# Invariant Metrics with Nonnegative Curvature on Compact Lie Groups

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1 Canad. Math. Bull. Vol. 50 (1), 2007 pp Invariant Metrics with Nonnegative Curvature on Compact Lie Groups Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp and Zhongtao Wu Abstract. We classify the left-invariant metrics with nonnegative sectional curvature on SO(3) and U (2). 1 Introduction 24 Every compact Lie group admits a bi-invariant metric, which has nonnegative sectional curvature. This observation is the starting point of almost all known constructions of nonnegatively and positively curved manifolds. The only exceptions are found in [1, 6, 12], and many of these exceptions admit a different metric which does come from a bi-invariant metric on a Lie group; see [10]. In order to generalize this cornerstone starting point, we consider the following problem: for each compact Lie group G, classify the left-invariant metrics on G which have nonnegative sectional curvature. The problem is well motivated because each such metric, g l, provides a new starting point for several known constructions. For example, suppose H G is a closed subgroup. The left-action of H on (G, g l ) is by isometries. The quotient space H\G inherits a metric of nonnegative curvature, which is generally inhomogeneous. Geroch proved [5] that this quotient metric can only have positive curvature if H\G admits a normal homogeneous metric with positive curvature, so new examples of positive curvature cannot be found among such metrics. However, it seems worthwhile to search for new examples with quasi- or almost-positive curvature (meaning positive curvature on an open set, or on an open and dense set). A second type of example occurs when H G is a closed subgroup, and F is a compact Riemannian manifold of nonnegative curvature on which H acts isometrically. Then H acts diagonally on (G, g l ) F. The quotient, M = H\(G F), inherits a metric with nonnegative curvature. M is the total space of a homogeneous F-bundle over H\G. Examples of this type with quasi-positive curvature are given in [9]. It is not known whether new positively curved examples of this type could exist. In this paper, we explore the problem for small-dimensional Lie groups. In Section 3, we classify the left-invariant metrics on SO(3) with nonnegative curvature. This case is very special because the bi-invariant metric has positive curvature, and therefore so does any nearby metric. The solution is straightforward and known to Received by the editors January 6, 2005; revised June 3, Supported in part by NSF grant DMS and NSF grant DMS AMS subject classification: 53C20. c Canadian Mathematical Society 2007.

2 Invariant Metrics With Nonnegative Curvature on Compact Lie Groups 25 some, but does not appear in the literature. In Section 4, we classify the left-invariant metrics on U (2) with nonnegative curvature. This classification does not quite reduce to the previous, since not all such metrics lift to product metrics on U (2) s double-cover S 1 SU(2). The only examples in the literature of left-invariant metrics with nonnegative curvature on compact Lie groups are obtained from a bi-invariant metric by shrinking along a subalgebra (or a sequence of nested subalgebras). We review this construction in Section 5. Do all examples arise in this way? In Section 6, we outline a few ways to make this question precise. Even for U (2), the answer is essentially no. Also, since each of our metrics on U (2) induces via Cheeger s method a left-invariant metric with nonnegative curvature on any compact Lie group which contains a subgroup isomorphic to U (2), our classification for small-dimensional groups informs the question for large-dimensional groups. 2 Background Let G be a compact Lie group with Lie algebra g. Let g 0 be a bi-invariant metric on G. Let g l be a left-invariant metric on G. The values of g 0 and g l at the identity are inner products on g which we denote as, 0 and, l. We can always express the latter in terms of the former: A, B l = φ(a), B 0 for some positive-definite self-adjoint φ: g g. The eigenvalues and eigenvectors of φ are called eigenvalues and eigenvectors vectors of the metric g l. There are several existing formulas for the curvature tensor of g l at the identity, which we denote as R: g g g g. Püttmann s formula [8] says that for all x, y, z, w g, (2.1) R(x, y)z, w l = 1 ( [φx, y], [z, w] 0 + [x, φy], [z, w] 0 4 ) + [x, y], [φz, w] 0 + [x, y], [z, φw] 0 1 ( ) [x, w], [y, z] l [x, z], [y, w] l 2 [x, y], [z, w] l 4 ( B(x, w), φ 1 B(y, z) 0 B(x, z), φ 1 B(y, w) 0 ), where B is defined by B(x, y) = 1 2 ([x, φy] + [y, φx]). Sectional curvatures can be calculated from this formula. An alternative formula by Milnor for the sectional curvatures of g l is found in [7]. If {e 1,...,e n } g is a g l -orthonormal basis, then the sectional curvature of the plane spanned by e 1 and e 2 is: n ( 1 (2.2) κ(e 1, e 2 ) = 2 α 12k ( α 12k + α 2k1 + α k12 ) k=1 1 4 (α 12k α 2k1 + α k12 ) (α 12k + α 2k1 α k12 ) α k11 α k22 ),

3 26 N. Brown, R. Finck, M. Spencer, K. Tapp, Z. Wu where α i jk = [e i, e j ], e k l are the structure constants. 3 Metrics on SO(3) Let G = SO(3), which has Lie algebra g = so(3). We use the following bi-invariant metric: A, B 0 = (1/2) trace(ab T ) for A, B g. A left-invariant metric g l on G has three eigenvalues, {λ 2 1, λ 2 2, λ 2 3}. The metric can be re-scaled so that λ 3 = 1. Proposition 3.1 A left-invariant metric on SO(3) with eigenvalues {λ 2 1, λ 2 2, 1} has nonnegative sectional curvature if and only if the following three inequalities hold: 2λ λ λ 4 1 2λ 2 1λ λ 4 2 0, (3.1) 2λ 2 1 3λ λ 2 1λ 2 2 2λ λ 4 2 0, 2λ 2 2 3λ λ 2 1λ 2 2 2λ λ λ 2 λ λ 1 λ 1 Figure 1: The values of λ 1 and λ 2 for which SO(3) has nonnegative sectional curvature, when λ 3 = 1 (left) and when λ i = 1 (right) The intersection of the graph in Figure 1 (left) with the identity function λ 1 = λ 2 is interesting. The eigenvalues {λ 2, λ 2, 1} yield nonnegative curvature if and only if λ 2 3/4. Re-scaled, this means that the eigenvalues {1, 1, µ 2 } yield nonnegative curvature if and only if µ 2 4/3. Metrics with two equal eigenvalues on SO(3) (or on its double-cover S 3 ) are commonly called Berger metrics. They are obtained by scaling the Hopf fibers by a factor of µ 2. The nonnegative curvature cut-off µ 2 4/3 is well known.

4 Invariant Metrics With Nonnegative Curvature on Compact Lie Groups 27 Proof Let g l be a left-invariant metric on SO(3). Let {u 1, u 2, u 3 } g denote an oriented g 0 -orthonormal basis of eigenvectors of g l, with eigenvalues {λ 2 1, λ 2 2, λ 2 3}. Then, { e 1 = u 1, e 2 = u 2, e 3 = u } 3 λ 1 λ 2 λ 3 is a g l -orthonormal basis of g. The Lie bracket structure of g is given by: (3.2) [u 1, u 2 ] = u 3, [u 2, u 3 ] = u 1, [u 3, u 1 ] = u 2. The structure constants, α i jk = [e i, e j ], e k l, are: α 123 = α 213 = λ 3 λ 1 λ 2, α 231 = α 321 = λ 1 λ 2 λ 3, α 312 = α 132 = λ 2 λ 1 λ 3, with all others equal to zero. Equation (2.2) gives: κ(e 1, e 2 ) = 1 λ ( 3 2 λ 1 λ 2 λ 3 λ 1 λ 2 + λ 1 λ 2 λ 3 + λ ) λ 1 λ 3 ( λ3 λ 1 + λ )( 2 λ3 + λ 1 λ ) 2. λ 1 λ 2 λ 2 λ 3 λ 1 λ 3 λ 1 λ 2 λ 2 λ 3 λ 1 λ 3 When λ 3 = 1, the inequality κ(e 1, e 2 ) 0 is equivalent to the first inequality of (3.1). The second and third inequalities are equivalent to κ(e 2, e 3 ) 0 and κ(e 1, e 3 ) 0, respectively. It remains to prove the following: if the planes spanned by pairs of {e 1, e 2, e 3 } are nonnegatively curved, then all planes are nonnegatively curved. It is immediate from Püttmann s formula (2.1) that R(e i, e j )e k = 0 whenever i, j, k are distinct. This implies that the curvature operator is diagonal in the basis {e 1 e 2, e 2 e 3, e 3 e 1 }. The result follows. To classify up to isometry the left-invariant metrics of nonnegative sectional curvature on SO(3) it remains to observe the following proposition: Proposition 3.2 Two left-invariant metrics on SO(3) are isometric if and only if they have the same eigenvalues. Proof Suppose first that f : (SO(3), g 1 ) (SO(3), g 2 ) is an isometry, where g 1 and g 2 are left-invariant metrics. We lose no generality in assuming that f (I) = I. The linear map 2 (g) 2 (g) induced by d f I : g g must send eigenvectors of the curvature operator of g 1 to eigenvectors of the curvature operator of g 2. Since these eigenvectors are wedge products of eigenvectors of the metrics, d f I : g g must send the eigenvectors of g 1 to the eigenvectors of g 2. Since d f I preserves lengths, the two metrics must have the same eigenvalues. Conversely, suppose that g 1 and g 2 are left-invariant metrics on SO(3) with eigenvectors E = {e 1, e 2, e 3 } and F = { f 1, f 2, f 3 }, respectively. Assume that the eigenvalues of the two metrics agree. The bases E and F can be chosen to be g 0 -orthonormal and to have the same orientation. There exists x SO(3) such that Ad x : g g sends E to F. It is straightforward to verify that C x : (SO(3), g 1 ) (SO(3), g 2 ) (conjugation by x) is an isometry.

5 28 N. Brown, R. Finck, M. Spencer, K. Tapp, Z. Wu 4 Metrics on U (2) Let G = U (1) SU(2). In this section, we classify the left-invariant metrics on G with nonnegative curvature. This is equivalent to solving the problem for U (2), which has the same Lie algebra. Let g 0 denote a bi-invariant metric on G. Suppose that g l is a left-invariant metric on G. The Lie algebra of G is g = u(1) su(2). The factors u(1) and su(2) are orthogonal with respect to g 0. If they are orthogonal with respect to g l, then g l is a product metric, so the problem reduces to the one from the previous section. We will see, however, that nonnegatively curved metrics need not be product metrics. This might be surprising, since when g l is nonnegatively curved, the splitting theorem [2, Ch. 4] implies that the pull-back of g l to the universal cover R SU(2) is isometric to a product metric. The subtlety is that the metric product structure and the group product structure need not agree. Further, the metric g l on G need only be locally isometric to a product metric, not globally. Let {u 1, u 2, u 3 } denote an orthonormal basis of eigenvectors of the restriction of g l to the su(2) factor. Let {λ 2 1, λ 2 2, λ 2 3} denote the corresponding eigenvalues, which we call restricted eigenvalues of g l. Notice that { e 1 = u 1 λ 1, e 2 = u 2 λ 2, e 3 = u 3 λ 3 } is g l -orthonormal. Let e 0 g span the u(1) factor. Let E 0 span the g l -orthogonal compliment of the su(2) factor, and be g l -unit length. Proposition 4.1 (G, g l ) has nonnegative curvature if and only if {λ 1, λ 2, λ 3 } satisfy the restrictions of Proposition 3.1, and one of the following conditions holds: (i) E 0 is parallel to e 0 (in this case, g l is a product metric); (ii) λ 1 = λ 2 = λ 3 (in this case, E 0 is arbitrary); (iii) λ 1 = λ 2 and E 0 span{e 0, e 3 } (or analogously if a different pair of λ s agree). We prove the proposition with a sequence of lemmas. Lemma 4.2 The map ad E0 : g g is skew-adjoint with respect to g l if and only if one of the three conditions of Proposition 4.1 holds. Proof Write E 0 = ae 0 + be 1 + ce 2 + de 3. In the g l -orthonormal basis {E 0, e 1, e 2, e 3 } of g, the structure constants α 0 jk = [E 0, e j ], e k l are as follows: α 012 = d λ 2 λ 1, α 021 = d λ 1 λ 2, α 023 = b λ 3 λ 2, α 032 = b λ 2 λ 3, α 031 = c λ 1 λ 3, α 013 = c λ 3 λ 1. Here we assume for convenience that g 0 is scaled such that the Lie bracket structure of the SU(2) factor is given by equation (3.2). The lemma follows by inspection, since ad E0 is skew-adjoint if and only if α 0i j = α 0 ji for all i, j.

7 30 N. Brown, R. Finck, M. Spencer, K. Tapp, Z. Wu locally isometric to untwisted (product) metrics. We end this section by explicitly exhibiting the local isometry between a twisted metric and a product metric. Suppose that g l is a twisted metric. Recall that {E 0, e 1, e 2, e 3 } is a g l -orthonormal basis of g. Let g l denote the pull-back of g l to the universal cover, R SU(2), of G. Let g denote the product metric on R SU(2) for which {e 0, e 1, e 2, e 3 } forms a g -orthonormal basis of g. Define f : (R SU(2), g ) (R SU(2), g l ) as follows: f (t, g) = flow from (0, g) for time t along E 0 = R e te 0 (0, g) = (at, ge tê 0 ), where E 0 = ae 0 + be 1 + ce 2 + de 3 and Ê 0 = be 1 + ce 2 + de 3. Proposition 4.5 If g l has nonnegative curvature, then f is an isometry. Proof First, (4.2) d f (t,g) (e 0 ) = d (a(t + s), ge (t+s)ê 0 ) = (a, (ge tê 0 )Ê 0 ), ds s=0 which is the value at f (t, g) of the left-invariant vector field E 0. So d f sends the left-invariant field e 0 to the left-invariant field E 0. Next, for i {1, 2, 3}, (4.3) d f (t,g) (e i g) = d f (t, e se i g) = d (at, e se i ge tê 0 ) = (0, e i (ge tê 0 )), ds s=0 ds s=0 which is the value at f (g, t) of the right-invariant vector field determined by e i. In other words, d f sends the right-invariant vector field determined by e i to itself. In summary, (4.2) implies that d f sends the g -unit-length field e 0 to the g l -unitlength field E 0. Further, (4.3) implies that d f sends the g -orthogonal compliment e 0 to the g l -orthogonal compliment of E 0 (both of which equal span{e 1, e 2, e 3 }). It remains to verify that the restriction d f : span{e 1, e 2, e 3 } span{e 1, e 2, e 3 } is an isometry. Under condition (ii) of Proposition 4.1, g restricts to a bi-invariant metric on the SU(2) factor. Under condition (iii), g restricts to a left-invariant and Ad H -invariant metric, where H = exp(span(ê 0 )). In either case, the metric has exactly enough right-invariance to give the desired result from (4.3). Propositions 3.2 and 4.5 together imply Corollary 4.6 Two left-invariant metrics with nonnegative curvature on U (1) SU(2) are locally isometric if and only if their restricted eigenvalues {λ 1, λ 2, λ 3 } are the same. These local isometries are really isometries between the universal covers. In general, they are not group isomorphisms. They do not generally descend to global isometries between twisted and product metrics on U (1) SU(2).

8 Invariant Metrics With Nonnegative Curvature on Compact Lie Groups 31 5 Review of Cheeger s Method In the literature, the only examples of left-invariant metrics with nonnegative curvature on compact Lie groups come from a construction which, in its greatest generality, is due to Cheeger [1]. In this section, we review Cheeger s method. Let H G be compact Lie groups with Lie algebras h g. Let g 0 be a leftinvariant and Ad H -invariant metric on G with nonnegative curvature. Let g H be a right-invariant metric on H with nonnegative curvature. The right diagonal action of H on (G, g 0 ) (H, g H ) is by isometries. The quotient is diffeomorphic to G and inherits a submersion metric, g 1, which is left-invariant and has nonnegative curvature: (G, g 1 ) = ((G, g 0 ) (H, g H ))/H. If g H is bi-invariant, then g 1 is Ad H -invariant. The metrics g 0 and g 1 agree orthogonal to h. To describe g 1 on h, let {e 1,...,e k } denote a g H -orthonormal basis of h. Let A and Ã denote the k-by-k matrices whose entries are a i j = g 0 (e i, e j ) and ã i j = g 1 (e i, e j ), respectively. The restrictions of g 1 and g 0 to h are shown in [1] to be related by the equation: (5.1) Ã = A(I + A) 1. A common special case occurs when g 0 is bi-invariant, and g H is a multiple, λ, of the restriction of g 0 to H. In this case, which was first studied in [4], g 1 restricted to h is obtained from g 0 by scaling by the factor t = λ/(1 + λ). In this case, g 1 is Ad H -invariant, and is therefore Ad K -invariant for any closed subgroup K H. So Cheeger s method can be applied again: (G, g 2 ) = ((G, g 1 ) (K, g K ))/K, and so on. Iterating Cheeger s method in this way is studied in [3]. In summary, whenever H 0 H 1 H l 1 H l = G is a chain of closed subgroups of G with Lie algebras h 0 h 1 h l 1 h l = g, one can apply Cheeger s method l-times. One chooses a starting bi-invariant metric g 0 on G and l constants {λ 0,...,λ l 1 }. The result is a new left-invariant metric with nonnegative curvature on G. The sub-algebras h i are separately scaled by factors determined by the λ s, with h i scaled a greater amount than h i+1. More precisely, the eigenvalues of the metric are a strictly increasing sequence t 0 < t 1 < < t l 1 < t l = 1. The λ s can be chosen to give any such strictly increasing sequence. The eigenspace of t 0 equals h 0. For i > 1, the eigenspace of t i equals the g 0 -orthogonal compliment of h i 1 in h i. 6 Do All Metrics Come From Cheeger s Method? Let G be a compact Lie group. Does every left-invariant metric on G with nonnegative curvature come from Cheeger s method? In this section, we outline several ways to precisely formulate this question, and address the cases G = SO(3) and G = U (2). First, one might ask whether all examples arise by starting with a bi-invariant metric g 0 on G and applying Cheeger s method to a chain of subgroups H 0 H l 1 G, each time choosing the metric on H i to be a multiple, λ i, of the restriction of g 0 to H i. The answer is clearly no, since not all metrics on SO(3) arise in this

9 32 N. Brown, R. Finck, M. Spencer, K. Tapp, Z. Wu fashion. But SO(3) is a very special case, since its bi-invariant metric has positive curvature. There are several ways to modify the question to reflect the guess that SO(3) is the only exception. What if, in each application of Cheeger s method, one allows more general rightinvariant metrics on the H i s? For example, if H i = SO(3) or SU(2), we allow any right-invariant metric with nonnegative curvature. With this added generality, the answer is still no, since metrics on SO(3) which are nonnegatively but not positively curved do not arise in this fashion: Proposition 6.1 If g 0 is a bi-invariant metric on SO(3) and g R is a right-invariant metric with nonnegative curvature on SO(3), then the following has strictly positive curvature: (SO(3), g l ) := ((SO(3), g 0 ) (SO(3), g R ))/ SO(3). Further, every positively-curved left-invariant metric g l can be described in this way for some bi-invariant metric g 0 and some positively curved right-invariant metric g R. Proof Let π : (SO(3), g 0 ) (SO(3), g R ) (SO(3), g l ) denote the projection, which is a Riemannian submersion. Let {e 1, e 2, e 3 } so(3) be a g 0 -orthonormal basis of eigenvectors of the metric g R, with eigenvalues denoted {λ 1, λ 2, λ 3 }. The horizontal space of π at the identity is the following subspace of so(3) so(3): (6.1) H (e,e) = {(V, W ) (V, W ), (e i, e i ) = 0 for all i = 1, 2, 3} = {(V, W ) V, e i 0 + λ i W, e i 0 = 0 for all i = 1, 2, 3}. Now let X 1 = (V 1, W 1 ), X 2 = (V 2, W 2 ) H (e,e) be linearly independent vectors. From (6.1), we see that V 1, V 2 are linearly independent as well. Since (SO(3), g 0 ) has positive curvature, κ(v 1, V 2 ) > 0, so κ(x 1, X 2 ) > 0. Finally, O Neill s formula implies that (SO(3), g l ) has strictly positive curvature. To prove the second statement of the proposition, notice that by (5.1), the eigenvalues { λ 1, λ 2, λ 3 } of the metric g l are determined from the eigenvalues {λ 1, λ 2, λ 3 } of the metric g R as follows: (6.2) λi = λ i 1 + λ i for each i = 1, 2, 3. Suppose that ( λ 1, λ 2 ) is an arbitrary pair such that the triplet { λ 1, λ 2, 1} strictly satisfies the inequalities of Proposition 3.1. Define λ 1 := a λ a(1 λ 1 ), λ a λ 2 2 := 1 + a(1 λ 2 ), λ 3 := a. For small enough a, the triplet {λ 1, λ 2, λ 3 } strictly satisfies the inequalities of Proposition 3.1. This is because, when scaled such that the third equals 1, the triplet ap-

10 Invariant Metrics With Nonnegative Curvature on Compact Lie Groups 33 proaches { λ 1, λ 2, 1} as a 0. If g R has eigenvalues {λ 1, λ 2, λ 3 }, then by equation 6.2, g l has the following eigenvalues: {( a ) ( a ) ( a )} λ1, λ2,. 1 + a 1 + a 1 + a Further, g R has positive curvature because the inner products which extend to right-invariant metrics with positive curvature are the same ones that extend to leftinvariant metrics of positive curvature. This is because a a 1 is an isometry between left- and right-invariant metrics determined by the same inner product. Thus, g R can be chosen such that g l is a multiple of any prescribed positively curved metric. Then, by scaling the bi-invariant metric g 0, this multiple can be made to be 1. Beginning with a bi-invariant metric g 0 on G = SU(2) U (1) and applying Cheeger s method, one cannot obtain product metrics for which the SU(2)-factor has nonnegative but not positive curvature, nor can one obtain twisted metrics of type (2). Notice that the only chains of increasing-dimension subgroups are U (1) G (possibly embedded diagonally), T 2 G (any maximal torus), U (1) T 2 G, and SU(2) G. To obtain more metrics, in addition to allowing general right-invariant metrics on the H i s as above, one could allow a more general starting metric g 0. For example, when G = SU(2) U (1), one could allow g 0 to be any left-invariant product metric with nonnegative curvature. Even with this added generality, it is straightforward to see that one cannot obtain twisted metrics of type (iii) for which the totally geodesic SU(2) has nonnegative but not positive curvature. Acknowledgement The fourth author is pleased to thank Wolfgang Ziller for suggesting Proposition 3.1 and for valuable feedback on the paper. References [1] J. Cheeger, Some examples of manifolds of nonnegative curvature. J. Differential Geometry 8(1973), [2] J. Cheeger and D. Gromoll, On the structure of complete open manifolds of nonnegative curvature. Ann. of Math. 96(1972), [3] W. Dickinson, Curvature properties of the positively curved Eschenburg spaces. Differential Geom. Appl. 20(2004), no. 1, [4] J. H. Eschenburg, Inhomogeneous spaces of positive curvature. Differential Geom. Appl. 2(1992), no. 1, [5] R. Geroch, Group-quotients with positive sectional curvatures. Proc. Amer. Math. Soc. 66(1977), no. 2, [6] K. Grove and W. Ziller, Curvature and symmetry of Milnor spheres. Ann. of Math. 152(2000), no. 1, [7] J. Milnor, Curvatures of left-invariant metrics on lie groups. Advances in Math. 21(1976), [8] T. Püttmann, Optimal pinching constants of odd dimensional homogeneous spaces. Ph. D. thesis, Ruhr-Universität, Germany, [9] K. Tapp, Quasi-positive curvature on homogeneous bundles. J. Differential Geom. 66(2003), no. 2, [10] B. Totaro, Cheeger manifolds and the classification of biquotients. J. Differential Geom. 61(2002), no. 3,

11 34 N. Brown, R. Finck, M. Spencer, K. Tapp, Z. Wu [11] N. Wallach, Compact homogenous Riemannian manifolds with strictly positive curvature. Ann. of Math. 96 (1972), [12] D. Yang, On complete metrics of nonnegative curvature on 2-plane bundles. Pacific J. Math. 171(1995), no. 2, Department of Mathematics Stanford University Stanford, CA U.S.A. Department of Mathematics Williams College Williamstown, MA U.S.A. Department of Mathematics Brown University Providence, RI College Road Princeton, MA U.S.A.

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