holds if and only if M has V-normalized null sectional curvature π2/a2. Here ĝ is the restriction to CVM of the metric on the TM. R̅ic denotes the quadratic form associated with the Ricci tensor of M and dμĝ is the canonical measure associated with ĝ.The authors used the previous result in proving several inequalities relating conjugate points along geodesics to global geometric properties [54]. Also, they have shown some classification results on certain compact Lorentzian manifolds without conjugate points along its null geodesics.Let(Mn,g) be a compact Lorentzian manifold admitting a timelike CKV field V [55]. If (Mn,g) has no conjugate points along its null geodesic, then(4.5)∫M[R̅ic(U)+S]hndμg≤0,
where h=[-g(V,V)]-1/2 so that g(U,U)=-1 with U=hV. Moreover, equality holds if and only if (M,g) has constant sectional curvature k≤0. If V is a timelike Killing vector field, then(4.6)∫MShndμg
and equality holds if and only if M is isomorphic to a flat Lorentzian n-torus up to a (finite) covering. In particular, U is parallel, the first Betti number of M is nonzero, and the Levi-Civita connection of g is Riemannian.Remark 4.2.
Recall the following classical Hopf theorem [56].
“A Riemannian torus with no conjugate points must be flat.”
As a Lorentzian analogue to Hopf theorem, Palomo and Romero [57] 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 [58] have obtained a sequence of integral inequalities for any (n≥3)-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(M,g) be an n-dimensional smooth manifold with a symmetric (0,2) tensor field g. Assume that g is degenerate on TM, that is, there exists a vector field ξ≠0, of Γ(TM), such that g(ξ,v)=0,forallv∈χ(TM). The radical distribution of TM, with respect to g, is defined by(5.1)RadTM={ξ∈Γ(TM);g(ξ,v)=0,∀v∈χ(TM)}
such that TM=Rad(TM)⊕orthS(TM), where S(TM) is a nondegenerate complementary screen distribution of RadTM in TM. Suppose dim(Rad(TM))=r≥1. Then, dim(S(TM))=n-r. As in case of semi-Riemannian manifolds, a vector field V on a lightlike manifold (M,g) is said to be a Killing vector field if £Vg=0. A distribution D on M is called a Killing distribution if each vector field belonging to D is a Killing vector field. Due to degenerate g on M, in general, there does not exist a unique metric (Levi-Civita) connection for M 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(M,g) with respect to g if and only if Rad(TM) is Killing.The above result also holds if(M,g) is a lightlike submanifold of a semi-Riemannian manifold (M̅,g̅) for which Rad(TM)=TM∩TM⊥ (see [59, page 169]).We refer to the following two books [60, 61] which include up-to-date information on extrinsic geometry of lightlike subspaces, in particular reference to a key role of Killing symmetry.Physical Interpretation
Physically useful are the lightlike hypersurfaces of spacetime manifolds which (under some conditions) are models as black hole horizons (see Carter [62], Galloway [63], and other, cited therein). To illustrate this use, let (M,g) be a lightlike hypersurface of a spacetime manifold (M̅,g̅). We adopt following features of the intrinsic geometry of lightlike hypersurfaces. Assume that the null normal ξ is not entirely in M, but is defined in some open subset of M̅ around M. This well-defines the spacetime covariant derivative ∇̅ξ, which, in general, is not possible if ξ is restricted to M as is the case of extrinsic geometry, where ∇̅ is the Levi-Civita connection on M̅. Following Carter [64], a simple way is to consider a foliation of M̅ (in the vicinity of M) by a family (Mu) so that ξ is in the part of M̅ foliated by this family such that, at each point in this region, ξ is a null normal to Mu for some value of u. Although the family (Mu) 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 (Mu) once evaluated at, say, Mu= constant. For simplicity, we denote by M=Mu= constant. Then the metric g is simply the pull-back of the metric g̅ofM̅toM,gij=←g̅ij, where an under arrow denotes the pullback to M. The “bending” of M in M is described by the Weingarten map:
(5.2)Wξ:TpM⟶TpMX⟶∇̅Xξ,
that is, 𝒲ξ associates each X of M the variation of ξ along X, with respect to the spacetime connection ∇̅. The second fundamental form, say B, of M is the symmetric bilinear form and is related with the Weingarten map by
(5.3)B(X,Y)=g(WξX,Y)=g(∇̅Xξ,Y).
Using £ξg(X,Y)=g(∇̅Xξ,Y)+g(∇̅Yξ,X) and B(X,Y) symmetric in (5.3), we obtain
(5.4)B(X,Y)=12£ξg(X,Y),∀X,Y∈TM,
which is well defined up to conformal rescaling (related to the choice of ξ). B(X,ξ)=0 for any null normal ξ and for any X∈TM implies that B has the same ξ degeneracy as that of the induced metric g.
Consider a class of lightlike hypersurfaces such that its second fundamental formB is conformally equivalent to its degenerate metric g. Geometrically, this means that (M,g) is totally umbilical in M̅ if and only if there is a smooth function σ on M such that
(5.5)B(X,Y)=σg(X,Y),∀X,Y∈Γ(TM).
It is obvious that above definition does not depend on particular choice of ξ. The name “umbilical” means that extrinsic curvature is proportional to the metric g. M is proper totally umbilical in M̅ if and only if σ is nonzero on M. In particular, M is totally geodesic if and only if B vanishes, that is, if and only if σ vanishes on M. It follows from (5.4) and (5.5) that
(5.6)£ξg=2σgon⁡M.
Thus, ξ is a conformal Killing vector (CKV) field in a totally umbilical M, with conformal function 2σ, which is Killing if and only if M is totally geodesic.
Now we need the following general result on totally umbilical submanifolds.Proposition 5.2 (Perlick [65]).
Let(M,g) be a totally umbilical submanifold of a semi-Riemannian manifold  (M̅,g̅). Then,   (1)
a null geodesic ofM̅ that starts tangential to M remains within M (for some parameter interval around the starting point),(2)
M is totally geodesic if and only if every geodesic of M̅ that starts tangential to M remains with in M (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 [66], which include Killing horizons [62] as a special case). A Killing horizon is defined as the union M=⋃Ms, where Ms is a connected component of the set of points forming a family of lightlike hypersurfaces Ms whose null geodesic (as per above proposition) generators coincide with the Killing trajectories of nowhere vanishing ξs. The isolated horizon (IH) of a stationary asymptotically flat black hole is represented by the Killing horizon if M 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.Physical model
Consider a 4-dimensional stationary spacetime(M̅,g̅) which is chronological, that is, M̅ admits no closed timelike curves. It is well known [39] that a stationary M̅ admits a smooth 1-parameter group, say G, of isometries whose orbits are timelike curves in M̅. Denote by M′ the Hausdorff and paracompact 3-dimensional Riemannian orbit space of the action G. The projection π:M̅→M′ is a principal R-bundle, with the timelike fiber G. Let T=∂t be the nonvanishing timelike Killing vector field, where t is a global time coordinate function on M′. Then, the metric g̅ induces a Riemannian metric gM′ on M′ such that
(5.7)M̅=R×M′,g̅=-u2(dt+η)2+π⋆gM′,
where η is a connection 1-form for the R-bundle π and
(5.8)u2=-g(T,T)>0.
It is known that a stationary spacetime (M̅,g̅) uniquely determines the orbit data (ℳ′,gM′,u,η) as described previously, and conversely. Suppose the orbit space M′ has a nonempty metric boundary ∂M′≠∅. Consider the maximal solution data in the sense that it is not extendible to a larger domain (ℳ′,gℳ′′,u′,η′)⊃(M′,gM′,u,η) with u′>0 on an extended spacetime ℳ. Under these conditions, it is known [39] that in any neighborhood of a point x∈∂M′, either the connection 1-form η degenerates, or u→0. The second case implies that the timelike Killing vector T becomes null and M′ degenerates into a lightlike hypersurface, say (M,g) of M̅. Moreover, lim(T)u→0=V∈χ(TM) is a global null Killing vector field of M.
In the following we quote a result on physical interpretation of an ADM spacetime (see Section3.2) which can admit a Killing horizon.Theorem 5.3 (see [67]).
Let (M̅,g̅) be an ADM spacetime evolved through a 1-parameter family of spacelike hypersurfaces Σt such that the evolution vector field is a null CKV field ξ on M̅. Then, ξ reduces to a Killing vector field if and only if the part of ξ tangential to Σt is asymptotic everywhere on Σt for all t. Moreover, ξ 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(M̅,g̅) which admits a timelike conformal Killing vector (CKV) field. Let (M,g) be a lightlike hypersurface of M̅ 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 M̅ 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 (M̅,G) related to a black hole spacetime (M̅,g̅) admitting a Killing horizon M generated by the null geodesic Killing field, with the conformal factor in G=Ω2g̅, where Ω is a nonvanishing function on M̅. Under this transformation, the Killing vector field is mapped to a conformal Killing field ξ provided ξi∇̅iΩ≠0. Since the causal structure and null geodesics are invariant under a conformal transformation, M still remains a null hypersurface of (M̅,G). Moreover, as per Proposition 5.2, the null geodesic of M̅ that starts tangential to M will remain within M. Also, its null geodesic generators coincide with the conformal Killing trajectories. Thus, M is a CKH in (M,G).Theorem 5.4 (Sultana and Dyer [70]).
Let(M,G) be a spacetime related to an analytic black hole spacetime  (M,g) admitting a Killing horizon  Σ0, such that the conformal factor in  G=Ω2g goes to a constant at null infinity. Then the conformal Killing horizon  Σ in  (M,G) 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 [71] 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.



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