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2015년 8월
교육학석사(수학)학위논문
The Completeness of Some Metrics on Lorentzian Warped
Product Manifolds with Fiber Manifold of Class (B)
조선대학교 교육대학원
수학교육전공
이 정 미
The Completeness of Some Metrics on Lorentzian Warped
Product Manifolds with Fiber Manifold of Class (B)
- 엽다양체가 (B)형인 로렌쯔 휜곱다양체위의 거리의 완비성 -
2015년 8월
조선대학교 교육대학원
수학교육전공
이 정 미
The Completeness of Some Metrics on Lorentzian Warped
Product Manifolds with Fiber Manifold of Class (B)
지도교수 정 윤 태
이 논문을 교육학석사(수학교육)학위 청구논문으로 제출함.
2015년 4월
조선대학교 교육대학원
수학교육전공
이 정 미
이정미의 교육학 석사학위 논문을 인준함.
심사위원장 조선대학교 교수 김 남 권 (인) 심사위원 조선대학교 교수 김 진 홍 (인) 심사위원 조선대학교 교수 정 윤 태 (인)
2015년 6월
조선대학교 교육대학원
CONTENTS
국문초록
1. Introduction ……… 1
2. Preliminaries ……… 3
3. Main results ………11
References ………19
국 문 초 록
The Completeness of Some Metrics on Lorentzian Warped Product Manifolds with Fiber Manifold of
Class (B)
이 정 미
지도교수 : 정 윤 태
조선대학교 교육대학원 수학교육전공
미분기하학에서 기본적인 문제 중의 하나는 미분다양체가 가지고 있는 곡률 함수 에 관한 연구이다.
연구방법으로는 종종 해석적인 방법을 적용하여 다양체 위에서의 편미분방정식을 유도하여 해의 존재성을 보인다.
Kazdan and Warner ([8], [9], [10])의 결과에 의하면 위의 함수 가 위 의 Riemannian metric의 scalar curvature가 되는 세 가지 경우의 타입이 있는 데 먼저
(A) 위의 함수 가 Riemannian metric의 scalar curvature이면 그 함수 가 적당한 점에서 일 때이다. 즉, 위에 nagative constant scalar curvature를 갖는 Riemannian metric이 존재하는 경우이다.
(B) 위의 함수 가 Riemannian metric의 scalar curvature이면 그 함수 가 항등적으로 ≡ 이거나 모든 점에서 인 경우이다. 즉, 위에서 zero scalar curvature를 갖는 Riemannian metric이 존재하는 경우이다.
(C) 위의 어떤 라도 scalar curvature를 갖는 Riemannian metric이 존재하 는 경우이다.
본 논문에서는 다양체 이 (B)에 속하는 compact Riemannian manifold 일 때, Lorentzian warped product manifold인 ∞ × 위에 함수 가 적당한 조건을 만족하면 Riemannian warped product metric의 scalar curvature 가 될 수 있는 지를 보였다. 우리의 방법은 상해∙하해 방법을 이용하여 증명하였다.
1
1. Introduction
In [11, 12], M.C. Leung have studied the problem of scalar curvature functions on Riemannian warped product manifolds and obtained partial results about the existence and the nonexistence of Riemannian warped metric with some prescribed scalar curvature function. In this paper, we also study the existence and the nonexistence of Lorentzian warped metric with prescribed scalar curvature functions on some Lorentzian warped product manifolds.
By the results of Kazdan and Warner ([8, 9, 10]), if is a compact Riemannian -manifold without boundary, ≥ , then belongs to one of the following three categories:
(A) A smooth function on is the scalar curvature of some Riemannian metric on if and only if the function is negative somewhere.
(B) A smooth function on is the scalar curvature of some Riemannian metric on if and only if the function is either identically zero or negative somewhere.
(C) Any smooth function on is the scalar curvature of some Riemannian metric on .
This completely answers the question of which smooth functions are scalar curvatures of Riemannian metrics on a compact manifold .
In [8, 9, 10], Kazdan and Warner also showed that there exists some obstruction of a Riemannian metric with positive scalar curvature (or zero scalar curvature) on a compact manifold.
In [11, 12], the author considered the scalar curvature of some Riemannian warped product and its conformal deformation of warped
2
product metric. And also in [6], authors considered the existence of a nonconstant warping function on a Lorentzian warped product manifold such that the resulting warped product metric produces the constant scalar curvature when the fiber manifold has the constant scalar curvature.
Ironically, even though there exists some obstruction of positive or zero scalar curvature on a Riemannian manifold, results of [6], say, Theorem 3.1, Theorem 3.5 and Theorem 3.7 of [6] show that there exists no obstruction of positive scalar curvature on a Lorentzian warped product manifold, but there may exist some obstruction of negative or zero scalar curvature.
In this paper, when is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future complete Lorentzian metrics on ∞ × with specific scalar curvatures, where is a positive constant. And we prove the existence of warping functions on Lorentzian warped product manifolds and the completeness of the resulting metrics with some prescribed scalar curvatures. The results of this paper are extensions of the results of Theorem 2.6, Corollary 2.7 and Corollary 3.7 in [7].
3
2. Preliminaries
First of all, in order to derive a partial differential equation, we need some definitions of connections, curvatures and some results about warped product manifolds.
Definition 2.1. Let ℱ denote the set of all smooth vector fields defined on , and let ℭ denote the ring of all smooth real-valued functions on
.
A connection
∇
on a smooth manifold is a function∇ ℱ × ℱ → ℱ
such that
(D1)
∇
isℭ
- linear in , (D2)∇
isℝ
- linear in ,(D3)
∇
∇
forf ∈ ℭ M
(D4)
∇
∇
, and(D5)
∇
∇
for all ∈ ℱ
.If
∇
satisfies axioms (D1)~(D3), then∇
is called the covariant derivative of with respect to for the connection∇
. If, in addition,∇
satisfies axioms (D4)~(D5), then∇
is called the Levi - Civita connection of , which is characterized by the Koszul formula ([13]).
A geodesic → is a smooth curve of such that the tangent
4
vector ′ moves by parallel translation along . In order words, is a geodesic if
(2.1)
∇
′′
(geodesic equation).A pregeodesic is a smooth curve which may be reparametrized to be a geodesic. Any parameter for which is a geodesic is called an affine parameter. If and are two affine parameters for the same pregeodesic, then for some constants ∈ ℝ A pregeodesic is said to be complete if for some affine parameterizion (hence for all affine parameterizations) the domain of the parametrization is all of ℝ.
The equation
∇
′′
may be expressed as a system of linear differential equations. To this end, we let be local coordinates on and let { ∂
∂ ∂
∂ } denote the natural basis
with respect to these coordinates.
The connection coefficients of
∇
with respect to are defined by∇
(connection coefficients).
Using these coefficients, we may write equation (2.1) as the system
(geodesic equations in coordinates).
Definition 2.2. The curvature tensor of the connection
∇
is a linear transformation valued tensor in Hom ℱ ℱ defined by :5
∇
∇
∇
∇
∇
. Thus, for ∈ℱ , ∇
∇
∇
∇
∇
.It is well-known that at depends only upon the values of
and at ([13]).
It ∈ is a cotangent vector at and ∈ are tangent vectors at , then one defines
for and smooth vector fields extending and , respectively.
The curvature tensor is a (1,3) - tensor field which is given in local coordinates by
⊗⊗⊗,
where the curvature components are given by
.Notice that and
Furthermore, if
and
then
6 and
Consequently, one has
Definition 2.3. From the curvature tensor , one nonzero tensor (or its negative) is obtained, by contraction. It is called the Ricci tensor. Its components are
The tensor is symmetric and its contraction
is called the scalar curvature ([1],[2],[3]).
Definition 2.4. Suppose Ω is a smooth, bounded domain in ℝ, and let : Ω × ℝ→ℝ be a Caratheodory function. Let ∈ Ω be given. Consider the equation
∆ ∈ Ω
on Ω
∈ Ω is a (weak) sub-solution if ≤ on Ω and
Ω
∇∇
Ω
≤ for all ∈∞Ω ≥
Similarly, ∈ Ω is a (weak) super-solution if in the above the reverse inequalities hold.
We briefly recall some results on warped product manifolds. Complete details may be found in ([2]) or ([13]). On a semi-Riemannian product manifold × , let and be the projections of × onto and , respectively, and let be a smooth function on
7
Definition 2.5. The warped product manifold × is the product manifold × furnished with metric tensor
∘
where and are metric tensors of and and and are pullback of and , respectively. In other words, if is tangent to at , then
Here is called the base of and the fiber ([13]).
We denote metric g by . In view of the following Remark 2.6 (1), we may also denote the metric by . The metric will be denoted by .
Remark 2.6. Some well known elementary properties of the warped product manifold × are as follows:
(1) For each ∈ the map × is an isometry onto .
(2) For each ∈ the map × is a positive homothetic map onto with homothetic factor
.
(3) for each ∈ , the horizontal leaf × and vertical fiber × are orthogonal at .
(4) The horizontal leaf × is a totally geodesic submanifold of
and the vertical fiber × is a totally umbilic submanifold of . (5) If ∅ is an isometry of , then × ∅ is an isometry of , and if is
8
an isometry of such that ∘ , then × is an isometry of .
Recall that vectors tangent to leaves are called horizontal and vector tangent to fibers are called vertical. From now on, we will often use a natural identification
× ≅ × ≅ ×
The decomposition of vectors into horizontal and vertical parts plays a role in our proofs. If is a vector field on , we define at by setting Then is -related to and -related to the zero vector field on . Similarly, if is a vector field of is defined by .
Lemma 2.7. If is a smooth function on , then the gradient of the lift
∘ of to is the lift to of gradient of on .
Proof. We must show that ∘ is horizonal and -related to
on . If is vertical tangent vector to , then
∘ ∘ , since .
Thus ∘ is horizonal. If is horizonal,
∘ ∘ ∘
Hence at each point, ∘ We simplify the notations by writing for ∘ and for ∘ For a covariant tensor
on , its lift to is just its pullback under the projection
→ . That is, if is a - tensor, and if ∈
9
then = ∈ Hence if is vertical, then
on For example, if is a smooth function on , the lift to of the Hessian of is also denoted by This agrees with the Hessian of the lift ∘ generally only on horizontal vectors. For detailed computations, see Lemma 5.1 in ([4]).
Now we recall the formula for the Ricci curvature tensor Ric on the warped product manifold × We write for the pullback by of the Ricci curvature of and similarly for
Lemma 2.8. On a warped product manifold × with dim let be horizontal and vertical.
Then
〈〉♯ where ♯
∆
〈 〉
and ∆ is the Laplacian on .
Proof. See Corollary 7.43 in ([13]).
On the given warped product manifold × we also write for the pullback by of the scalar curvature of and similarly for From now on, we denote by ∇
10
Lemma 2.9. If is the scalar curvature of × with dim
then
(2.1)
∆
〈∇ ∇〉 where ∆ is the Laplacian on .
Proof. For each ∈ × let
be an orthonormal basis for Then by the natural isomorphism
is an orthonormal set in We can choose
on such that
forms an orthonormal basis for Then
〈
〉
which implies that
forms an orthonormal basis for By Lemma 2.8 (1) and (3), for each and
and
∆
∇ ∇
for ≤ ≤ , for ≤ ≤
Hence, for and ,
∆
∇ ∇
which is a nonlinear partial differential equation on × for each ∈
11
3. Main results
Let be a Riemannian manifold of dimension and let :
∞ → be a smooth function, where is a positive number. The Lorentzian warped product of and ∞ with warping function is defined to be the product manifold ∞ × ′ with
(3.1) ′
Let R() be the scalar curvature of . Then the scalar curvature
of ′ is given by the equation
(3.2)
″ ′
for ∈ ∞ and ∈ (For details, cf. [5] or [6]) If we denote
then equation (3.2) can be changed into
(3.3)
′′
In this paper, we assume that the fiber manifold is nonempty, connected and compact Riemannian -manifold without boundary. Then, by Theorem 3.1, Theorem 3.5 and Theorem 3.7 in [6], we have the following proposition.
12
Proposition 3.1. If the scalar curvature of the fiber manifold is an- arbitrary constant, then there exists a nonconstant warping function on
∞ such that the resulting Lorentzian warped product metric on
∞ × produces positive constant scalar curvature.
Proposition 3.1 implies that in Lorentzian warped product there is no obstruction of the existence of metric with positive scalar curvature.
However, the results of [8] show that there may exist some obstruction about the Lorentzian warped product metric with negative or zero scalar curvature even when the fiber manifold has constant scalar curvature.
Remark 3.2. Theorem 5.5 in [14] implies that all timelike geodesics are future (resp. past) complete on ∞ ∞ × if and only if
∞
∞
∞
∞ for some and Remark2.58 in [1] implies that all null geodesics are future (resp. past) complete if and only if
∞
∞
∞
∞ for some (cf.
Theorem 4.1 and Remark 4.2 in [4]. In this reference, the warped product metric is ′
If admits a Riemannian metric of zero scalar curvature, then we let
in equation (3.3), where ∈ is a constant, and we have
≤
Therefore, from the above fact, Remark 3.2 implies the following:
13
Theorem 3.3. For ≥ let ∞ × be the Lorentzian warped product -manifold with compact -manifold. Suppose that is in class (B), then on there is a future geodesically complete Lorentzian metric of negative scalar curvature outside a compact set.
We note that the term achieves its maximum when
. And when
and admits a Riemannian metric of zero scalar curvature, we have
If is the function of only t-variable, then we have the following proposition whose proof is similar to that of Lemma 1.8 in [12].
Proposition 3.4. If then there is no positive solution to equation (3.3) with
≤
for ≥
where 1 and are constants.
Proof. See Proposition 3.4 in [7].
In particular, if then using Lorentzian warped product it is impossible to obtain a Lorentzian metric of uniformly negative scalar curvature outside a compact subset. The best we can do is when
or
where the scalar curvature is negative but goes to zero at infinity.
14
Proposition 3.5. Suppose that and ∈∞∞
Assume that for there exist an (weak) upper solution and a (weak) lower solution such that ≤ . Then there exists a (weak) solution of equation (3.3) such that for
≤ ≤
Proof. See Theorem 2.5 in [7].
Theorem 3.6. Suppose that Assume that
∈ ∞ ∞ is a function such that
≤
for
where and
are constants. Then equation (3.3) has a positive solution on ∞ and on the resulting Lorentzian warped product metric is a future geodesically complete metric.
Proof. Since , put
. Then ″
. Hence
″
≤
≤
Therefore is our (weak) upper solution. Since and
≤
, we take the lower solution where the constant
) will be determined later. Then ″ .
15 Hence
″ ≥
≥
if is sufficiently close to
. Thus is a (weak) lower solution and for large . Hence proposition 3.5 implies that equation (3.3) has a (weak) positive solution such that
for large . And since is sufficiently close to
,
Therefore
∞
∞
≥
∞
∞
≥
∞
∞ and
∞
∞
≥
∞
∞
∞
16
which, by Remark 3.2, implies that the resulting warped product metric is a future geodesically complete one.
Theorem 3.7. Suppose that Assume that
∈∞∞ is a function such that
where are positive constants. If
, then equation (3.3) has a positive solution on ∞ and on the resulting Lorentzian warped product metric is not a future geodesically complete metric.
Proof. Put , where
is a positive constant. Then
″ Hence
″
≥
≥
for large and such that ≥
. Thus, for large , is a (weak) lower solution.
Since and ≥
, we take the upper solution
where the constant
will be determined later. Then
″ Hence
17
″ ≤
≤
if is sufficiently close to
. Thus is a (weak) upper solution and for large . Hence Proposition 3.5 implies that equation (3.3) has a (weak) positive solution such that for large
. And since is sufficiently close to
,
Therefore
∞
∞
≤
∞
∞
∞ and
∞
∞
≤
∞
∞
∞
which, by Remark 3.2, implies that the resulting warped product metric is not a future geodesically complete one.
Remark 3.8 In case that we see that the function
∙
∙
is a fiducial point whether the resulting warped
18
product metric is geodesically complete or not. Note that
is a solution of equation (3.3) when
∙
∙
. In case that
, we know that the resulting warped product metric is a geodesically complete one.
19
References
[1] T. Aubin,
Nonlinear analysis on manifolds. Monge – Ampere equations,
Springer – Verlag New York Heidelberg Berlin, 1982.[2] J. K. Beem and P. E. Ehrlich,
Global Lorentzian Geometry,
Pure and Applied Mathematics, Vol.67, Dekker, New York, 1981.[3] J. K. Beem, P. E. Ehrlich and K. L. Easely,
Global Lorentzian Geometry
, Pure and Applied Mathematics, Vol. 202, Dekker, New York, 1996.[4] J. K. Beem, P. E. Ehrlich and Th. G. Powell, ″Warped product manifolds in relativity
″ Selected Studies (Th.M. Rassias, G.M. Rassias, eds.), North- Holland
, 1982, 41-56.[5] F. Dobarro and E. Lami Dozo, ″Positive scalar curvature and the Dirac operator on complete Riemannian manifolds″
Publ. Math. I.H.E.S.
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Tsukuba J. Math.
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10(1975), 113-134.[9] J. L. Kazdan and F. W. Warner, ″Existence and conformal deformation of metrics with prescribed Guassian and scalar curvature″
Ann. of Math.
101(1975), 317-331.
[10] J. L. Kazdan and F. W. Warner, ″Curvature functions for compact 2 - manifolds″
Ann. of Math.
99(1974), 14-74.[11] M. C. Leung, ″Conformal scalar curvature equations on complete
20
manifolds″
Comm. in P.D.E.
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[13] B. O’Neill,
Semi-Riemannian geometry with applications to relativity,
Academic Press, New York, (1983).[14] T. G. Powell, ″Lorentzian manifolds with non-smooth metrics and warped products″
