https://doi.org/10.5831/HMJ.2018.40.3.447
NONCONSTANT WARPING FUNCTIONS ON EINSTEIN LORENTZIAN WARPED PRODUCT
MANIFOLDS
Yoon-Tae Jung, Eun-Hee Choi and Soo-Young Lee∗
Abstract. In this paper, we consider nonconstant warping func- tions on Einstein Lorentzian warped product manifolds M = B ×f2
F with an 1−dimensional base B which has a negative definite met- ric. As the results, we discuss that on M the resulting Einstein Lorentzian warped product metric is a future (or past) geodesically complete one outside a compact set.
1. INTRODUCTION AND PRELIMINARIES
In [1], A.L. Besse, the author study a new compact Einstein mani- fold using the warped product. Then the author asked the following:
“Does there exist an Einstein warped product manifold with noncon- stant warping function?”
Let M = B ×f2 F be an Einstein warped product space. In [5] and [7], the authors proved that if B is a compact, then M is simply a Riemannian product space. In [6], the author proved that there does not exist a compact Einstein warped product space with nonconstant warping function, if the scalar curvature is nonpositive or the base is of 2−dimensional.
In this paper, we consider the following question:
Question If the base manifold B is 1−dimensional, does there ex- ist an Einstein Lorentzian warped product manifold with nonconstant warping function such that the resulting metric is a future (or past) geodesically complete one outside a compact set?
Received January 31, 2018. Revised June 5, 2018. Accepted June 20, 2018.
2010 Mathematics Subject Classification. 53C15, 53C21, 53C25, 58D17
Key words and phrases. warping function, Lorentzian warped product manifold, scalar curvature, Einstein manifold, Ricci curvature, Ricci tensor.
The first author was supported by Chosun University Research Fund 2017.
*Corresponding author
Definition 1.1. Let (B, gB) and (F, gF) be two manifolds. Let gB
be a metric tensor of B and gF be a metric tensor of F. We denote by π and σ the projections of B × F onto B and F, respectively.
For a positive smooth function f on B the warped product manifold M = B ×f2 F is the product manifold M = B × F furnished with the metric tensor g defined by g = π∗(gB) + (f ◦ π)2σ∗(gF). We denote by π∗ and σ∗ the pullback π and σ, respectively. Here B is called the base of M and F the fiber([3]).
Proposition 1.2 (See Proposition 9.106 in [1], p.266). The Ricci curvature Ric of the warped product manifold M = B ×f2 F satisfies
(i) Ric(V, W ) = RicF(V, W ) + g(V, W ) ∆f
f − (p − 1)||df ||2 f2
π
, (ii) Ric(X, V ) = 0,
(iii) Ric(X, Y ) = RicB(X, Y ) − p
fHf(X, Y )
for any vertical vectors V, W and any horizontal vectors X, Y, where Hf and ∆f denote by the Hessian of f and the Laplacian of f for gB. We denote by RicF and RicB the Ricci curvatures of (F, gF) and (B, gB), respectively. We denote by RicB and RicF the lifts to M of Ricci curvatures of B and F, respectively. Let p be a dimension of F ([8]).
Corollary 1.3 (See Corollary 9.107 in [1], p.267). The warped prod- uct M = B ×f2 F is Einstein manifold (with Ric = λg) if and only if gF, gB and f satisfy
(i) (F, gF) is Einstein (with RicF = λ0gF), (ii) ∆f
f − (p − 1)kdf k2 f2 +λ0
f2 = λ, (iii) RicB− p
fHf = λgB.
Proof. Obviously, (i) gives a condition on (F, gF) alone, whereas (ii) and (iii) are two differential equations for f on (B, gB)([1]).
Remark 1.4. Using Corollary 1.3 (ii) and (iii), we denote by q = dimB and dimF = p > 1. Then we may replace the unique equation (1.1) RicB−p
fHf =1
2[ sB+ 2p∆f
f − p(p − 1)kdf k2 f2 + pλ0
f2− (p + q − 2)λ ]gB, where sB is a scalar curvature of B.
Remark 1.5. Theorem 5.5 in [9] implies that all timelike geodesics are future (resp. past) complete on (−∞, +∞) ×v N if and only if
R+∞
t0
v 1+v
12
dt = +∞ (resp. Rt0
−∞
v 1+v
12
dt = +∞) for some t0. On the other hand, Remark 2.58 in [2] implies that all null geodesics are future (resp. past) complete if and only if R+∞
t0 v12dt = +∞ (resp.
Rt0
−∞v12dt = +∞) for some t0 (cf. Theorem 4.1 and Remark 4.2 in [4]. In this reference, the warped product metric is g0= −dt2+ v(t)g).
In this paper, we study equation (1.1). Recalling that ∆f = −trace(Hf) and kdf k2 = −f0 2 on B = (a, b) with the negative definite metric gB = −dt2, then equation (1.1) implies that we have equation
(1.2) f02= λ
pf2− λ0
p − 1.
From now on, we study the nonconstant solution of equation (1.2).
Above all, in case that if f is a constant, then f0 = 0. Hence we have f = ±
s pλ0
(p − 1)λ, where λ p · λ0
p − 1 > 0. It is trivially a simple product manifold. In case that if f is a nonconstant, then we consider the following sections depending on the signs of λ0 and λ.
2. FIBER MANIFOLD WITH λ0 > 0
In this section, we study the solution of equation (1.2) for λ0 > 0.
Theorem 2.1. Suppose that λ0 > 0. If λ is a constant, then there exists a solution f of equation (1.2):
(i) For λ > 0, f = s
pλ0
(p − 1)λcosh(±
s λ p t +
s λ
p c), where c is a constant.
(ii) For λ ≤ 0, there does not exist a solution of equation (1.2).
Proof. (i) For λ > 0. We changed the equation (1.2) and rewrite as,
Z df
qλ
pf2−p−1λ0
= ± Z
dt, (2.1)
Putting λ
p = a > 0 and λ0
p − 1 = b > 0. Equation (2.1) implies that we have
Z df
paf2− b = ± Z
dt.
By using trigonometric substitution, f =
√
√b
asec θ, df =
√
√b
asec θ tan θdθ, then we obtain Z
√1
asec θdθ = ± Z
dt.
Upon integration, we have ln | sec θ + tan θ | = ±√
a t +√
a c, where c is constant. Then we become
ln |√
a f +p
af2− b| = ±√
a t +√
a c + ln
√ b, where c is a constant. Thus we obtain f =
√
√b
acosh(±√
a t +√ a c) , where c is a constant. Therefore we have nonconstant warping function
f = s
pλ0
(p − 1)λcosh(±
s λ p t +
s λ p c), where c is a constant.
(ii) For λ ≤ 0, Equation (1.2) implies that f02 < 0, which is a contra- diction. Hence there does not exist a solution of equation (1.2).
From the results of Theorem 2.1, we have the following corollary.
Corollary 2.2. Suppose that λ0 > 0 and λ > 0. Then there exist an Einstein Lorentzian warped product metric on M with a nonconstant warping function such that the resulting metric is a future (or past) geodesically complete one.
Proof. For λ > 0 and λ0 > 0, Theorem 2.1 implies that there exists a nonconstant warping function f (t) =
s pλ0
(p − 1)λcosh(±
s λ p t +
s λ p c) on (−∞, ∞), where c is a constant. Since f (t) → ∞ as t → ±∞ on (−∞, ∞), for some t1,
Z +∞
t1
f (t)2 1 + f (t)2
12 dt ≥
Z +∞
t1
√1
2 dt = +∞
and Z +∞
t1
f (t)dt ≥ Z +∞
t1
1 dt = +∞.
Therefore Remark 1.5 implies that the resulting metric is a future geodesi- cally complete one.
On the other hand, by similar methods, the resulting metric is a past geodesically complete one. Thus on M the resulting metric is future and past geodesically complete.
For special case with λ0 = p − 1, λ = p, and c = 0, we have f (t) = cosh(±t ).
3. FIBER MANIFOLD WITH λ0 = 0
In this section, we study the solution of equation (1.2) for λ0 = 0.
Theorem 3.1. Suppose that λ0 = 0. If λ is a constant, then there exist solutions f of equation (1.2):
(i) For λ > 0, f = e±
qλ p t+c
, where c is a constant.
(ii) For λ = 0, f = c, where c is a constant.
(iii) For λ < 0, there does not exist a solution of equation (1.2).
Proof. For λ0= 0, equation (1.2) implies that
(3.1) f02 = λ
pf2. (i) For λ > 0, we have f0= ±
s λ
p f . Multiplying both sides of equation by 1
f and an integration gives ln |f | = ± s
λ
p t + c, where c is constant.
Therefore we have f = e±
qλ p t+c
, where c is a constant.
(ii) For λ = 0. Equation (3.1) implies that f02 = 0 then f0 = 0.
Upon integration, we have f = c, where c is a constant.
(iii) For λ < 0, equation (3.1) implies that f02 < 0, which is a contradiction. Thus there does not exist a solution of equation (3.1).
From the above results, we have the following corollary.
Corollary 3.2. Suppose λ0 = 0 and λ > 0. Then there exist an Einstein Lorentzian warped product metric on M with a nonconstant warping function such that the resulting metric is a future (or past) geodesically complete one.
Proof. For λ0 = 0 and λ > 0, Theorem 3.1 implies that there exists a nonconstant warping function f (t) = e
qλ p t+c
on (−∞, ∞), where c is a constant. Since f (t) → ∞ as t → ∞, for some t1,
Z +∞
t1
f (t)2 1 + f (t)2
12 dt ≥
Z +∞
t1
√1
2 dt = +∞
and
Z +∞
t1
f (t)dt ≥ Z +∞
t1
1 dt = +∞.
Hence Remark 1.5 implies that the resulting metric is a future geodesi- cally complete, but not past geodesically complete.
On the other hand, if we choose f (t) = e−
qλ p t+c
on (−∞, ∞), where c is a constant, then we prove that the resulting metric is past geodesically complete, but not future geodesically complete.
The following example shows that our results are similar with well- known special cases for Einstein warped product manifold [1].
Example 3.3. From above theorem 3.1, we have nonconstant warp- ing functions of equation (1.2) depending on special constant.
For λ0 = −0, λ = p, and c = 0, we have f (t) = e±t.
4. FIBER MANIFOLD WITH λ0 < 0
In this section, we consider equation (1.2) for λ0 < 0.
Theorem 4.1. Suppose that λ0 < 0. If λ is a constant, then there exist solutions f of equation (1.2)
(i) For λ > 0, f = s
−pλ0
(p − 1)λsinh(±
s λ p t +
s λ
p c), where c is a constant.
(ii) For λ = 0, f = ± s
−λ0
p − 1 t + c, where c is constant.
(iii) For λ < 0, f = s
pλ0
(p − 1)λsin(±
s
−λ p t +
s
−λ
p c), where c is a constant.
Proof. (i) For λ > 0. Putting λ
p = a > 0 and − λ0
p − 1 = b > 0. By a proof similar to Theorem 2.1 (i), equation (2.1) implies that we have
Z 1
paf2+ bdf = ± Z
dt.
By using trigonometric substitution, f =
√
√b
a tan θ and df =
√
√b
a sec2θdθ, then we become
√1 a
Z
sec θdθ = ± Z
dt.
Integrating the both sides of equation, then we have ln | sec θ + tan θ | = ±√
a t +√ ac, where c is a constant. Then we obtain f =
√b
√asinh(±√
a t +√ a c), where c is a constant. Therefore we have
f =
s −pλ0
(p − 1)λsinh(±
s λ p t +
s λ p c), where c is a constant.
(ii) For λ = 0. Equation (1.2) implies that f02 = −λ0
p − 1 and f0 =
± s
−λ0
p − 1. Upon integration, then we have f = ±
s
−λ0 p − 1 t + c, where c is a constant.
(iii) For λ < 0. Putting −λ
p = a > 0 and − λ0
p − 1 = b > 0 . By a proof similar to Theorem 2.1 (i) or Theorem 4.1 (i), equation (2.1) implies that
Z df
p−af2+ b = ± Z
dt.
By using trigonometric substitution, f =
√
√b
a sin θ and df =
√
√b
a cos θdθ, then we become Z 1
√adθ = ± Z
dt.
Integrating the both sides of equation, we have θ = ±√
a t +√ a c, where c is a constant. Then we obtain f =
√a
√
b sin(±√
a t +√ a c), where c is a constant. Therefore we have
f = s
pλ0
(p − 1)λsin(±
s
−λ p t +
s
−λ p c), where c is a constant.
Remark 4.2. From results of Theorem 4.1, we have nonconstant warping functions on B as follows:
(i) For λ0 < 0 and λ > 0, f (t) = s
−pλ0
(p − 1)λsinh(
s λ p t +
s λ p c) on (−c, ∞) or f (t) =
s −pλ0
(p − 1)λsinh(−
s λ p t +
s λ
p c) on (−∞, c), where c is a constant.
(ii) For λ0 < 0 and λ = 0, f (t) = s
−λ0
p − 1 t + c on (− c q−λ0
p−1
, ∞)
or f (t) = − s
−λ0
p − 1 t + c on (−∞, c q−λ0
p−1
), where c is constant.
(iii) For λ0 < 0 and λ < 0, f (t) = s
pλ0 (p − 1)λsin(
s
−λ p t +
s
−λ p c) on ( 2nπ
q−λ p
−c, 2nπ + π q−λ
p
−c) or f (t) = s
pλ0
(p − 1)λsin(−
s
−λ p t +
s
−λ p c) on (c − 2nπ + π
q−λ p
, c − 2nπ q−λ
p
), where c is a constant and n is an integer.
We discuss the completeness on (a, ∞), where a is a positive constant.
Corollary 4.3. Suppose λ0 < 0 and λ ≥ 0. Then there exist an Einstein Lorentzian warped product metric on M with a nonconstant warping function on (a, ∞) such that the resulting metric is a future geodesically complete one.
Proof. For λ0 < 0 and λ > 0, Theorem 4.1 implies that there exists a nonconstant warping function f (t) =
s −pλ0
(p − 1)λsinh(
s λ p t +
s λ p c), where c is a constant. Since f (t) → ∞ as t → ∞, for some t1,
Z +∞
t1
f (t)2 1 + f (t)2
12 dt ≥
Z +∞
t1
√1
2 dt = +∞
and
Z +∞
t1
f (t) dt ≥ Z +∞
t1
1 dt = +∞.
Therefore Remark 1.5 implies that the resulting metric is a future geodesically complete one.
And for λ0 < 0 and λ = 0, Theorem 4.1 implies that there exists nonconstant warping function f (t) =
s
−λ0
p − 1 t+c, where c is a constant.
Since f (t) → ∞ as t → ∞, for some t1, Z +∞
t1
f (t)2 1 + f (t)2
12 dt ≥
Z +∞
t1
√1
2 dt = +∞
and
Z +∞
t1
f (t) dt ≥ Z +∞
t1
1 dt = +∞.
Thus Remark 1.5 implies that the resulting metric is a future geodesi- cally complete one.
Remark 4.4. By a proof similar to Corollary 4.3, we have similar results about the past geodesically completeness on (−∞, b). And for λ0 < 0 and λ > 0, we can discuss the existence of nonconstant warping function on only a finite interval B = (a, b).
The following examples show that our results are similar with well- known special cases for Einstein warped product manifold [1].
Example 4.5. From above theorem 4.1, we have nonconstant warp- ing functions of equation (1.2) depending on special constant.
(i) For λ0= −(p − 1) and λ = p, f (t) = sinh(±t + c), (ii) for λ0 = −(p − 1) and λ = 0, f (t) = ±t + c,
(iii) for λ0 = −(p − 1) and λ = −p, f (t) = sin(±t + c), where c is a constant.
References
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[3] J.K. Beem, P.E. Ehrlich and K.L. Easley, Global Lorentzian Geometry (2nd ed.), Marcel Dekker, Inc., New York(1996).
[4] J.K. Beem, P.E. Ehrlich and Th.G. Powell, Warped product manifolds in relativ- ity, Selected Studies (Th.M.Rassias, eds.), North-Holland, 1982, 41-56.
[5] D.-S. Kim, Einstein warped product spaces, Honam Mathematical J. 22(1) (2000), pp.107-111.
[6] D.-S. Kim, Compact Einstein warped product spaces, Trends in Mathematics, Information center for Mathematical Sciences, 5(2), December (2002), 1-5.
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[8] B. O’Neill, Semi-Riemannian Geometry, Academic, New York(1983).
[9] T.G. Powell, Lorentzian manifolds with non-smooth metrics and warped products, ph. D. thesis, Univ. of Missouuri-Columbia(1982).
Yoon-Tae Jung
Department of Mathematics, Chosun University, Kwangju 61452, Korea.
E-mail: [email protected] Eun-Hee Choi
Department of Mathematics, Chosun University, Kwangju 61452, Korea.
E-mail: [email protected] Soo-Young Lee
Department of Mathematics, Chosun University, Kwangju 61452, Korea.
E-mail: [email protected]
