Einstein manifolds with skew torsion Ilka Agricola∗ & Ana Cristina Ferreiray ∗ Philipps-Universit¨atMarburg (Germany) y Centro de Matem´aticada Universidade do Minho (Portugal) 1. Abstract Proposition 2: Under the action of the orthogonal group, the Theorem 2: Let G be a Lie group equipped with a bi-invariant metric. r-curvature tensor decomposes as Consider the 1-parameter family of connections with skew torsion t We present a notion of Einstein manifolds with skew torsion on Riemannian r r Y = t[X; Y ]: r r 1 r ⊙ s ⊙ 1 X manifolds for all dimensions, initiated by the doctoral work of the second R = W + − (Z g) + − g g + dH n 2 n(n 1) 12 rt 6 author in dimension four. Then for t = 0 and t = 1, is Einstein (it is in fact flat) and for t = 0 and t =6 1, rt is Einstein if and only if rg is Einstein. So it is natural to set: Nearly K¨ahlerin dimension 6 2. Overview S6 ⊂ R7 has an almost complex structure J inherited from the \cross prod- uct" on R7. J is not integrable and it is still an open problem whether or not Definition: Let (M; g; H) be a Riemannian manifold equipped with S6 admits a complex structure. Here is the analogous picture for the 2-sphere • Torsion, and in particular skew torsion, has been a topic of interest to both a three-from H. We say that (M; g; H) is Einstein with skew torsion in R3: mathematicians and physicists in recent decades. H if the trace free part of the Ricci-tensor • The first attempts to introduce torsion in general relativity go back to the r 1920's with the work of E.´ Cartan. More recently, torsion makes its appearance r r s Z = Ric − g; in string theory, where the basic model for type II consists of a Riemannian n manifold, a connection with skew torsion, a spinorial field and a dilaton func- where r is the metric connection with skew torsion H, vanishes. tion. • From the mathematical point of view, skew torsion has played a signifi- cant role in the work of Bismut and his local index theorem for non-K¨ahler We also have the following remarkable property: manifolds. Skew torsion is also an important feature in generalized geometry, where there are two natural connections with skew torsion that come from the exterior derivative of the B-field. Theorem 1: If (M; g; H) is Einstein with parallel skew torsion H r • Torsion is also ubiquitous in the theory of non-integrable geometries. The then the scalar curvature s is constant. Even though J is not integrable, rgJ =6 0, we have that J is an example of idea is to choose a G-structure so that the G-connection with torsion admits a nearly K¨ahlerstructure: rg J(X) = 0 (g is the round metric). desired parallel objects, in particular spinors, interpreted as supersymmetry X For any nearly K¨ahlermanifold, the Gray connection transformations. As a first step in this investigation, T. Friedrich and S. Ivanov proved that many non-integrable geometric structures admit a unique 1 g g r Y + (r J)JY invariant connection with parallel totally anti-symmetric torsion, thus being a 5. r -Einstein and r-Einstein X 2 X natural replacement for the Levi-Civita connection. is a connection with totally skew symmetric torsion. In dimension 6, all nearly K¨ahlermanifolds are Einstein and of constant type so it can be easily proved An interesting question is what is the relation between the standard Einstein that the Gray connection is Einstein with skew torsion (as observed by T. condition and the Einstein condition with skew torsion. By looking at the Friedrich and S. Ivanov '02). 3. Metric connections with skew torsion formulas for the Ricci tensors, we see that if (M; g) is Einstein then (M; g; H) being Einstein depends only on the algebraic type of the three-form H. Up G2 T-manifolds in dimension 7 7 to dimension 6, we have a classification provided by Schouten which can be A G2 structure can be seen as a triple (M ; g; !) where ! is a 3-form of gen- Definition, existence and uniqueness easily illustrated by the following graphs. eral type at any point. A G2 T -manifold - G2 manifold with (skew) torsion - • Let (M; g) be a Riemannian manifold. Suppose that r is a connection on is such that there exists a one-form θ such that d ∗ ! = θ ^ ∗!. In this case, TM and let T be its (1,2) torsion tensor. If we contract T with the metric there exists a unique connection with totally skew symmetric torsion which we get a (0,3) tensor which we still call the torsion of r . If T is a three-form preserves both the metric g and the three-form !: then we say that r is a connection with skew-symmetric torsion. ( ) • 1 1 Given any three-form H on M then there exists a unique metric connection r = rg + − ∗ d! − g(d!; ∗!)! + ∗(θ ^ !) : with skew torsion H which is defined explicitly by 2 6 g 1 Furthermore, its torsion is proportional to ! and thus it is r-parallel. (T. g(rXY; Z) = g(r Y; Z) + H(X; Y; Z) X 2 Friedrich and S. Ivanov '02) g where r is the Levi-Civita connection. Let (M; g; !) be a G2 T -manifold which is also nearly parallel, that is, d! = λ ∗ !, for some λ 2 R. Then the M7 is Einstein and r is Einstein with skew torsion. Einstein-Sasaki in odd dimensions 4. Motivation and definition Let (M2k+1; g; '; ξ; η) be a Sasaki manifold. Then it admits a connection r with skew torsion H such that The obvious way to define the Einstein equations with skew torsion is simply rg = rη = r' = 0 to set Ricr = λg for some function λ. This, however, presents two immediate problems. The first is that the function λ might not be constant, and also given by the torsion H = η ^ dη and also we have that H is r-parallel. r that Ric might not be symmetric since In this particular instance, it is not too difficult to check that a Sasaki-Einstein r r g 1 1 ∗ metric never gives that is Einstein with skew torsion. Ric (X; Y ) = Ric (X; Y ) − g(H(e ;X);H(e ;Y )) − d H(X; Y ); 2 4 i i 2 However, if we consider the Tanno deformation gt = tg+(t −t)η⊗η, t > 0, of 2k+1 1 g in the direction of ξ, then (M ; gt; ξt; '; ηt), where ξt = ξ and ηt = tη, where feig is an orthonormal frame of the tangent bundle; and requiring H t to be co-closed is, a priori, too restrictive. is also a Sasaki manifold and we can prove: The standard Einstein equations of Riemannian geometry can be obtain by a variational argument. They are the critical points of the Hilbert functional Theorem 3: Let (M; g; ξ; '; η) be an Einstein-Sasaki manifold. Con- Z It can be checked by direct computation that type I. works in dimension 3, sidering the 1-parameter family (M; gt; ξt; '; ηt) described above, there g g 7−! (s − 2Λ) dvolg; and types IV. and V. work in dimension 6. Types II. and III. never work, so in exists a t > 0 such that the connection M particular an Einstein metric never gives an Einstein metric with skew torsion 1 where Λ is the cosmological constant. rt = rgt + η ^ dη in dimensions 4 and 5. 2 t t So one way of obtaining Einstein equations with skew torsion is to look for the critical points of the following functional is rt-Ricci flat (but not flat). Z Z ( ) ( ) 3 7−! r − g − 3k k2 − Example: Consider the 3-dimensional sphere S and take g to be the (g; H) s 2Λ dvolg = s H 2Λ dvolg: 3 g M M 2 round metric. Then (S ; g) is Einstein with s = 6. Consider a global 3 frame fe1; e2; e3g of TS , f any non-constant smooth function and define Acknowledgements the three-form H by 1 ^ 2 ^ 3 Proposition 1: The critical points of the functional H = 2fe e e : The first author was supported by DFG grant 92/3-1. Z ( ) Then the connection given by r = rg + 1H is Einstein with skew torsion The second author would like to thank the Differential Geometry and Analysis r 2 L(g; H) = s − 2Λ dvolg and group at Philipps University - where this project was carried out - for its hos- r 3 M s = sg − kHk2 = 6 − 6f(x)2 pitality. Acknowlegments are also due to the Research Center of Mathematics r 2 at the University of Minho. are given by pairs (g; H) such that the Ricci tensor Ric is symmetric which is clearly not a constant. This research was funded by the Portuguese Foundation for Science and and satisfies the equation Technology (FCT) - project PTDC/MAT/118682/2010 - and the Deutscher r 1 r Akademischer Austausch Dienst (DAAD). −Ric + s g − Λg = 0: 2 References: [1] I. Agricola, The Srn´ılectures on non-integrable geometries with torsion, 6. Special geometries Arch. Math., Brno, 42(5):5-84, 2006. In the context of non-integrable geometries, requiring that H is r parallel is [2] I. Agricola, A.C. Ferreira, Einstein manifolds with skew torsion, in prepa- a natural condition and this will then imply that H is co-closed.