On completeness of certain families of semi-Riemannian manifolds

Letters in Mathematical Physics 53:103-117, 1994. 103 (~) 1994 Kluwer Academic Publishers. Printed in the Netherlands.

On Completeness of Certain Families of Semi-Riemannian Manifolds*

ALFONSO ROMERO AND MIGUEL SANCHEZ Departamento de Geometrfa y Topologfa, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain. e-mail: [email protected] [email protected]

(Received: 2 April 1993; revised version: 26 July 1993)

Abstract. Semi-Riemannian manifolds with a suitable set of conformal symmetries are shown to be complete. Locally warped products are studied and warped-completeness is introduced. In the case of definite and complete basis, several assumptions on the growth of the warping function yield some of the three kinds of completeness. The case of 1-dimensional basis (including a known family of relativistic space-times) is specially studied. Null warped-completeness is related to the completeness of a certain conformal metric on the basis. Several examples and counter-examples explaining the main results are also given.

Mathematics Subject Classifications (1991). 53C22, 53C50.

1. Introduction

The following facts about completeness in semi-Riemannian manifolds (i.e. con- nected manifolds with non-degenerate metrics) are well known:

(a) Timelike, spacelike and null completeness are logically independent ([1, Th. 5.4]). Nevertheless, the equivalence of the three kinds of completeness has been proved in the locally symmetric case [5] (recall that a semi-Riemannian symmetric space is always complete).

(b) A conformal factor yielding a timelike and null complete metric can be found on any globally hyperbolic space-time ([1, Th. 5.5]), see also references therein for related results).

(c) An indefinite manifold (i.e. semi-Riemannian but non-definite) which is a warped product of definite complete manifolds may be incomplete (Beem- Buseman's example, [7, Example 7.41]). It is also well known in the compact case:

(d) A compact homogeneous semi-Riemannian manifold is complete, which was proved by Marsden in [6] (see also [7, Prop. 9.39]).

(e) A compact fiat Lorentzian manifold is complete, according to a theorem by Carri~re [2] (see [3] for a previous particular result for a flat Lorentzian toms).

We remark that there are non-compact fiat homogeneous Lorentzian manifolds which are incomplete ([7, Remark 9.37]). Two open questions are:

* This work was supported in part by a DGICYT Grant PB91-0731.

104 ALFONSO ROMERO AND MIGUEL S,~NCHEZ

(i) whether there is dependence among the three kinds of completeness on compact indefinite manifolds, in particular, if null completeness implies completeness (a proof is given in [9], but has been critiqued in [8]);

(ii) decide if there exists a conformal factor on a compact indefinite manifold such that one of the two conformally related metrics be complete, and the other one incomplete.

(f)

A certain relation between these two questions follows from the following fact:

If two indefinite manifolds are conformal, and the conformal factor f~ satisfies 0 < inf(fl) _< sup(f~) < oo, then one metric is null complete if (and only if) the other one is too (you can see this as a consequence of (3.5) later, even though it is easy to give a direct proof). Thus, an affirmative answer to (ii) would yield incomplete but null complete compact indefinite manifolds.

In Section 2 we carry out a study of the completeness of semi-Riemannian manifolds with a suitable set of conformal symmetries. Assuming that these have a certain causal behaviour, the completeness is achieved in Proposition 2.1. As a consequence of the proof, several semi-Riemannian manifolds globally conformal to those given in (d) and (e) are found to be complete (Corollary 2.3 and Remark 2.2). Some examples and counter-examples which will be useful in the next section, are also constructed and explained.

The main purpose of this paper, developed in Section 3, is to study the completeness of local warped semi-Riemannian metrics, specially those in which the basis and fiber are complete. The main tool is to reduce the extendibility of geodesics to the extendibility of their projections on the basis (Lemma 3.1). Thus, the con- cept of warped-completeness of (B, 9B, f ) , with (B, gB) a semi-Riemannian manifold and f > 0 a smooth function on B, is introduced in a natural way in Definition 3.3. If (B, 9B) were definite and complete, several conditions on the decreasing of f yield the (timelike, null or spacelike) warped-completeness of (B, gB, f ) , see Theorem 3.9. Examples related with this result are also shown in Remarks 3.7, 3.10 and 3.11, Example 3.12, and Counter-example 3.13. Thus, the case dim B --- 1 (which includes the Robertson-Walker relativistic space-times) is specially considered (Theorem 3.14, Corollary 3.15). The case (B, 9B) indefinite is studied in the last part of this paper. First, Proposition 2.1 is extended in Proposition 3.16. Warped incompleteness is shown to appear when (B, 9s) is indefinite (Counter-example 3.17) even under more restrictive assumptions than in Theorem 3.9. Finally, Theorem 3.19 points out a certain relation between the null warped-completeness of (B, 9B, f ) and the completeness of (1/f2).gB. This result provides a link (Remark 3.20) between the problem of the warped- completeness for compact indefinite manifolds and the open questions (i) and (ii) above.

COMPLETENESS OF SEMI-RIEMANNIAN MANIFOLDS 105

2. Semi-Riemannian Manifolds with Conformal-Killing Fields

Let (M, g) be a semi-Riemannian manifold. A tangent vector ~p to M at p E M is spacelike ifg({p, {p) > 0 or{p = 0, null ifg({p, {p) = 0 and {p • 0, and timelike ifg({p, {p) < 0. Two semi-Riemannian manifolds (M, g) and (M' , 9') are said to be conformal if there is a diffeomorphism ¢: M + M ' such that ¢*g' = #.g, where # is a smooth function on M. So, a conformally flat manifold will, in this paper, mean a semi-Riemannian manifold which is (globally) conformal to a flat one. A conformal-Killing vector field is a vector field ~ such that the Lie derivative o fg with respect to { satisfies, for a (smooth) function ~r on M:

L~g = a.g (2.1)

Note that, clearly, if ~ is a Killing vector field (or - 0) for 9 then L~ (~2.9) = ~ (f~). g for any conformal factor f~ > 0.

Next, we are going to state our main result in this section, which is specially useful in the compact case.

PROPOSITION 2.1. Let ( M, g) be a manifold with an indefinite metric of index 8. (M, g) is complete if there exist s timelike conformal-Killing vector fields El,.", ~s satisfying:

(i) the Gram matrix {g(~i, ~j)} has inverse {giJ } , and Ei,j(gij ) 2 is bounded on M;

(ii) the functions ai satisfying (2.1)for the corresponding ~i, are bounded; (iii) the associated Riemannian metric gR is complete, where gR is given by

gR(X, Y) = 9(X, Y), gR(A, B) = -g (A , B), gR(X, A) = 0

for any A, B C Span{~l~. . . , ~s) and X, Y belonging to its 9-orthogonal complement.

Proof We are going to see that any geodesic 7 : [0, b [~ M, 0 < b < oo, is extendible beyond b. By (iii) it suffices to show that 9R(7', 7') is bounded. But gR(3/, '7') = C - 2. ~i,jg z3 .g((i, "T').g((j, 7'), where C = 9(7 ~, 7') is a constant. From (i) it suffices to observe that condition (ii) implies that 9(~i, 7 ~) is bounded, 1 < i < s. Taking (2.1) into account we have:

-~ g({i, 7') = ½C(~ri o 7) (2.2)

so, (d/dt)9(~i, 3") and, as a consequence, g(~i, 7') are bounded. []

Remark 2.2. The properly discontinuous subgroups of rigid motions of L 2 consist of pure translations or their compositions with reflections on straight lines. Therefore, if a compact 2-dimensional Lorentzian manifold, which may be assumed to be time oriented, is conformally flat, then it admits a timelike conformal-Killing

106 ALFONSO ROMERO AND MIGUEL $,~NCHEZ

vector field. Hence, it is under the assumptions in Proposition 2.1 and it is complete (compare with [3]).

Completeness is also obtained if M is compact and, for every p in M, there exist 8 (= index of M) conformal-Killing vector fields which are timelike and independent at p. Thus, we have the next generalization of Marsden's theorem:

COROLLARY 2.3. A compact semi-Riemannian manifold which is conformal to a homogeneous semi-Riemannian manifold is complete.

Now, several manifolds endowed with (conformal) Killing vector fields are constructed.

EXAMPLE 2.4. (1) Consider a principal fiber bundle P ( B , G) with basis B and structural group G and let w: T P ~ ~j be any connection on P (~ denotes the Lie algebra of G). Let gB be a complete Riemannian metric on B and suppose ~ has a bi- invariant non-degenerate metric g~ (if ~ is assumed to be semisimple, we may take its Killing form). Let f > 0 be any smooth function on B such that inf ( f ) > 0, and define on P the (generalized Kaluza-Klein) metric gY = 7r~gB + ( f o ZrB )2w* g~. If we take s (= index of gg) timelike left-invariant vector fields { A l , . . . , As } in ~, then the corresponding fundamental vector fields {A~, . . . , AT} on P are Killing fields under the assumptions of Proposition 2.1, so gY is complete.

(2) Consider on ~2n the semi-Riemannian metric g of index n given, in the usual coordinate system, by gii = a, gi+n,i = gi,i+,~ = r , g~+,~,i+n = -6 , 1 < i < n, and zero for the remainder of the components of g, where a, fl and ~ only depends on the coordinates (X~+l , . . . , x2n), and satisfy (a) inf(at5 +/32) > 0, (b) a, fl and /~ are bounded and (c) sup a < 0. Note that the Killing vector fields O/Oxi, 1 <_ i < n, are under the hypothesis of Proposition 2.1 (condition (iii) holds because the infimum of the eigenvalues of the Gram-matrix {gij } is greater than 0). So, for any conformal factor f~ = exp 2w such that w and Ow/Ox~, 1 < i < n, are bounded the metric ft.g is complete. This family of metrics includes, after straightforward changes of coordinates, metrics on ~2 such as:

gl = f~(Xl, x2)(exp(2w2(x2)) dx 2 - exp(2wl(Xl)) dx22)

or

gz = f~(x,, x2)(dXl 2 - exp(2u(xm - x2)) dx2),

where the functions f~, ~;1, ~z and several of their derivatives are assumed to be conveniently bounded, and inf(u) > 0.

Remark 2.5. If (M, g) is any semi-Riemannian manifold with index s and M is assumed to be compact, then any set of s pointwise independent timelike conformal-Killing vector fields satisfies (i) to (iii) in Proposition 2.1, which guaran- tees the completeness of g. Moreover, we can allow the fields to be non-timelike or


dependent in a finite subset of points of M, and (M, g) remains complete. But this condition cannot be improved; so, recall that the Clifton-Pohl toms is incomplete even though it has a Killing vector field without any zero (there are infinite points on which is null, spacelike and timelike). Next, slight modifications in Example 2.5(2) permit us to construct incomplete compact Lorentzian manifolds such that each one admits a Killing vector field which is non-spacelike at any of its points, and timelike almost everywhere.

COUNTER-EXAMPLE 2.6. Put n = 1 in Example 2.5(2), assume that ~, /3 and 6 are periodic with commensurable periods, so, inducible in a 2-torus T, and delete condition (c), that is, suppose now that there is Y0 E I~ such that a(y0) = 0. Thus we have a null geodesic t --+ (zl(t), Y0), where x1(t) is a solution of the differential equation:

a'(Yo), ,,2~., : 0 ( 2 . 3 )

Observe that if at(yo) ~ 0 then this geodesic is incomplete. Assume that, instead of (c), the following condition is now satisfied: (c) 'a < 0, a(yo) = O, and a(t) < 0, at any t E ]Y0, Y0 + T[, where T is the period of a. In this case the Killing vector field O/Oxl is non-spacelike at any point. Although the geodesics obtained from (2.3) are now complete, we are going to check that there are incomplete geodesics. Choose a E 1R and b E ]Yo, Yo + T[ and note that for each e C { - 1 , 1} there is a null geodesic 7(t) = (xl(t), x2(t)) starting at (a, b) which satisfies:

x](t) = -~(t) + E((e(t).~(t)~(t) +/32(t))(1/z)] xi(t) (2.4a)

eII x~2(t) = (~(t)5(t) + ~z(t))O/2) (2.4b)

where t~(t), /3(t) and ~(t) mean a(x2(t)), /3(x2(t)) and 6(x2(t)), respectively, and H is the constant g(7 ~, O/Oxl). From (2.4b) and our assumptions on g we have that x2 = x2(t) is a diffeomorphism from R onto R. Now, take ~ such that e(/3 o x 21)(y0) < 0. It is easy to see, by using (2.4a), that the null geodesic obtained in this way cannot be defined at t = 0. It is not difficult to show that there are also spacelike and timelike incomplete geodesics.

3. Warped Products

Let E(B, F) be a (differentiable) fiber bundle with basis B and fiber F. Denote by 7rB: E ~ B the canonical projection and, for each fiber-chart ~: 7r~l(U) U x F (U is an open subset of B which will be called trivializing), denote by 7rF the projection onto F. Let 9B, Yr be semi-Riemannian metrics on B, F respectively,

108 ALFONSO ROMERO AND MIGUEL S,~NCFIEZ

and let f > 0 be a smooth function on B. A metric gf on E is defined to be the local warped product of gB and gF with warping function f if there is a covering of B consisting of trivializing open subsets of B such that in the corresponding open subsets of E, with the usual identifications, gf = 7r;3ge + ( f o 7rB)27r)gF. Let 7 be a curve in E. It is easy to show that 7 is a geodesic with respect to g f if and only if it satisfies:

( D ) ( C ) V B f i n B 7~ = ( f o 7B) 3

d-~ 7 ~ = - ~ ( f o T B ) 7 ~- i n F

(3.1a)

(3.1b)

(the last one in each fiber chart) where 7B = 7rB o 7, 7F = a'F o 7, D/dt denote the corresponding covariant derivatives, v B f is the gradient of f in (B, gB) and the constant C is equal to ( f o 7B)ngF(7), 7 ) ) , independent of the chosen fiber chart (compare with [7, Prop. 7.38]).

IfgB is definite we introduce r /6 { - 1, 1 } such that r/gB be always Riemannian, and II II denotes the r/gB-norm.

LEMMA 3.1. Assume that ( B, gB) and (F, gF) are semi-Riemannian, the second one complete, f > 0 a smooth function on B and (E(B, F), g f) as defined above. Let 7: [0, b[--+ E , b < do, be a geodesic of g I. The following conditions are equivalent:

(1) 7 is extendible as a geodesic past b. (2) 7B is continuously extendible to b. (3) 7~3([0, b[) lies in a compact subset of TB.

Moreover, if gB is assumed to be definite, the previous conditions are also equivalent to:

(4) 7B([0, bD lies in a compact subset of B.

Proof. Clearly (1) implies (2) and (3). To prove (2) =~ (1), observe that 7F is (for every fiber chart) a pregeodesic of gF. Now, take a trivializing open subset of B which contains limt~bTB(t), and let h be a reparametrization such that 7F o h is a geodesic. Then there exist A 6 1~ - {0} such that h' = A ( f o 7B o h) 2. So, we have that h' is bounded on [0, b[, which implies that 7F, and hence 7, are continuously extendible to b. To prove (3) ::¢, (2) we consider an arbitrary Riemannian metric g' on B. (3) implies that 7B has finite #'-length and, therefore, (2) is satisfied. Assume now that gB is definite. It is obvious that (3) =~ (4), and the converse follows easily from:

,riG (1/2) (3.2)


where D is the constant gY(7', 7'). []

So, in order to study the extendibility of 7 it is enough to consider the extendibility of the solution of equation (3. la), taking into account the value of C and the initial condition of 7B. Therefore, the next result easily follows:

PROPOSITION 3.2. Let (B, gB ) be a semi-Riemannian manifold and let f > 0 be a smooth function on B. The following conditions are equivalent:

(1) There is a complete manifold with an indefinite metric (Fo, gFo) and a fiber bundle Eo( B, Fo) such that ( Eo( B, Fo), gf ) is complete (resl~ timelike, null or spacelike complete).

(2) Every ( E( B, F), gY) is complete (resp. timelike, null or spacelike complete) for any complete manifold with indefinite metric ( F~ gF ).

If gB is assumed to be definite, condition (1) can be replaced by:

(1)' There is a complete definite manifold (Fo, gPo), (dim F0 > 0 )wi th -~gFo Riemannian, and a fiber bundle E0(B, F0) such that (E0(B, Fo), gY) is complete (resp. timelike, null or spacelike complete).

DEFINITION 3.3. Let (B, gB) be a semi-Riemannian manifold and let f > 0 be a smooth function on B. (B, gB, f ) is said to be warped-complete (resp. timelike, null or spacelike warped-complete) if every (E(B , F) , gY) is complete (resp. timelike, null or spacelike complete) for every complete semi-Riemannian manifold (F, gF). From now on, given (B, gB, f ) , a solutionTB of the differential equation (3.1a) (for some constant C) will be called a geodesic projection on (B, gB, f ) . Such a 7B will be called timelike (resp. null; spacelike) if the constant D = gB(TtB, 7~3) + C / ( f o 7B) 2 is negative (resp. 0; positive). [Observe that a constant curve in B is a null-geodesic projection.]

So, Lemma 3.1 states that completeness of geodesic projections is equivalent to warped-completeness.

A. DEFINITE BASIS

Next, we are going to consider (B, gB) definite, and will try to impose on f reasonable conditions to get (timelike, null or spacelike) warped-completeness of (B, gs, f ) .

DEFINITION 3.4. A geodesic projection 7B on (B, gB, f ) is said to be definite if ~7C _> 0, otherwise, 7B is said to be indefinite.

Next, we are going to assume that gB is complete. In this case we get, by using Lemma 3.1 (see 3.2), that any definite geodesic projection on (B, gB, f ) is complete. So, we are going to pay attention only on those that are indefinite.

110 ALFONSO ROMERO AND MIGUEL S./~NCHEZ

LEMMA 3.5. Let G:[y0, yl[---+])~, (x)[, -oo < Yo < Yl <_ c~, )~ > O, be a measurable function. I f there exists Eo in ] - )~, oc[ such that f~ l (G(y) + Eo) -(1/2) dy = co, then for each E in [-)~, oo[,

fy yl

(G(y) -t- E) (-1/2) dy = 0

DEFINITION 3.6. Let qo: [a, b[~]0, oo[, - ~ < a < b _< ~ , be a continuous function. We call it 1-complete (resp. 2-complete; 3-complete) if there is a constant

b E0 > 0 such that f : (Eo + (1/qo(y)2)) -(1/2) dy = oc (resp. f : ¢p(y) dy - co; V

is unbounded or fb qo(y) dy = oo).

Remark 3.7. (1) From Lemma 3.5, if ~ is 1-complete and M2 = lim supt._.b ~(t) 2, then for all E > ( - 1 / M 2) we have fb (E+

(1/qD(y)2))-O/2) dy = ~ , where aE E [a, oo[ is chosen such that E+(1/q~(y)2) > 0 on all [aE, oo[. The same assertion is true if E = - ( I / M 2 ) , whenever

there exists some aE under the same above assumption. In particular, f b (E + (1/qo(y)2))-(1/2) dy = o~ for all E > 0. (2) It is clear that 1-completeness ~ 2- completeness ~ 3-completeness, and that if ~p is bounded then the converses also hold. If qo is unbounded, clearly the last converse does not hold. On the other hand, even if b = o¢ we could find a 2-complete differentiable qo which is not complete. In fact, take a sequence of open intervals In = ]an, bn[, where 0 < an < bn < an+l, limn~oo an = c~ and Zn(br~ - an) = 2. Put B = U~ I~, and choose another

I ! sequence I~ = [a~, bin] such that an < a,~ < b~ < bn, and ~n(b~ - a~) = 1. Put BI = U,~ 1~ and consider the function T: B I ~ [0, 1[, T(x) = Lebesgue measure of [0, x] a B ~, for all x E B ~. Now, we can obtain the function ~ as claimed, taking any positive differentiable extension (easily built by using a partition of the unity) on [0, co[ of the functions (1 - T) -1 on B' and x ~ (1 + x2) -1 on ~ - B.

Fix x0 E B and let d: B ~ [0, oo[ be the r/gB-distance function from the point x0. If f* is a smooth function on B we define fief(r) = in f{ f* (y ) /d (y ) = r} at any r in [0, oo [. We are going to state the main result of this subsection, but first we need the next integral inequality:

LEMMA 3.8. I f p: [0, b[~ B, 0 < b < oo, is any smooth curve then:

~d d( p( s ) ) ~0 s (p(o)) f i* f ( r )dr < f*(P(S)) II P'(g)[I dg

Proof. We first remark that the continuous function d could be non-differentiable. Denote by gl(s) the left member of the above inequality, and by g2(s) the right one. Then, at any s in [0, b[:

lim sup[ gl(8 q- h ) - g l ( s ) . ] h--.0 h


lim sup [ iao p(s + h) - a o p(s)l] < fin f( do p(s)) - h--+O [ h ]

_<

where the last inequality is clear from Idop(s + h) - dop(s)I <_ f~+h N P'(S) N d~. Now the required result follows from the following generalization of the clas-

sical mean value theorem: 'Let 91, 92: [0, b[--+ ~ be two continuous functions, 91 (0) = 92(0) such that:for each sequence {h~} --+ 0 and each s in [0, b[ satisfying that the limits li = l i m n ~ [ (gi( s + hn ) - 9~( s) ) / hn], i = 1,2, exist (finite or not), we have ll <_ 12. Then gl <_ 92 in [0, b[.' []

THEOREM 3.9. Let (B, 9B) be a complete definite manifold, xo E B, and let f > 0 be a smooth function on B.

(i) I f B is compact then (B, gB, f ) is warped-complete. (ii) Assume that B is not compact, then we have."

(a) if finf is 1-complete then ( B, gB, f ) is warped-complete; (b) in the case rl = - 1 (resp. r / = 1)iffinf is 2-eomplete then (B, 9B, f ) i s

spacelike (resp. timelike) and null warped-complete; (c) in the case ~1 = - 1 (resp. ~7 = 1) if finf is 3-complete then (B, 9B, f ) is

spacelike (resp. timelike) warped-complete; (d) if f depends only on the distance from xo then the converses of (a), (b)

and (c) are also true.

Proof. (i) Trivial from Lemma 3.1(4). (ii) Let 7B: [0, b[-+ B, b < ~ , be an indefinite geodesic projection on (B, 9B, f ) . We are going to see that 7B is extendible past b. Take f* = (17D - rl(C/f2)) (-1/2) in Lemma 3.8. Thus, from (3.2), we obtain:

/a(-y,(s)) ( E + dr _<s (3.3) ( - ~C)(-l/2) aa(~.(o)) \ finf(r) 2 ) )

where E = - ( D / C ) . According to Lemma 3.1, if 7B were inextendible to b then there would be a sequence {t~} --+ b such that {d o 7B(tn)} --+ c~. Thus, (b) and (c) are straightforward from (3.3), and statement (a) easily follows taking into account Remark 3.7(1). For (d), let p: [0, ~[--+ B be a geodesic of gB such that p(0) = x0 and d(p(s)) = s at any s E [0, oo[. Choose E0 C IR and put t(s) = f~(Eo+(1/finf(r)2))(-1/2) dr (recall t ha t f = finfod). I fs( t ) isthe inverse of the function t (s) (when it exists), then 7B (t) = p (s (t) ) is a geodesic proj ection on (B, gB, f ) with 17C = - 1 and D = 17E0. Finally, if E0 is conveniently chosen then the converses are directly obtained by using Lemma 3.5. []

Remark 3.10. We are going to show how 1- or 2-completeness of finf may depend on the chosen point x0 in B. Take B = IR and put I~ = [an, bn], 'n E Z,

112 ALFONSO ROMERO AND MIGUEL S,/~NCHEZ

where an = n + (1/21hi+l), bn = n + 1 - (1/21~1+2). Let f:II~ ---+ ~ be smooth and such that 0 < f < 1, f l ±2. = 1, f l z2n+i = 1/21~1. Now, take x0 = 0 and then clearly f0 °° liar(r) dr < c~, but i fx0 = 1/2, then we have f ~ fi~f(r)dr = ~ . On the other hand, this example also shows that if f is not a function of the distance from x0 then the converses of (a), (b) and (c) could not be true [it is not difficult to show this by using a triple (B, gB, f ) such that, for any point x0 in B, finf is not 1, 2 or 3 complete].

Remark 3.11. If the behaviour of f in f ( r ) at infinity is like rp, p >_ - 1 , then (B, gB, f ) is warped-complete, but i f p < - 1 we cannot guarantee it (compare with the Beem-Buseman's example). If in f ( f ) > 0 or, more generally, if there exists ~ > 0 such that the subset {r E [0, ~[/finf(X) > ~} has infinite measure, then finf is 1-complete and warped-completeness is achieved. This can be used to construct many complete (and incomplete) warped metrics on M s × S ~-s, where

(basis) is the negative definite s-dimensional hyperbolic space and S n-s (fiber) is a (n - s)-dimensional sphere; in particular, the pseudo-sphere S~ can be seen as one of those warped products for certain warping function f with inf ( f ) = 1, so the completeness of S~ is proved (from a quite different method to the usual one). Observe that every (conformal) Killing field on the fiber of a warped product can be seen as a (conformal) Killing field on the product manifold (compare with Section 2).

EXAMPLE 3.12. It is not difficult to find local warped products which are not (global) warped products. For example, consider the metric on the principal fiber bundle _P(B, G) given in Example 2.4(1). If the connection ~ were flat then the metric g] is a local warped metric; if P(B, G) is not a trivial bundle then this local warped metric is not a global one. [We point out that the way to find all the principal fiber bundles which allow a fiat connection can be seen as follows: Let B be a manifold and let M be a non-trivial regular covering of B with covering projection 7r: M ~ B. Remember that M is a principal fiber bundle on B with the deck transformation group Aut(Ml~ ) as structural gmp ([4, p. 61]). Now, consider a monomorphism ¢ from Aut(M[~) into a Lie group G, and for each transition function guy: U fq V -+ Aut(Ml~ ) define fluv = ~ o guy. These functions can be used as transition functions of a principal fiber bundle P(B, G), and the natural monomorphism from M into P maps the trivial connection on M into a flat connection on P ([4, p. 79]). Conversely, if P(B, G) is a principal fiber bundle with a flat connection ~, the holonomy bundle through uo E 19, P(uo), is a regular covering space of B ([4, p. 93]), which allows us to recover P(B, G) as before.]

A natural extension of warped products are twisted products. They are defined as follows: given two semi-Riemannian manifolds (B, gB), (F, gF) and a smooth function h > 0 on B × F, the twisted product (/3 × F, gh) is the product manifold B × F furnished with the metric tensor gh = 7r~3gB + h27r~gF • If (B, gB) and (F, gF) are Riemannian, B complete and F compact, then any twisted product (/3 / F, gh ) is complete.


Next, we are going to see that, even if B and F are definite and compact, a twisted product (B x F, gh) may be incomplete:

COUNTER-EXAMPLE 3.13. Let B be the unit circle in C, and take F = B, gB the usual metric on B, and gF = --gB. Consider h: B x F --+ ~ given by h(exp(i01), exp(i02)) = exp(sen(Ol - ~2)) (compare with the metric 92 in Exam- ple 2.4(2)). A direct computation shows that this twisted product is incomplete (nevertheless, we can see it by choosing as new coordinates ¢i = ½(t91 + v~2), x2 = 1(01 - ~2), and comparing with Counter-example 2.6).

Although (B, gB) was assumed to be complete in Theorem 3.9, several of the ideas there developed can be used to study the case (B, gB) incomplete. So, we are going to see the case dim B = 1. Observe previously that, from (3.1a), any geodesic projection 713 is constant if and only if there is 8 such that 7~(8) --- 0 and VB f(7/3 (s)) = 0, or 7~ (8) = 0 and C = 0. Therefore, if 7/3 is not constant then the set of zeroes of 7~ consists of isolated points. Recall now that if dim B = 1 then B is either S 1 (and the warped-completeness of (B, 9B, f ) is then trivial for all f ) or (B, rl9/3 ) is isometric to an interval ]c, d[, - e e _< c < d _< oo, with its usual metric. Thus, we are going to consider the last case with ~ = - 1 (so, the results can be more easily applicated to relativistic space-times). For a (indefinite) geodesic projection with - D < 0 and initial condition through so, if there is z2+ > 7B(s0) (resp. 9- < 7B(80)) such that D / C = 1 / f (~+) 2 (resp. D / C = 1 / f ( f t - )2 ) then 7/3 cannot be extended beyond 9+ (resp. ~_ ). This easily follows if f increases at .~+ (resp. decreases at ~_ ). On the other hand, if f has a local maximum at ~3+ (resp. at zj_ ) then the range of 7B cannot reach the point if+ (resp. if_), otherwise, as reasoned before, 7/3 is constant. Now, fix a non-constant inextendible geodesic projection 7B and a point so such that 7~(s0) ~ 0. Let C, D be the constants considered up till now, and put A+ = {y E]e, d[/TB(s0 ) < y, C / f ( y ) 2 = D}, A_ = {y E ]e, d[/TB(SO) > Y, C / f ( p ) 2= D}. We take

y+ = m i n ( A + ) i f A T ~ ( ~ , and y + = d i f A + = 0 (3.3a)

y_ = m i n ( A _ ) i f A ~ , and y_ = c i f A _ = ¢ (3.3b)

The announced results in the case B = ]c, d[ are summarized as follows:

THEOREM 3.14. Let 7/3 be an inextendible geodesic projection on (B = ]c, d[, g/3, f ) defined on an interval I, and let s~ E I .

(i) 7B is constant (and hence complete) if and only i f T~ ( s 1 ) = 0 and f ' (7/3 (s 1 ) ) = 0, or 7~(s l ) = O a n d C = O.

(ii) If TB is not constant, let sO E I ande E { - 1 , 1} such that e.'l'~(So) > O, and choose y+ , y_ as in (3.3). We have: (la) If y_ = c then 7B is extendible to -e .oo i f and only if."

~'YB(~°) ( - D + ( f@y)2) ) - (1 /2) dy = oo (3.4a)

114 ALFONSO ROMERO AND MIGUEL S,~NCHEZ

(lb) If y+ = d then 7B is extendible to s.oo if and only if"

~d (_D+ ( C ))-(1/2) B(~o) ~ dy = oo (3.4b)

(2) 7B is complete if and only if f satisfies: if y_ = e then (3.4a), if y+ = d then (3.4b) (so, there is no condition if c < y_, y+ < d)

(3) The range of 7 B is contained in [y_, y+], it contains ]y_, y+[, and y+ (resp. y_) belongs to it if and only if y+ < d, fl(y+) ~ 0 (resp. c < y_, f l(y_ ) ~ 0). Moreover, 7B is a diffeomorphism onto its image if and only if this image is ]y_, y+[.

Fix e C ]c, d[ and define f+:[e, d [~ ~, f - : [ - e , - c [ ~ ~, f+(x) = f(x), y - ( y ) = f ( - y ) , for all x, y in the corresponding domains. Observe that the 1-, 2- or 3-completeness of f+ and f - do not depend on the chosen point e in ]e, d[. f will be called 1-complete (resp. 2-complete, 3-complete) if f+ and f - are 1-complete (resp. 2-complete, 3-complete) in the sense of Definition 3.6. We end this subsection with the following consequence of the last theorem (compare with [1, Th. 2.57], and [7, p. 356]).

COROLLARY 3.15. (1) Each of the following conditions is equivalent to the timelike warped-completeness of(B, gB, f ) :

(a) (B, gB, f ) is warped-complete. (b) f is 1-complete. (c) There is a timelike indefinite geodesic projection which is complete.

Moreover, if B ~ ~ then there are neither indefinite geodesic projections nor timelike geodesics tangent to B which are complete.

(2) Each of the following conditions is equivalent to the null warped-completeness of(B, gB, f ) :

(a) ( B, gB, f ) is null and spacelike warped-complete. (b) f is 2-complete. (c) There is a non-constant null geodesic projection which is complete.

(3) (B, gB, f ) is spacelike warped-complete if and only if f is 3-complete.

Note that if f is bounded then all the previous conditions are equivalent.

B. INDEFINITE BASIS

In the last part of this paper we are going to analyze what we can say on the warped-completeness of (B, gB, f ) if (B, gB) is an indefinite manifold. First we have the next generalization of Proposition 2.1, which is also specially useful in the compact case:


PROPOSITION 3.16. Let ( B , gB ) be an indefinite manifold and {El , . . . , ~s} as in Proposition 2.1. Then for all smooth function f on B with inf(f) > 0 and (~(f) bounded, 1 < i < s, (B, 9B, f ) is warped-complete.

Proof Let 713: [0, b[--+ B, b < 0% be a geodesic projection on (B, 9/3, f ) . From (2.1) and (3.1a) we have:

g/3(~i, 7b) - ( f o,./B)3

for all i, 1 < i < s, which is bounded on [0, b[. So, the result follows by reasoning as in Proposition 2.1, and using Lemma 3.1. []

Nevertheless, if (B, 9/3) is indefinite we cannot guarantee warped-completeness of (B, 9/3, f ) under the assumptions of Theorem 3.9, even if f satisfies 0 < inf(f) < sup(f) < oo (see Remark 3.11). The next counter-example also shows that the condition ( i ( f ) bounded in Proposition 3.16 is necessary.

COUNTER-EXAMPLE 3.17. Let 90 be the usual Lorentz metric on R2. Let u, h: R --+ ~ be two smooth functions such that: (i) u~(0) ~ 0, (ii) h > 0, and (iii) f0 °° h(y) -1/2 dy < c~. We are going to show that, setting f ( z l , z2) = exp(u(h(zl ) - h(z2))), the triple (R2, 90, f ) is not warped-complete. In order to achieve it observe that the non-constant geodesic projections 7e2(t) = (Zl(t), x2(t)), t E I, 0 E I, on (~2, 90, f ) with xl(t) = x2(t) satisfy f~l(o)(B +

Ah(~)) -1/2 d~ = t(x), where A = 2C. u'(0) exp(-2u(0)) , and B = x](0) z - Ah(zl(O)). Now, put x~(0) # 0, choose conveniently C and then 7~2(t) cannot be defined on all ~.

Remark 3.18. Consider u: IR --+ iR periodic with u~(0) # 0, h the identity function on JR, and construct f as in Counter-example 3.17 (clearly, h no longer satisfies the crucial condition (iii) above). Not only the metric 9o but also the warping function f can be induced in a two-toms T 2. The warped-completeness of the corresponding triple, (T 2, g0, f ) , is guaranteed by Proposition 3.16. If we consider now a non-constant geodesic projection 7/3 on ( ~ z g0, f ) with xl(t) = xz(t) as above, it induces a (complete) geodesic projection ~/3 on T 2. It is easy to see that "~ (~) does not lie in any compact subset of T T 2. [Remember that if an inextendible geodesic of a semi-Riemannian manifold is incomplete then its velocities do not lie in any compact subset of the tangent bundle. Thus, constructing local warping products on (T 2, 90, f ) with a compact complete fiber (F, 9F)(--u'(O).gV non- Riemannian), we obtain that the converse is not true, even in compact manifolds.]

As announced in Section 1, we point out a certain relation between the null-warped completeness of a triple (B, 9/3, f ) and the completeness of the metric (1/f2).9/3 on B.

THEOREM 3.19. Let (B, 9/3) be a semi-Riemannian manifoldand let f > 0 be a smooth function on B.

116 ALFONSO ROMERO AND MIGUEL SANCHEZ

(1).If s u p ( f ) < c~ and (B, gB, f ) i s null warped-complete then (1/f2) .gB is complete.

(2) I f i n f ( f ) > 0 and (1/ f2) .gB is complete, then (B, gB, f ) is null warped- complete.

Proof First, take any smooth function w on B and put g* = e 2w gB. Consider a geodesic a ( t ) of g* and let 6~(s) = a(t(s)) be a reparametrization of o~ such that dt/d8 = k. exp(2w(~(8 ) ) ) with k a non-zero constant. A direct computation shows

(3.5)

where C' = g*(a ' , a ' ) . Now, take e 2~ = 1 / f 2 in (3.5). Comparing the new formula with (3.1a), and taking into account that the assumptions on f imply bounds for the reparametrizations of ~, the theorem is easily proved. []

It should be noted that if we take 0 = 0 in (3.5) then the assertion (f) in Section 1 is fol lowed now.

Remark 3.20. Suppose now that (B , gB) is compact. From Theorem 3.19 it fol lows that (B , gB, f ) is nul [warped-comple te if and only if (1/ f2) .gB is complete. Therefore, if (B , #B) is also complete but (B, gB, f ) is not null warped-complete then an affirmative answer to (ii), Section 1, is found. It seems natural to ask if: (iii) there exists a warped-incomplete triple (B , gB, f ) such that (B , gB) is (indefinite), complete and compact. So, an affirmative answer to (ii), Section 1, would imply an affirmative answer to (iii), and, on the other hand, for (f), Section 1, an affirmative answer to (iii) would imply a negative answer to (i), Section 1.

Acknowledgement

The authors would like to express their thanks to Y. Carri~re for his valuable comments on Section 2.

References

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3. Furness, P. M. D. and Fedida, E.: Sur les structures pseudo-riemannienes plates des vari6t6s compactes, J. Nigerian Math. Soc. 5 (1986), 63-78.

4. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. I, Wiley Interscience, 1963.

5. Lafuente, J.: A geodesic completeness theorem for locally symmetric Lorentz manifolds, Rev. Mat. Univ. Complutense Madrid 1 (1988), 101-110.


6. Marsden, J.: On completeness of homogeneous pseudo-Riemannian manifolds, Indiana Univ. Math. 22 (1973), 1065-1066.

7. O'Neill, N.: Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983. 8. Romero, A. and S~inchez, M.: On completeness of geodesics obtained as a limit, Z Math. Phys.

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On completeness of certain families of semi-Riemannian manifolds

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