Riemannian Manifolds with Almost Constant Scalar Curvature

Liviu Nicolescu and Gabriel Pripoae

Abstract

We find sufficient conditions for a Riemannian manifold with almost constant scalar curvature s (i.e. satisfying (R(X,Y)Z)s = 0) to have constant scalar curvature. In particular, this is true for compact or conformally flat manifolds, or for hypersurfaces in spaceforms. Examples are given which show a quite exotic behaviour of the scalar curvature.

Mathematics Subject Classification : 53B20, 53B21, 53C21

Key words : scalar curvature, conformally flat manifolds, constant sign sectional curvature, Codazzi tensors, Yamabe-like problem

1 Introduction

Defined by E.Cartan [4] as a generalization of locally symmetric manifolds, semisymmetric manifolds are characterized by the condition (involving the curvature tensor) RR = 0, where R(X, Y) acts on the tensorial algebra like a derivation, for every vector fields X and Y. Semi-symmetric manifolds were classified by Z.Szabo (see [14] for further details and references).

When contracting the previous formula, we obtain RQ = 0 , where Q is the Ricci tensor. The manifolds which satisfy this condition were studied by several authors (K. Nomizu [9], S. Tanno [15], Sekigawa and Takagi [13], G. Pripoae and the author [8] ).

This process may continue (even if not by contraction !), for the class of Riemannian manifolds satisfying Rs = 0, where s is the scalar curvature; that is, R(X, Y)Zs = 0, for every vector fields X, Y and Z. We say that the scalar curvature is almost constant.

Despite its (maybe too) formal appearence, this property has a strong geometric flavour: generically, it means that the scalar curvature is or constant, or its level hypersurfaces are invariant submanifolds. Moreover, the almost constant scalar curvature seems to be useful in at least two difficult topics. First, we have the posibility to formulate an extended Yamabe problem, ”intermediating” between the constant and the arbitrary prescribed scalar curvature:

Balkan Journal of Geometry and Its Applications, Vol.4, No.2, 1999, pp. 83-92 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(*) Given a Riemannian manifold (M, g), is there a metric g_{1}, conformal to g, with almost constant scalar curvature s_{1} ?

When M is compact and s_{1} is asked to be constant, we recover the classical problem of Yamabe, with positive answer given by Schoen ([12]). As we prove in Theorem 5, on compact manifolds there do not exist any metric with non-constant almost constant scalar curvature, so there Problem (*) is irrelevant.

On noncompact manifolds, existence and non-existence results were proven for constant ([1]) or prescribed ([6]) scalar curvatures in some conformal class. A common feature of these results are some strong non-positivity hypothesis put on the sectional and the Ricci curvatures of the apriori metric. In §5 we consider an example of Riemannian manifold with non-constant almost constant sectional curvature, nonconformally flat and without the previously quoted negativity conditions. So, there is no doubt that Problem (*) is non-trivial on non-compact manifolds but, as yet, it is not clear if the system of PDE’s governing it is workable.

Secondly, the almost constant scalar curvature condition is equivalent to the fact that the Hessian of the scalar curvature is a Codazzi tensor. It is known ([5]) that the eigenspace distributions of a Codazzi tensor are integrable, and the integral manifolds are totally umbilical. In §4 we prove that these leaves are also invariant submanifolds, so all their second fundamental forms are also Codazzi tensors. For them we can repeat the construction, considering their integral manifolds; this process of (generic) reduction stops only when every umbilical leaf is totally geodesic.

We also provide examples of (warped product) Riemannian manifolds having (nonconstant) almost constant scalar curvature (§2). In order to distinguish between the constant and the (non-constant) almost constant cases, we prove (§3):

Each of the following conditions are sufficient for an almost constant scalar curvature to be constant : (1) M is compact; (ii) M is conformally flat; (iii) the Ricci tensor is non-degenerated; (iv) the sectional curvature has constant sign; (v) M is a hypersurface in a space form.

2 Examples

Let (M, g) a connected, n-dimensional Riemannian manifold. We denote by ∇ the Levi-Civita connection of g, and by R the curvature tensor, defined for every vector fields X, Y by

R(X, Y) = [∇_{X}, ∇_{Y}] − ∇_{[X,Y]}.

Consider Q the Ricci tensor of (0,2)-type and r the associated (1,1)-tensor field, defined by g(rX, Y) = Q(X, Y). The trace of Q is the scalar curvature s. If s is almost constant, we say that (M, g) is an ACSC-manifold. Remarks. (i) Every two-dimensional Riemanian manifold with almost constant scalar curvature has constant curvature.

(ii) In the case n = 3 , the curvature tensor may be expressed as

R(X, Y) = \frac{s}{2}X ∧ Y − rX ∧ Y − X ∧ rY.

If Rs = 0, by contracting twice we obtain rX(s) = 0, and Y (s)X − X(s)Y = 0 for every vector fields X, Y on M. Thus s is constant.

So, in what follows, we may suppose the dimension of manifolds be greater than three. Examples. (i) A Riemannian manifold with parallel ds has almost constant scalar curvature s. It is known that any differentiable manifold admits a metric with constant scalar curvature (see [12] for the compact and [3] for the non-compact cases).

(ii) Let A be an open half plane in R^{2} and f : A → R a non-vanishing linear function. Consider the warped product metric g on M = A ×_{f} R^{2}, given by

g = (dx^{1})^{2} + (dx^{2})^{2} + f^{2}((dx^{3})^{2} + (dx^{4})^{2}).

Then the scalar curvature s = −2|df|^{2}/f^{2} is a non-constant function defined on A.

On another hand, the curvature tensor vanishes for all vector fields on M, but for those tangent to the second plane. This proves that R(X, Y)Zs = 0 , for every vector fields X, Y, Z on M.

So, (M, g) is ACSC-manifold with non-constant scalar curvature (and ds nonparallel).

(iii) More generally, let (B, g_{B}) and (F, g_{F}) be two Riemannian manifolds (the ”base” and the ”fiber” respectively) and f a differentiable function on B. Denote the dimension of F by d and the projections on B and F by π and σ respectively. Consider the warped product manifold B×_{f}F, with the metric

g = π^{∗}(g_{B}) + (f ∘ π)^{2}σ^{∗}(g_{F}).

Then

s = s_{B} − \frac{2d}{f}Δ_{B}f + \frac{s_{F}}{f^{2}} − d(d − 1)\frac{|df|^{2}}{f^{2}}

where s_{B} and s_{F} are the scalar curvatures of B and F and Δ_{B} is the Laplacian of g_{B}. Denote by X, Y, Z and by U, V, W arbitrary vector fields tangent to the base and to the fiber, respectively. Denote H_{s} the Hessian of f on B. Then, the condition Rs = 0 writes

[^{B}R(X, Y)Z] s = 0 , H_{f}(X, Y)V (s_{F}) = 0 , (∇_{X}gradf)s = 0

[^{F}R(V, W)U + \frac{|df|^{2}}{f^{2}} g(V, U)W − g(W, U)V ] (s_{F}) = 0

A sufficient condition for B ×_{f} F to be an ACSC-manifold is: ”s_{F} is a constant function and f satisfies the (previous) first and third differential equations on B”.

Finally, we remark that a Riemannian product of ACSC-manifolds is an ACSCmanifold too.

3 Basic properties of ACSC-manifolds

Consider a Riemannian manifold M, with scalar curvature s. Denote ξ the vector field associated to the one-form ds. Then we have some general properties. Lemma 1. The following assertions are equivalent :

(i) Rs = 0;

(ii) R(X, Y)ξ = 0; (iii) R(ξ, X)Y = 0;

(iv) δQ(R(X, Y)Z) = 0;

(v) H_{s} is a Codazzi tensor.

Proof. The condition Rs = 0 writes g(ξ, R(X, Y)Z) = 0, hence the first three assertions are equivalent.

For the next equivalence we recall the classical identity ds = 2δQ, where δ is the divergence operator. We have

g(R(X, Y)Z, ξ) = X[(∇_{Y} ds)Z] − (∇_{Y} ds)(∇_{X}Z) − Y [(∇_{X}ds)Z]+
+(∇_{X}ds)(∇_{Y} Z) − (∇_{[X,Y]}Z)(s) = (∇_{X}H_{s})(Y, Z) − (∇_{Y} H_{s})(X, Z)

which proves the last equivalence. □ Lemma 2. If Rs = 0 , then

(i) (rX)s = 0 ;

(ii) r(∇_{ξ}ξ) = r(ξ) = 0;

(iii) (δQ)(rX) = 0;

(iv) (∇_{X}∇_{Y} r)ξ = (∇_{Y} ∇_{X}r)ξ;

(v) (∇_{ξ}R)(X, Y, Z) = R(∇_{Y} ξ, X)Z − R(∇_{X}ξ, Y)Z.

Proof. By contracting in Lemma 1,(i) we get (i) and (iii). Formula (ii) results from g(rξ, X) = g(ξ, rX) = 0.

Consider the second identity of Bianchi

(1)  ∑_{XY Z}(∇_{X}R)(Y, Z, V) = 0

for arbitrary vector fields X, Y, Z, V. We replace X = V = ξ and we deduce R(Y, Z)∇_{ξ}ξ = 0, for every Y and Z. By contraction, we obtain (ii).

If in (1) we replace X = ξ, from the previous relations we get (v).

Finally, we use the identity

(∇_{X}∇_{Y} Q)(Z, W) − (∇_{Y} ∇_{X}Q)(Z, W) = Q(R(Z, W)X, Y) − Q(R(Z, W)Y.X)

and we obtain (iv) . □ Remarks. The definition of ACSC-manifolds might look a bit formal. The following geometric interpretation shows that this new class has a non-trivial behaviour:

(i) Consider an ACSC-manifold whose scalar curvature has no critical points. Then, the level sets of the scalar curvature foliate M. From Prop.1, (iii) we see that each such hypersurface is invariant (that is, the curvature of M, restricted to each leaf, remains tangent to the respective leaf).

Generically, this property characterizes the ACSC-manifolds. Indeed,let (M, g) be a Riemannian manifold and suppose the scalar curvature has no critical points (so, we exclude the constant scalar curvature case and we consider only the generic regular points of s). If all the level sets of s are invariant hypersurfaces, then (M, g) is an ACSC-manifold.

This characterization is similar to a result by Ogiue [10]: a Riemannian manifold has constant (sectional) curvature if and only if all its submanifolds are invariant.

(ii) For the larger class of manifolds satisfying only the condition rs = 0, we have a similar characteristic property: the leaves of the respective foliation are Ricci-invariant hypersurfaces (i.e. the restriction of the Ricci tensor ofM, to each leaf, remains tangent to the leaf).

The next two results are easy consequences of the lemmas. Proposition 3. An ACSC-manifold with non-degenerate Ricci tensor has constant scalar curvature. Proposition 4. An ACSC-manifold with non-vanishing sectional curvature has constant scalar curvature.

The next result shows there are no interesting compact ACSC-manifolds. Theorem 5. A compact ACSC-manifold has constant scalar curvature. Proof. Apply the Bochner-Lichnerowicz formula to the scalar curvature function s

\frac{1}{2}Δ(| ds |^{2}) = | H_{s} |^{2} + dΔs(ξ) − Q(ξ, ξ)

So, on the compact manifold M we have the integral formula

∫_{M} | H_{s} |^{2}dσ = 0.

We deduce H_{s} = 0, so the differential of s is a parallel 1-form. If s is not constant, then (generically) M decomposes as a product with a line factor, which leads to a contradiction.□

When a new geometric condition appears, it is interesting to investigate it in the framework of the submanifolds theory.The next result shows that the ACSC condition is irrelevant for a large class of hypersurfaces. Theorem 6. If a hypersurface in a spaceform is an ACSC-manifold, then it has constant scalar curvature. Proof. Let (M, g) be an immersed hypersurface in a spaceform of constant curvature c. Denote by A the Weingarten tensor of M. The Gauss equation gives the curvature tensor of M

R(X, Y)Z = c{g(Y, Z)X − g(X, Z)Y } + g(AY, Z)AX − g(AX, Z)AY.

By hypothesis, Rs = 0, so

c{X(s)Y − Y (s)X} + AX(s)AY − AY (s)AX = 0.

Fix a point of M, and e_{1}, ... , e_{n} a basis of eigenvectors of A, corresponding to the eigenvalues λ_{1}, ... ,λ_{n} respectively. The last formula becomes

(1)  (c − λ_{i}λ_{j})e_{j}(s) = 0

for every indices i and j.

Case I: c < 0. For i = j in (1), we obtain e_{j} = 0, for every j.

Case II: c > 0. If all the eigenvalues are equal to \sqrt{c}, then M is a totally umbilical hypersurface; a standard result implies M has constant (sectional) curvature.

Consider i fixed, and another indice j such that the corresponding eigenvalues are distinct. Then at least one of the two quantities (c − (λ_{i})^{2}) and (c − λ_{i}λ_{j}) is non-null. From formula (1) we derive e_{i}(s) = 0.

Case III: c = 0. Consider a dense open subset U of M, on which the number of distinct eigenvalues of A is locally constant. On U, the eigenvalue functions x → λ_{i}(x) are smooth, and the corresponding eigenspace distributions V_{λ_{i}} are smooth and with constant rank.

Suppose V_{0} (the eigenspace distribution corresponding to the null eigenvalue) has rank k (k ≥ 1); if λ_{1} = ... = λ_{k} = 0, then formula (1) implies e_{i}(s) = 0 for i > k.

One knows V_{0} is a totally geodesic distribution. If (N, h) is an integral manifold of V_{0}, it is flat. So, the scalar curvature of M, restricted to N, vanishes. We deduce s is constant along the distribution V_{0}, so is constant on the whole U. By continuity, s is constant on M. □

The next step is to investigate the relevance of the ACSC property in the conformal geometry. Theorem 7. A conformally flat ACSC-manifold has constant scalar curvature. Proof. If (M, g) is conformally flat, we have the well-known formula, given by the vanishing of the conformal curvature tensor

(2)  R(X, Y)Z − \frac{s}{(n − 1)(n − 2)}\{g(X, Z)Y − g(Y, Z)X\}−

−\frac{1}{n − 2}\{g(Y, Z)rX − Q(X, Z)Y − g(X, Z)rY + Q(Y, Z)X\} = 0.

The condition Rs = 0 becomes

(3)  \frac{s}{(n − 1)}\{X(s)Y − Y (s)X\} − X(s)rY + Y (s)rX = 0

since rX(s) = 0 by Prop.2,(ii).

Suppose that the gradient ξ of the scalar curvature is non-null. In (3), we replace X = ξ and obtain

(4)  rY = \frac{s}{n − 1}\{Y − \frac{Y (s)}{ξ(s)}ξ\}.

Suppose λ be a non-null eigenvalue of r and Y a corresponding eigenvector. Then, from (4) we deduce Y (s) = 0.

If Y is an eigenvector of the null eigenvalue of r, then (4) implies Y is a multiple of ξ. Hence, the multiplicity of the null eigenvalue is one.

Let λ be a non-null eigenvalue or r and Y a corresponding eigenvector. Since Y cannot be colinear with ξ, from (4) it follows λ = \frac{s}{n − 1}. Hence, all the n − 1 non-null

eigenvalues of r are equal to \frac{s}{n − 1}.

For X, Y, Z orthogonal to ξ, there exists a function k such that (2) writes

R(X, Y)Z = k{g(Y, Z)X − g(X, Z)Y }.

If k = 0, then the curvature tensor vanishes, hence a contradiction.

On another hand, for each point p where k ≠ 0, the set

{R(x, y)z | x, y, z ∈ T_{p}M}

spannes the whole tangent space T_{p}M. From R(s) = 0 we deduce s is constant.□

4 A Codazzi digression

Consider an ACSC-manifold. Suppose ds is non-parallel, so the Hessian H_{s} is a nonnull Codazzi tensor. Let U be a dense open set in M such that on U the number of the eigenvalues for H_{s} is locally constant. Denote by λ such an eigenvalue function and by V_{λ} the corresponding eigenspace distribution. From the general theory of Codazzi tensors (for exemple, ch.16,C in [2]), it is known that each distribution V_{λ} is integrable, and has umbilical leaves. Moreover, we have Proposition 8. For an arbitrary Codazzi tensor on M, the leaves of eigenspace distributions are invariant submanifolds, so all the second fundamental forms are Codazzi tensors. Proof. Let b a (0,2)-tensor of Codazzi type. Let X ∈ V_{α}, Y ∈ V_{β}, Z ∈ V_{γ}. One knows ([5]) that for α ≠ γ, and for β ≠ γ we have R(X, Y)Z = 0. From the symmetries of the curvature tensor we derive easily that R(X, Y)Z ∈ V_{α}, for X, Y, Z ∈ V_{α} and for X, Z ∈ V_{β}, Y ∈ V_{α} with α ≠ β.

So, for every integral manifold N of V_{α}, the curvature tensor of (M, g) leaves invariant the tangent spaces of N. By the Codazzi-Mainardi equation, for every vector field W ∈ V_{β}, orthonormal to N, the second fundamental form h^{W} is a Codazzi tensor on N. □ Remarks. (i) Considering an arbitrary non-null Codazzi tensor field b, we can perform an ”umbilical reduction” of the manifold M, in a neighboorhood of each generic point p ∈ M; so, we can obtain a totally umbilical leaf N containing p, for each eigenspace distribution of b. If every N is totally geodesic, the algorithm stops. If there exists at least a leaf N, with non-null second fundamental form h, this one is again a Codazzi tensor, but on N. Now we can repeat the reduction, starting with h on N, instead with b on M.

(ii) If the Codazzi tensor is just H_{s}, then Prop.8 implies that the Ricci tensor Q of (M, g) vanishes on each product V_{α} × V_{β}, with different α and β.This remark, together with the Weitzenböck formula ([2], p.436)

δ∇H_{s} − H_{Δs} = R^{0}(H_{s}) − H_{s} ∘ Q

prove the Corollary 9. For an ACSC-manifold, H_{Δs} = δ∇H_{s}, on every product of different eigenspace distributions.

5 A Yamabe digression

In [1], Aviles and McOwen solved the Yamabe problem for noncompact complete Riemannian manifolds, with upper bounded negative scalar curvature, and found conformally metrics with constant negative scalar curvature. Jin ([6]) started with noncompact, complete Riemannian manifolds with upper bounded negative sectional curvature and Ricci curvature bounded from below by a negative constant. For a prescribed function K, satisfying various bound conditions, he proved existence or non-existence theorems for conformally related metrics having scalar curvature K.

Consider Exemple 2 from §2. In view of Theorem 7, it is clear that the ACSCmanifold (M, g) is not conformally flat. Moreover, its sectional, scalar and Ricci curvatures can reach arbitrary negative values. So, we have a non-trivial solution for Problem (*), which cannot be recovered from the previously quoted results. Remark. Let (M, g) be a Riemannian manifold, f a real differentiable function on M and the metric g_{1} = e^{2f}g. We denote by

A_{f} = H_{f} − df ∘ df + (\frac{|df|^{2}}{2} ) g

and by a_{f} its associated (1,1)-tensor field. Let h_{f} the associated (1,1)-tensor for the Hessian H_{f}. It is known that the (1,3)-curvature tensor R^{1} and the scalar curvature s^{1} on (M, g_{1}) are related to those on (M, g) by

R^{1} = R − g ⊗ a_{f} − A_{f} ⊗ Id + a_{f} ⊗ g + Id ⊗ A_{f}

and

s^{1} = e^{−2f}(s − 2(n − 1)tracea_{f}))

where

2tracea_{f} = −Δf + (n − 2)|df|^{2}.

Imposing the condition R^{1}s^{1} = 0 leads to a third order system of nonlinear PDE’s, whose dominant term is (in local coordinates)

(R^{1})_{jkl}^{i} \frac{∂}{∂x^{i}} Δf.

Integrating it is much more chalenging than the classical Yamabe problem, whose solutions (constant s^{1}) are (maybe) the simplest.

6 Comments

Remarks. (i) The notion of ACSC-manifold can be defined also in semi-Riemannian geometry (i.e. for non-degenerate metrics), without major difficulties. Most of our results admit extensions to this more general case, with some precautions due to the signature and to the possible degeneracy of induced metric tensors.

(ii) The class of ACSC-manifolds is strictly contained in the class of Riemannian manifolds satisfying rs = 0 (for example, choose an appropriate function f in the warped products of §2).

(iii) Consider an ACSC-manifold M and, for every point p ∈ M, define the subspace D_{p} spanned bt the set {R(x, y)z | x, y, z ∈ T_{p}M}. If, for each point p, D_{p} ”filles” the tangent space T_{p}M, then the scalar curvature is constant.

But, in general, the distribution p → D_{p} may have non-constant rank. Denote D^{⊥} its complementary distribution. (In general, D and D^{⊥} may be singular and even non-differentiable). If s is constant ”along” D^{⊥}, then again s is constant on M.

This fact and the examples of §2 suggest that, on ACSC-manifolds, the obstruction to the constancy of s is contained in the distribution D^{⊥}.

(iv) In the proof of Theorem 7 we need not (M, g) to be conformally flat; it suffices that the conformal curvature tensor C satisfy the condition Cs = 0.

A similar result is true for the projective curvature tensor P : if an ACSC-manifold satisfies also P s = 0, then its scalar curvature is constant.

(v) Proposition 4 can be improved: if p is a regular point of s, then the sectional curvature of every 2-plane π containing ξ_{p} vanishes. Acknowledgements. A version of this paper was presented at the Workshop on Electromagnetic Flows and Dynamics, Oct 15-19, 1999, University Politehnica of Bucharest.

Supported by MEN Grant No 21815/28.09.1999, CNCSU-31.

References

[1] P. Aviles, R. McOwen, Conformal deformations to constant negative scalar curvature on noncompact Riemannian manifolds, J. Diff. Geom. 27 (1988), 225-239.

[2] A. Besse, Einstein Manifolds, Springer Verlag, Berlin, 1987.

[3] J. Bland, M. Kalka, Negative scalar curvature metrics on non-compact manifolds, Trans. AMS 316 (1989), no.2, 433-446.

[4] E. Cartan, Leçons sur la géométrie des espaces de Riemann, 2nd ed., Paris, 1946.

[5] A. Derdzinski, C.L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc. 47 (1983), 15-26.

[6] Z. R. Jin, Prescribing scalar curvatures on the conformal classes of complete metrics with negative curvature, Trans. AMS, 340 (1993), 785-810.

[7] A. Lichnerowicz, Courbure, nombres de Betti et espaces symétriques, Proc. Internat. Congress Math., Cambridge, 1950, AMS, vol.II (1952), 216-223.

[8] L. Nicolescu, G. Pripoae, Sur les espaces de Riemann satisfaisant à RQ = 0, C.R. Acad. Sci. Paris, 323 (1996), 389-392.

[9] K. Nomizu, On hypersurfaces satisfying a certain condition on the curvature tensor, Tohoku Math J., 20 (1968), 46-59.

[10] K. Ogiue, On invariant immersions, Annali di Mat., 1969, 387-397.

[11] B. O’Neill, Semi-Riemannian Geometry, Academic Press, N.Y., 1983.

[12] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984), 479-495.

[13] K. Sekigawa, H. Takagi, On the conformally flat spaces satisfying a certain condition on the Ricci tensor, Tohoku Math J., 23 (1971), 1-11.

[14] Z. Szabo, Structure theorems on Riemannian spaces satisfying R(X, Y)R = 0 , I. The local version, J.Diff.Geom., 17 (1982), 531-582 II. Global versions, Geom. Dedicata, 19 (1985), 65-108.

[15] S. Tanno, A class of Riemannian manifolds satisfying R(X, Y)R = 0, Nagoya Math. J., 42 (1971), 67-77.

Faculty of Mathematics, University of Bucharest, Str. Academiei no. 14, Cod 70109 Bucharest, Romania e-mail: lnicol@geometry.math.unibuc.ro

gpripoae@math.math.unibuc.ro