Compact Manifolds with Positive Einstein Curvature - IngentaConnect

Geometriae Dedicata 108: 205–217, 2004. 

Ó 2004 Kluwer Academic Publishers. Printed in the Netherlands. 

205 

Compact Manifolds with Positive 

Einstein Curvature 

M.-L. LABBI 

Department of Mathematics, College of Science, University of Bahrain, 

Isa Town 32038, Bahrain. e-mail: labbi@sci.uob.bh 

(Received: 6 October 2003; accepted in final form: 18 February 2004) 

Abstract. In this paper we study positive Einstein curvature which is a condition on the 

Riemann curvature tensor intermediate between positive scalar curvature and positive sectional 

curvature. We prove some constructions and obstructions for positive Einstein curvature 

on compact manifolds generalizing similar well known results for the scalar curvature. 

Finally, because our problem is relatively new, many open questions are included. 

Mathematics Subject Classifications (2000). 53C21, 53C42. 

Key words. positive curvature, Riemannian submersion, surgery, connected sum, minimal 

submanifold, fundamental group, group action. 

1. Introduction 

After the scalar curvature and sectional curvature, the 1-curvature (which coincides 

with the Einstein curvature) is the most important special case of the p-curvature. Its 

positivity was introduced and studied in our previous works [1, 2] but only in the 

general context of the p-curvature and so it deserves a separate study. Here in this 

paper, from one side we survey and improve (whenever possible) our previous 

general results when applied to the special case p ¼ 1. On the other side, we prove 

new results including a generalization of a famous theorem of Schoen and Yau about 

positive scalar curvature. 

1.1. THE EINSTEIN CURVATURE: DEFINITION AND PROPERTIES 

Let ðM; gÞ be a Riemannian manifold of dimension nP3 and let R and Ric denote 

respectively its Riemann curvature (0,4)-tensor and the Ricci (0,2)-tensor. 

The Einstein tensor denoted by E is a combination of the metric tensor g and the 

Ricci tensor as follows: 

E ¼ 1 2 sg Ric; 

where s denotes the scalar curvature function. 

Recall that the Ricci curvature r is the function defined on the unit tangent bundle 

UM of ðM; gÞ by rðvÞ ¼Ricðv; vÞ. Similarly we define The Einstein curvature e to be 

the function defined on UM by 

eðvÞ ¼2Eðv; vÞ ¼s 2rðvÞ;

206 

M.-L. LABBI 

where we multiplied by the constant 2 to make it equal to the p-curvature [1, 2] with 

p ¼ 1, that is, the Einstein curvature precisely coincides with the average of the 

sectional curvatures in the directions orthogonal to v 

eðvÞ ¼ X i; j2I 

Rðe i ; e j ; e i ; e j Þ; 

where fe i ; i 2 Ig is any orthonormal basis for the orthogonal supplement ð

COMPACT MANIFOLDS WITH POSITIVE EINSTEIN CURVATURE 207 

EXAMPLE. If ðM; gÞ is an n dimensional manifold with constant curvature k (resp. 

with constant Ricci curvature r) then the Einstein curvature is also constant and 

equals 

e ¼ nðn 2Þk ðresp: e ¼ðn 2ÞrÞ: 

1.2. POSITIVE EINSTEIN CURVATURE 

Here, in this paper we are particularly interested in the study of the positivity 

properties of the Einstein curvature on compact manifolds. First of all, remark that 

from the definition and property 5 we have positive sectional curvature ) positive 

Einstein curvature ) positive scalar curvature. 

However, the relation between positive Einstein curvature and positive Ricci 

curvature is not so simple: 

(1) The example of the Riemannian product S 4 H 2 of the standard 4-sphere with 

the standard 2-hyperbolic space, shows that the positivity of the Einstein curvature 

does not imply the positivity of the Ricci curvature. Furthermore, there 

exist compact manifolds with positive Einstein curvature and do not admit any 

metric with positive Ricci curvature, for example S 1 S n , nP3. 

(2) On the other hand, positive Ricci curvature does not also imply positive Einstein 

curvature (at least algebraically, that is for the same metric). Below is a 

counter example: 

Let p : S 2nþ1 ! P n C be the Hopf fibration, and let the total space be endowed 

with the canonical standard metric say g. The complex projective space is endowed 

with the natural metric g such that the fibration p is a Riemannian 

submersion. 

We modify the metric g on the sphere by multiplying it by t 2 R in the 

directions tangent to the fibers (vertical directions), see Section 2. Let us denote 

by g t the new metric on S 2nþl . The Ricci curvature of the new metric is as follows 

(see [3] p. 250 and 253) 

Ric t ðU; VÞ ¼2nt 2 gðU; VÞ; 

Ric t ðX; YÞ ¼ð 2t þ 2n þ 2ÞgðX; YÞ; 

Ric t ðX; UÞ ¼0; 

where U and V (resp. X and Y) denote tangent vectors to the fibre (resp. normal 

vectors). Consequently, the scalar curvature of the metric g t is s t ¼ 

2nð2n þ 2 tÞ and therefore the Einstein curvature of g t is as follows: 

E t ðU; VÞ ¼2ntð2n þ 2 3tÞgðU; VÞ; 

E t ðX; YÞ ¼ð4ðn 2 1Þ 2ðn 2ÞÞgðX; YÞ; 

E t ðX; UÞ ¼0:

208 

M.-L. LABBI 

From which is clear that the metric g t is with positive Ricci curvature if 

2 

3 ðn þ 1Þ < t < n þ 1 but ðeÞ tðUÞ is negative. 

(3) If k 1 Ok 2 O Ok n denote the eigenvalues of Ricci at a point m 2 M then the 

positivity of the Einstein curvature at m is equivalent to the following pinching 

condition on the eigenvalues 

k n < Xn 1 

k i : 

i¼1 

In particular, a Ka¨hlerian manifold with positive Ricci curvature has positive 

Einstein curvature, because for a Ka¨hlerian manifold each eigenvalue of Ricci 

has multiplicity at least 2. 

Furthermore, for the same reason the converse holds in dimension 4. 

However, it remains an open question to see if in general a manifold with positive 

Ricci curvature admits a metric with positive Einstein curvature. 

EXAMPLES 

(1) Manifolds with positive sectional curvature are with positive Einstein curvature 

(if the dimension of the manifold is P3). 

(2) The standard Riemannian product S 1 S n ; nP3 is with positive Einstein curvature. 

(3) If a compact Riemannian manifold M is with positive scalar curvature then the 

Riemannian product S 2 M is with positive Einstein curvature, where the radius 

of the sphere is sufficiently small (see Section 2.1). 

(4) If M is an arbitrary compact manifold, then the product metric on S 3 M is 

with positive Einstein curvature, where the radius of the sphere is sufficiently 

small (see Section 2.2). 

(5) The Riemannian standard product S p H q , ðp; q > 0Þ, of the standard sphere 

with the standard hyperbolic space is with positive Einstein curvature if and 

only if pPq þ 2. 

(6) Let G be a compact, connected Lie group with a bi-invariant metric b,then 

ðG; bÞ is with positive Einstein curvature unless G is a torus. 

(7) Let H be a closed subgroup of G. There exists a unique metric h on the 

quotient G=H such that p : ðG; bÞ !ðG=H; hÞ is a Riemannian submersion (it 

is called the normal homogeneous metric). Then ðG=H; hÞ is with non-negative 

Einstein curvature that is eP0. Furthermore, e 0 if and only if G=H is a 

torus. 

Remarks. (1) Recall that if M is an arbitrary compact manifold then the product 

S 2 M admits a metric with positive scalar curvature. Consequently, Example 3 

shows that in some sense the positivity of the Einstein curvature in the class of all 

compact manifolds with positive scalar curvature behaves like the positivity of the 

scalar curvature in the class of all compact manifolds.


(2) As a consequence of property 3 in Section 1.1, the positivity of the Einstein 

curvature can be interpreted geometrically by 

volðS n 2 ðrÞÞ < vol e ðS n 2 ðrÞÞ 

for all small (n 2)-spheres as in previous property 3. 

(3) Note that in dimension 2 the Einstein curvature is always 0 so that we cannot 

have positive Einstein curvature, however in dimension 3 positive Einstein curvature 

is equivalent to positive sectional curvature. 

2. Constructions of Metrics with Positive Einstein Curvature 

2.1. USING RIEMANNIAN PRODUCTS 

Let ðM 1 ; g 1 Þ and ðM 2 ; g 2 Þ be two Riemannian manifolds. Every unit tangent vector v 

to ðM 1 ; g 1 ÞðM 2 ; g 2 Þ can be decomposed as follows. 

v ¼ cos av 1 þ sin av 2 ; 

where v 1 , v 2 are unit tangent vectors to M 1 , M 2 , respectively. 

By a straightforward computation we obtain 

2Eðv; vÞ ¼scal 1 þ scal 2 cos 2 ar 1 ðv 1 Þ 2 þ sin 2 ar 2 ðv 2 Þ; 

where we indexed by i the invariants of the metric g i (for i ¼ 1; 2). So the 

1-curvature of the product is positive if and only if 2E 1 ðv 1 ; v 2 Þþscal 2 > 0 and 

2E 2 ðv 2 ; v 2 Þþscal 1 > 0. Consequently, we have the following 

PROPOSITION 2.1. In each of the following cases, The Riemannian product 

ðM 1 ; g 1 ÞðM 2 ; g 2 Þ is with positive Einstein curvature: 

(1) The two manifolds are with positive Einstein curvature. 

(2) The two manifolds are both with nonnegative Einstein curvature and with positive 

scalar curvature. 

Also, by amplifying the scalar curvature of one of the factors (by multiplying the 

metric by a positive real number) and assuming a suitable positivity condition on the 

other factor we can appear positive Einstein curvature on the product, as it is shown 

by the following proposition: 

PROPOSITION 2.2. (1) The product of any compact manifold with positive scalar 

curvature by the standard two-dimensional sphere admits a metric with positive Einstein 

curvature. 

(2) The product of a compact manifold with nonnegative Einstein curvature and 

positive scalar curvature with a compact manifold with positive scalar curvature admits 

a metric with positive, Einstein curvature. 

Remark. It would be interesting to see if in general the product of two manifolds each 

one with positive scalar curvature admits a metric with positive Einstein curvature.

210 

M.-L. LABBI 

2.2. USING RIEMANNIAN SUBMERSIONS 

Let ðM; gÞ and ðB; gÞ be two Riemannian manifolds, and let p : ðM; gÞ !ðB; gÞ a 

Riemannian submersion. We define, for every t 2 R, a new Riemannian metric g t on 

the manifold M by multiplying the metric g by t 2 in the vertical directions. Recall 

that 8m 2 M, we have a natural orthogonal decomposition of the tangent space at m 

T m M ¼V m H m ; 

where V m is the tangent to the fiber at m and H m is the horizontal space, so that 

g t jV m ¼ t 2 g; 

g t jH m ¼ g; 

g t ðV m ; H m Þ¼0: 

Note that in this case, p : ðM; g t Þ!ðB; gÞ is still a Riemannian submersion with the 

same horizontal and vertical distributions [3]. 

In the following we will index by t all the invariants of the metric g t , and in the 

case case t ¼ 1 we omit the index 1. 

Also, We make under a hat ‘’ (resp. under a check ‘’) the invariants of the fibers 

with the induced metric (resp. of the basis B). 

The following lemma gives an estimate of the sectional curvature K t of the new 

metric g t (see [1] for the proof): 

LEMMA 2.3. Let P be a 2-plane in T m M then: 

(1) If P \H m 6¼f0g, we have K t ðPÞ ¼Oð1Þ is bounded. 

(2) If P \H m ¼f0g, there exists an orthonormal basis ðE 1 ; E 2 Þ of P such that its 

orthogonal projection ðU 1 ; U 2 Þ onto V m is an orthogonal basis. Furthermore, we 

have 

 

K t ðPÞ ¼K t ðE 1 ; E 2 Þ¼ a2 1 a2 2 bKðU 

t 2 1 ; U 2 ÞþO 1 

; 

t 

where a i ¼ g t ðE i ; U i Þ, for i ¼ 1; 2. 

Consequently, we can estimate the scalar curvature s t , the Ricci curvature r t and the 

Einstein curvature ðeÞ t 

of the metric g t of M as follows: 

Let F 1 ¼ a 1 U 1 þ b 1 X 1 be an arbitrary g t -unit tangent vector to M at m, where 

U 2V m and X 1 2H m are g t -unit vectors. We complete F 1 into an orthonormal basis 

of T m M by the use of F 2 ¼ b 1 U 1 þ aX 1 ; U 2 ; ...; U m 2V m and X 2 ; ...; X n 2H m , 

where m ¼ dim V m , n ¼ dim H m . Therefore, 

s t ¼ X K t ðU i ; U j Þþ X K t ðU i ; X j Þþ X 

i6¼j 

i;j 

i6¼j 

¼ 1 X 

 

^KðU 

t 2 i ; U j ÞþO 1 

t 

i6¼j 

 

K t ðX i ; X j Þ 

¼ 1 t 2 ^s þ O 1 t


and 

r t ðF 1 Þ¼K t ðF 1 ; F 2 Þþ Xm 

i¼2 

K t ðF 1 ; U i Þþ Xn 

j¼2 

K t ðF 1 ; X j Þ 

¼ K t ðU 1 ; X 1 Þþ Xm a 2 

1 

t b i¼2 

2 KðU 1 ; U i ÞþO 1 

t 

 

¼ a2 1 

t ^rðU 2 1ÞþO 1 

; 

t 

so that 

ðeÞ t 

ðF 1 Þ¼s t 2r t ðF 1 Þ 

¼ 1 

t ^s 2 2 a2 1 

t ^rðU 2 1ÞþO 1 

t 

 

¼ a2 1 

t 2 ð^s 2^rðU 1ÞÞ þ 1 t 2 ð1 a2 1 Þ^s þ O 1 t 

 

¼ a2 1 

t ^eðU 2 1Þþ b2 1 

t ^s 2 þ O 1 

: 

t 

Recall that a 2 1 þ b2 1 ¼ 1 and ^e > 0 implies that ^s > 0, therefore it is now clear that 

there exists t 0 such that 8tOt 0 we have ðeÞ t 

ðF 1 Þ > 0. Thus we have proved the 

following theorem: 

THEOREM 2.4. Suppose a compact manifold M admits a Riemannian metric g such 

that its restriction on the fibers (the induced metric) is with positive Einstein curvature 

then the manifold M admits a Riemannian metric with positive Einstein curvature. 

The following corollary is a direct consequence of the previous theorem: 

COROLLARY 2.5. The product S p M of an arbitrary compact manifold M with a 

sphere S p , pP3 admits a Riemannian metric with positive Einstein curvature. 

Remark. This theorem does not require the sectional curvature of the fibers to be 

nonnegative, so it improves our similar result on the p-curvature in the case p ¼ 1 [1]. 

2.3. USING GROUP ACTIONS 

The following theorem generalizes a similar result [1] in the case of free actions of 

groups and when p ¼ 1: 

THEOREM 2.6. If a compact manifold M admits a free and smooth action of the 

group SU(2) or SO(3) then the manifold M admits a Riemannian metric with positive 

Einstein curvature.

212 

M.-L. LABBI 

Proof. The canonical projection M ! M=G is in this case a smooth submersion. 

Let the fibers be equipped with a bi-invariant metric from the group G via the 

canonical inclusion GT m M. 

Using any G-invariant metric on M, we define the horizontal distribution to which 

we lift up an arbitrary metric from the basis M=G. Thus we have defined a metric on 

M such that the projection M ! M=G is a Riemannian submersion. 

Finally, since the group G ¼ SUð2Þ or SO(3) with a bi-invariant metric is with 

positive Einstein curvature then so are the fibers with the induced metric, and we 

conclude using Theorem 2.4. ( 

Since every compact connected non-Abelian Lie group contains SU(2) or SO(3) 

then we have the following. 

COROLLARY 2.7. If a compact manifold M admits a free and smooth action of a 

compact connected non-Abelian Lie group then M admits a Riemannian metric with 

positive Einstein curvature. 

Remark. In the case of nonfree actions, the previous results are not true, in fact S 2 

and S 2 S 1 (and probably S 2 T n , see Section 4) admit nontrivial and effective 

action of SO(3) but they do not admit any metric with positive Einstein curvature. 

Nevertheless, it is proved in [1] that 

THEOREM 2.8. If a compact manifold admits a nontrivial and smooth action of a 

compact connected simple Lie group of rank P2 then it admits a Riemannian metric 

with positive Einstein curvature. 

2.4. USING CONNECTED SUMS AND SURGERIES 

In [2], we proved the following stability theorem in the class of manifolds with 

positive Einstein curvature: 

THEOREM 2.9. If a compact manifold M is obtained from a manifold N by surgery 

in codimension P4, and N admits a metric with positive Einstein curvature then so 

does M. 

In particular, the connected sum of two manifolds with dimension P4 and having 

each one a metric with positive Einstein curvature has a metric with positive Einstein 

curvature. 

Using the previous theorem and the ideas of Gromov and Lawson [4], we proved 

in [2] the following consequences: 

THEOREM 2.10. (1) A compact 2-connected manifold of dimension P7 admits a 

metric with positive Einstein curvature if and only if it admits a metric with positive 

scalar curvature.


(2) Every compact, nonspin and simply connected manifold M of dimension P7 such 

that p 2 ðMÞ Z 2 admits a metric with positive Einstein curvature. 

Remark. It follows from the work of Sha and Yang [5] and the previous theorem 

that a compact simply connected manifold of dimension 4 is homeomorphic to a 

Riemannian manifold with positive Einstein curvature if and only if it is homeomorphic 

to a manifold with positive Ricci curvature and consequently if and only if 

it is homeomorphic to a manifold with positive scalar curvature. 

3. The Fundamental Group of Manifolds with Positive Einstein Curvature 

Since for any compact manifold M, the product S 3 M always admits a metric with 

positive Einstein curvature, then any group can be made as the fundamental group 

of a compact manifold of dimension P7 with positive Einstein curvature. 

More generally, there are no obstructions on the fundamental group of compact 

manifolds of dimension P5 with positive Einstein curvature as it is shown by the 

following theorem: 

THEOREM 3.1. Let G be a finitely presented group. Then for every nP5, there exists 

a compact n-manifold M with positive Einstein curvature such that p 1 ðMÞ ¼G. 

Proof. Let G be a group which has a presentation consisting of k generators 

x 1 ; x 2 ; ...; x k and 1 relations r 1 ; r 2 ; ...; r l : 

Let S 1 S n 1 be endowed with the standard product metric which is with positive 

Einstein curvature ðn 1P3Þ. Remark that the fundamental group of S 1 S n 1 is 

infinite cyclic. Hence, by taking the connected sum N of k-copies of S 1 S n 1 ,we 

obtain an orientable compact n-manifold with positive Einstein curvature. By Van 

Kampen’s theorem, The fundamental group of N is a free group on n-generators, 

which we may denote by x 1 ; x 2 ; ...; x k . 

We now perform surgery 1-times on the manifold N, Killing in succession the 

elements r 1 ; r 2 ; ...; r l . The result will be a compact, orientable n-manifold M with 

positive Einstein curvature (since the surgery is of codimension n 1P4) such that 

p 1 ðMÞ ¼G, as desired. ( 

4. Obstructions to Positive Einstein Curvature 

In celebrated papers [6] and [7], Schoen and Yau used minimal surfaces techniques to 

obtain obstructions to positive scalar curvature. The key point of their work is the 

following: 

THEOREM 4.1. Let ðM; gÞ be a compact Riemannian manifold with positive scalar 

curvature such that dim M ¼ mP3. If V is a compact smooth ðm 1Þ-dimensional 

immersed submanifold of M with trivial normal bundle, and if V is a local minimum of 

the ðm 1Þ-volume, then V admits a metric with positive scalar curvature.

214 

M.-L. LABBI 

This theorem can be generalized to the case of positive Einstein curvature as follows: 

THEOREM 4.2. Let ðM; gÞ be a compact Riemannian manifold with positive Einstein 

curvature such that dim M ¼ mP4. If V is a compact smooth ðm 2Þ-dimensional 

immersed submanifold of M with globally flat normal bundle, and if V is a local 

minimum of the ðm 2Þ-volume, then V admits a metric with positive scalar curvature. 

Remark. By a globally flat normal bundle we mean that the normal bundle to V 

has two globally parallel and orthonormal sections. 

Proof. Let V be a compact smooth n-dimensional immersed submanifold of M 

with codimension 2 and with globally flat normal bundle. 

Let x be an arbitrary point in V, we denote by B x the second fundamental form of 

V in M at the point x. Recall that 

B x : T x V T x V ! N x V; 

where T x V (resp. N x VÞ denotes the tangent space of V at x (resp. the normal space of 

V at x). Below we will make under a bar the invariants of ðM; gÞ. Let ðe 1 ; e 2 ; ...; e n Þ 

be an arbitrary orthonormal basis of T x V, and let W be an arbitrary normal vector 

field on V, then since the Riemannian submanifold V is a local minimum for the 

ðm 2Þ-volume we have (see for example [8]), 

X n 

i¼1 

Z 

V 

B x ðe i ; e i Þ¼0; 

fkr N Wk 2 hB 2 W; Wi hRðWÞ; Wig d volP0; ð2Þ 

where r N is the normal connection of the normal bundle and 

kr N Wk 2 ¼ Xn 

kr N e i 

Wk 2 ; 

i¼1 

hB 2 W; Wi ¼ Xn 

i;j¼1 

hRðWÞ; Wi ¼ Xn 

j¼1 

hBðe i ; e j Þ; Wi 2 ; 

Rðe j ; W; e j ; WÞ: 

Now, Gauss equation implies that 

Kðe j ; e k Þ¼ Kðe j ; e k ÞþhB x ðe j ; e j Þ; B x ðe k ; e k Þi hB x ðe j ; e k Þ; B x ðe j ; e k Þi 

and therefore 

X n 

Kðe j ;e k Þ¼ Xn 

Kðe j ;e k Þþ Xn 

2 

X n 

B x ðe j ;e k Þ 

hB 

 

 

x ðe j ;e k Þ;B x ðe j ;e k Þi 

j;k¼1 

j;k¼1 

¼ Xn 

j;k¼1 

Kðe j ;e k Þ 

j¼1 

X n 

j;k¼1 

j;k¼1 

hB x ðe j ;e k Þ;B x ðe j ;e k Þi ¼ Xn 

j;k¼1 

Kðe j ;e k Þ kBk 2 ; 

ð1Þ


where we used Equation (1). Consequently, if W 1 , W 2 are two orthonormal and 

parallel vector fields to V, we have 

hB 2 W 1 ; W 1 iþhRðW 1 Þ; W 1 i 

¼kBk 2 

¼ 1 2 kBk2 þ 1 2 kBk2 

¼ 1 2 

X n 

j;k¼1 

X n 

¼ 1 2 

j;k¼1 

hB 2 W 2 ; W 2 iþ Xn 

Kðe j ; e k Þ 

hB 2 W 2 ; W 2 iþ Xn 

Kðe j ; e k Þþ Xn 

j¼1 

Kðe j ; W 1 Þ 

hB 2 W 2 ; W 2 iþ Xn 

X n 

j;k¼1 

j;k¼1 

j¼1 

j¼1 


! 

Kðe j ; e k Þ þ 1 2 kBk2 



1 

2 scalV þ 

þ 1 2 kBk2 

hB 2 W 2 ; W 2 i 

¼ 1 2 eðW 2Þ 

1 

2 scalV þ 1 2 ðhB2 W 1 ; W 1 i hB 2 W 2 ; W 2 iÞ 

Therefore Equation (2) becomes 

Z 

feðW 2 Þ scal V þðhB 2 W 1 ; W 1 i hB 2 W 2 ; W 2 iÞg d volO0 

V 

Since e > 0 and the last term can be supposed to be P0 (if not just we change W 1 by 

W 2 ) then R V scalV d vol > 0 and consequently V admits a metric with positive scalar 

curvature, as desired. 

5. Examples and Remarks 

(1) This result is not true if we replace positive Einstein curvature by positive scalar 

curvature, in fact the torus T n can be embedded in the required manner in 

S 2 T n which is with positive scalar curvature (we suppose that the radius of S 2 

is sufficiently small) but T n does not admit any metric with positive scalar 

curvature. 

(2) The manifold V ¼ S 2 S 1 can be embedded as in the theorem in S 2 S 2 S 1 

which is with with positive Einstein curvature, but the submanifold V does not 

admit any metric with positive Einstein curvature. This example shows that we 

cannot replace in the theorem positive scalar curvature by positive Einstein 

curvature for the sub-manifold V.

216 

M.-L. LABBI 

(3) The previous theorem remains true if we replace codimension 2 by codimension 

1 (that is for hypersurfaces) since positive Einstein curvature implies positive 

scalar curvature and then we apply the Shoen–Yau Theorem. However, the 

manifold V ¼ S 2 S 1 can be embedded as a minimal and stable hypersurface in 

S 3 S 1 which is with positive Einstein curvature but V does not admit any 

metric with positive Einstein curvature. 

6. Final Remarks and Open Problems 

(1) Conjecture: The product S 2 T 2 does not admit any metric with positive Einstein 

curvature. 

More generally, the product S 2 T n , nP2 does not admit any metric with 

positive Einstein curvature. The answer will be yes if one is able to prove that 

under every Riemmanian metric on S 2 T n always there exists a minimal stable 

immersion with globally flat normal bundle of the torus T n . 

(2) Another possible way to obtain obstructions to positive Einstein curvature is to 

find adequate vanishing theorems by the use of general Weitzenbok formulas 

and Dirac operators. 

It is known (see for example Appendix D in the famous book of Lawson and 

Michelsohn on spin geometry) that if X is a Spin c manifold with associated line 

bundle k on which a connection with curvature 2-form X is fixed, and if D is the 

Dirac operator on a bundle S of complex spinors on X with the canonical 

connection. Then the corresponding Weitzenbok formula can be written as 

follows: 

D 2 ¼r rþ 1 4 s þ i 2 X; 

where s denotes the scalar curvature of X. 

The term of order 0 in the previous formula is positive if s 2kXk > 0. A 

condition which a priori looks like positive Einstein curvature! 

(3) It is plausible that every manifold admits a metric with negative Einstein curvature. 

It would be interesting to have an adaptation of the proof of the similar 

result (due to Lohkamp, see [9]) concerning the existence of negative Ricci 

curvature on any manifold. 

References 

1. Labbi, M. L.: Actions des groupes de Lie presques simples et positivite´ de la p-courbure, 

Ann. Fac. Sci. Toulouse 5(2) (1997), 263–276. 

2. Labbi, M. L.: Stability of the p-curvature positivity under surgeries and manifolds with 

positive Einstein tensor, Ann. Global Anal. Geom. 15(4) (1997), 299–312. 

3. Besse, A. L.: Einstein Manifolds, Springer-Verlag, New York, 1987. 

4. Gromov, M. and Lawson, H. B.: The classification of simply connected manifolds of 

positive scalar curvature, Ann. Math. 111 (1980), 423–434.


5. Sha, J. P. and Yang, D.: Positive Ricci curvature on compact simply connected 

4-manifolds, Proc. Sympos. Pure Math. (3) 54 (1993), 529–538. 

6. Schoen, R. and Yau, S. T.: On the structure of manifolds with positive scalar curvature, 

Manuscripta Math. 28 (1979), 159–183. 

7. Schoen, R. and Yau, S. T.: Existence of incompressible minimal surfaces and the topology 

of three dimensional manifolds with non-negative scalar curvature, Ann. Math. 110 (1979), 

127–142. 

8. Lawson, H. B. and Bourguignon, J. P.: Formules de variations de l’aire et applications, 

Aste´risque No. 154–155 (1987), 8; 51–71 (1988), 349. 

9. Lohkamp, J.: Metrics of negative Ricci curvature, Ann. Math. (2) 140(3) (1994), 655–683.

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