EINSTEIN METRICS. Andrzej Derdzinski. The Ohio State University, Columbus, Ohio, USA. August 8, 2009

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1 The Ohio State University, Columbus, Ohio, USA August 8, 2009 Workshop on Riemannian and Non-Riemannian Geometry Indiana University - Purdue University, Indianapolis August 8-9, 2009 these notes are posted at andrzej/indianapolis.pdf

2 An Einstein metric on a manifold M of dimension n 1: any pseudo-riemannian metric g on M with Ric = κg for some κ IR (the Einstein constant of g). One refers to such (M, g) as an Einstein manifold, and calls it Ricci-flat if κ = 0. Obviously, κ = Scal/n. If n 2, one may equivalently require that Ric = κg for a function κ : M IR, as κ must be constant by a version of Schur s lemma (the Bianchi identity for Ric). For n = 2, one always has Ric = κg with a function κ (the Gaussian curvature of g). p.1

3 MOTIVATION naturality: the simplest (nonlinear) eigenvalue condition on g ( Einstein metrics are the harmonic oscillators of Riemannian geometry ); optimal (Riemannian) metrics: e.g., on closed surfaces or compact complex manifolds with c 1 < 0; the Ricci flow: Einstein metrics are its fixed points (modulo scaling); general relativity: vacuum solutions of the Einstein equations with or without a cosmological constant (κ 0 or κ = 0). p.2

4 CURVATURE DECOMPOSITION A pseudo-riemannian manifold (M, g) of dimension n 1. The Levi-Civita connection of (M, g), its curvature tensor R, Ricci tensor Ric, Einstein (traceless Ricci) tensor Ein. Irreducible components of R under the action of the pseudo-orthogonal group: R = S + Z + W curvature = scalar + traceless Ricci + Weyl. In dimensions n 3 one has W = 0. For n 2, Einstein metrics are characterized by Z = 0. andrzej/indianapolis.pdf p.3

5 ACTION ON 2-FORMS In terms of the curvature operators one has R, S, Z, W : [T M] 2 [T M] 2, S = Scal n(n 1) (times Id), Z = {Ein, } n 2 (the anticommutator). tr W = 0. p.4

6 THE HODGE STAR Let n = 4. Up to a sign change, g has three possible sign patterns: ++++, ++, +++ Riemannian, neutral, Lorentzian. The Hodge star operator : [T M] 2 [T M] 2 (if M is oriented): (e 1 e 2 ) = ε 3 ε 4 e 3 e 4 whenever e j are orthonormal and ε j = g(e j, e j ). p.5

7 (ANTI) SELF-DUALITY For oriented (M, g), with n = 4, we have 2 = Id (Riemannian/neutral), 2 = Id (Lorentzian). Lorentzian: [T M] 2 is a complex 3-plane bundle. Riemannian/neutral: [T M] 2 = Λ + M Λ M, where the real 3-plane bundles Λ ± M are the ±1-eigenspace bundles of (the bundles of self-dual and anti-self-dual 2-forms). andrzej/indianapolis.pdf p.6

8 THE SINGER-THORPE MATRIX If n = 4 and (M, g) is oriented, [W, ] = 0. Lorentzian: W : [T M] 2 [T M] 2 is complex-linear. Riemannian/neutral: Λ ± M are W-invariant. One calls (M, g) self-dual if W = 0, where W ± denote the restrictions of W to Λ ± M, trivially extended to the other summand. We have tr W ± = 0. As shown by Singer and Thorpe (1969), relative to the decomposition [T M] 2 = Λ + M Λ M, R = W Scal Z. Z W Scal p.7

9 THE SIMPLEST EXAMPLES flat = Ricci-flat (here belongs the case n = 1); closed surfaces of constant Gaussian curvature: round S 2 and IRP 2, flat tori and Klein bottles, higher genus hyperbolic surfaces; spaces of constant curvature (in dimensions n 3 these are the only Einstein manifolds); Riemannian homogeneous spaces with an irreducible isotropy representation (e.g., CP n and IHP n with their standard metrics); p.8

10 FURTHER SIMPLE EXAMPLES torsionfree connections with the property that the Ricci tensor Ric is symmetric, -parallel and nondegenerate. Here g = Ric is a (pseudo-riemannian) Einstein metric and denotes its Levi-Civita connection. This is an equivalent definition of non-ricci-flat pseudo- Riemannian Einstein metrics, up to rescaling. andrzej/indianapolis.pdf p.9

11 A SPECIAL CASE Let H be a connected Lie group, with the Lie algebra h (consisting of all left-invariant vector fields on H). The Killing form B : a left-invariant symmetric 2-tensor on h, with B(v, w) = tr (Ad v)(ad w) for v, w h, where, as usual, Ad v = [v, ] : h h. By requiring that v w = 1 [v, w] whenever v, w h, 2 we define a torsionfree connection with the Ricci tensor Ric = 1 B, so that Ric = 0. 4 p.10

12 Namely, B( u v, w) + B(v, u w) = 0 due to bi-invariance of B, i.e., skew-symmetry of B([u, v], w) in u, v, w. Therefore: For any semisimple Lie group H, the Killing form B of H is a bi-invariant Einstein metric on H. In the above example, B is also locally symmetric in the sense that R = 0. Local symmetry implies that Ric = 0, and so, in the Riemannian case, locally symmetric manifolds are (locally) products of locally symmetric Einstein manifolds. p.11

13 CALABI S CONJECTURE (1954) Let M be a compact complex manifold admitting a Kähler metric, and let c 1 be the first Chern class of M. If c 1 < 0, then M also admits a Kähler-Einstein metric, and such a metric is unique up to a constant factor. If c 1 = 0, every positive cohomology class ω H 2 (M, IR) contains a unique Kähler form representing a Ricci-flat Kähler metric. The first part of the conjecture was proved, independently, by Aubin (1976) and Yau (1977); the second, by Yau (1977). andrzej/indianapolis.pdf p.12

14 THE CHEN-LEBRUN-WEBER METRIC Chen, LeBrun and Weber (2008) proved the following theorem. The compact complex surface obtained from CP 2 by blowing up two distinct points admits an Einstein metric g with positive Einstein constant, which is globally conformal to a Kähler metric. The Chen-LeBrun-Weber metric g itself is not a Kähler metric; in fact, the underlying complex surface does not admit any Kähler-Einstein metric. A Kähler metric g globally conformal to g can be written down explicitly: g = W + 2/3 g, where is the g-norm and W + is the self-dual Weyl tensor of g. p.13

15 THE THORPE-HITCHIN INEQUALITY For any compact orientable Einstein four-manifold (M, g), τ(m) 2 3 χ(m), and the inequality is strict except when either g is flat, or (M, g) admits, for some r {1, 2, 4}, an r-fold Riemannian covering by a K 3 surface with a Ricci-flat Kähler metric. The inequality itself is due to Thorpe (1969), while the equality case was settled by Hitchin (1974). p.14

16 CURVATURE AND CHARACTERISTIC NUMBERS The Thorpe-Hitchin inequality (without the equality part) follows from the curvature expressions for χ(m) and τ(m), valid for any compact Riemannian four-manifold (M, g) (oriented in the case of τ(m)): 192π 2 χ(m) = 24 W 2 12 Ein 2 + Scal 2, where denotes the L 2 norm, and These equalities give 12π 2 τ(m) = W + 2 W 2,. 96π 2 [2χ(M) + 3τ(M)] = 48 W Ein 2 + Scal 2. andrzej/indianapolis.pdf p.15

17 ALEKSEEVSKY S CONJECTURE (1975) Alekseevsky made the following (still wide-open) conjecture: Whenever G/K is homogeneous Riemannian Einstein manifold with negative Einstein constant, K must be a maximal compact subgroup of G. Here is a special case (also open): Every connected Lie group carrying a left-invariant Riemannian Einstein metric with negative Einstein constant is solvable. p.16

18 Another unsolved special case: SL(n, IR), for any n 3, admits no left-invariant Riemannian Einstein metric. However, Leite and Dotti (1982) showed that SL(n, IR), for every n 3, does admit a left-invariant Riemannian metric of negative Ricci curvature. If Alekseevsky s conjecture is true, the task of classifying homogeneous Riemannian Einstein manifolds will be nearly completed. p.17

19 PETROV S CANONICAL FORMS Suppose that IK = IR or IK = C, F is a 3-dimensional vector space over IK,, is a nondegenerate IK-bilinear symmetric form on F, if IK = IR, the sign pattern of, is +++ or +. W : F F is IK-linear, traceless and, -self-adjoint. We set δ = 1 if, has the sign pattern + (so that IK = IR), and δ = 1 otherwise. Petrov (1950) described canonical forms of such pairs (,, W ): andrzej/indianapolis.pdf p.18

20 δ 0 0 λ 0 0, = 0 δ 0, W = 0 µ 0, λ, µ, ν IK, λ + µ + ν = 0, ν λ 0 ε, = 0 δ 0, W = 0 2λ 0, λ IK, ε = ± 1, λ δ 0, = 0 δ 0, W = 0 0 1, p 0 0, = 0 0 1, W = 0 p q, p, q IK = IR, q q p The first three cases, with IK = C (and δ = 1) are usually referred to as Petrov s types I, II, III. p.19

21 THE PETROV-SEGRE CLASSES 1, 100, 11, 111, 2, 21, 3. The dimensions of the different eigenspaces of W are listed in decreasing order, without commas to separate them. Subclass 2 + : Second line with λ = 0 and ε = +1, Subclass 2 : Second line with λ = 0 and ε = 1, Subclass 21 + : First line with δ = 1 and µ = λ 0, ν = 2λ, Subclass 21 : First line with δ = 1 and ν = λ 0, µ = 2λ. p.20

22 CLASSES VS. GENERA In the Riemannian/neutral cases, instead of just Petrov-Segre clases, one needs Petrov-Segre genera, that is, pairs of (sub)classes, separated by a slash, to account for W + and W. There are three Riemannian Petrov-Segre genera: 3/3, 3/21, 21/21, six Lorentzian Petrov-Segre classes: 3, 21, 2, 111, 11, 1, and as many as forty-five neutral genera, for instance 3/3, 3/21 +, 21 /2,..., 1/1. andrzej/indianapolis.pdf p.21

23 THE LOCALLY-SYMMETRIC CASE Here are the classes/genera realized by locally-symmetric Einstein four-manifolds. Riemannian (Cartan, 1926): 3/3, 3/21, 21/21. Geometrically: spaces of constant curvature; nonflat spaces of constant holomorphic sectional curvature; products of two surfaces with equal nonzero constant Gaussian curvatures. Lorentzian (Petrov, 1950): 3, 21, 2. Neutral (Cahen and Parker, 1980): 3/3, 3/21 ±, 21 ± /21 ±, 21 + /21, 3/2 ±, 2 ± /2 ±, 2 + /2. p.22

24 CURVATURE-HOMOGENEITY A pseudo-riemannian manifold (M, g) is called curvature homogeneous if the algebraic type of its metric/curvature pair (g, R) is the same at all points, i.e., if for any x, y M some isomorphism T x M T y M takes (g x, R x ) to (g y, R y ). Local homogeneity implies curvature-homogeneity, but not vice versa. Problem: Describe all curvature-homogeneous Einstein fourmanifolds of any metric signature. p.23

25 THE RIEMANNIAN CASE The answer (D., 1983): here, curvature-homogeneity local symmetry. The classes/genera (Cartan, 1926): 3/3, 3/21, 21/21. Jensen (1969) showed that, for Riemannian Einstein fourmanifolds, local homogeneity is equivalent to local symmetry. andrzej/indianapolis.pdf p.24

26 THE DIAGONALIZABLE CASE More generally (D., 2003), if R, or W, is complex-diagonalizable: the manifold in question must be locally homogeneous, and either locally symmetric, or (for both indefinite sign patterns) locally isometric to a specific Lie group with a left-invariant metric. p.25

27 THE LORENTZIAN CASE If W is diagonalizable, the preceding result applies. This settles the question for classes 3, 21, 111. Brans (1971) described all such Lorentzian metrics which also belong to class 1 (that is, the Petrov type III) Classes 2 and 11 remain to be classified. p.26

28 THE NEUTRAL CASE, GENUS 3/1 One also speaks here of type III SDNE manifolds, which is short for self-dual neutral Einstein four-manifolds of Petrov type III. Another terms used is type III Jordan-Osserman manifolds of dimension four. The local structure of all such manifolds, at points in general position, can be described as explained below. There are two separate description, corresponding to two possible situations: the Walker case (Díaz-Ramos, García-Río and Vázquez- Lorenzo, 2006), and the strictly non-walker case (D., 2009). andrzej/indianapolis.pdf p.27

29 Every type III SDNE manifold carries a distinguished twodimensional null distribution V, which, in addition, is integrable and has totally geodesic leaves. Namely, W + acting on self-dual 2-forms has rank 2 at every point x, and hence its kernel is one-dimensional. Thus, we may declare V x to be the nullspace of some, or any, self-dual 2-form at x spanning Ker W + x. Given a type III SDNE manifold (M, g), we say that g is a Walker metric if the null distribution V is parallel; g is strictly non-walker if the fundamental tensor of V (which measures its deviation from being parallel) is nonzero everywhere. p.28

30 Usually, by a Walker metric one means a pseudo-riemannian metric g, on the underlying manifold of which there exists a nonzero, g-null, g-parallel distribution. Our terminology is justified by the following fact: A type III SDNE manifold (M, g) admits a two-dimensional null parallel distribution compatible with the orientation if and only the null distribution V defined earlier is parallel. Comments: for any type III SDNE manifold (M, g), the null distribution V is compatible with the orientation; if a two-dimensional null parallel distribution compatible with the orientation exists on (M, g), it must coincide with V. p.29

31 TWO CONSTRUCTIONS Two constructions of type III SDNE manifolds are known. One, always leading to Walker metrics, was discovered by Díaz- Ramos, García-Río and Vázquez-Lorenzo (2006). The other construction, always resulting in strictly non-walker metrics, will be outlined here. A special case of Díaz-Ramos, García-Río and Vázquez-Lorenzo s construction was first described by García-Río, Kupeli, Vázquez- Abal and Vázquez-Lorenzo (1999). andrzej/indianapolis.pdf p.30

32 UNIVERSAL MODELS The two constructions just mentioned yield universal models: at points in general position, every type III SDNE manifold is locally isometric to one of the examples resulting from these constructions. Universality of the Walker examples is a result of Díaz-Ramos, García-Río and Vázquez-Lorenzo (2006). Next step: an outline of the two constructions, preceded by a definition. p.31

33 RIEMANN EXTENSIONS Let be a connection on a manifold Σ (of any dimension). We denote by π : T Σ Σ the bundle projection of the cotangent bundle of Σ. The Patterson-Walker Riemann extension metric on T Σ is the neutral-signature pseudo-riemannian metric g on the manifold M = T Σ, defined by requiring that all vertical and all -horizontal vectors be g -null, while g x (ξ, w) = ξ(dπ x w) for any x M = T Σ, any vertical vector ξ Ker dπ x = T y Σ, with y = π(x), and any w T x M. p.32

34 UNIVERSAL MODELS: THE WALKER CASE (Díaz-Ramos, García-Río and Vázquez-Lorenzo, 2006) Given: a surface Σ, a connection on Σ, and a twice-covariant symmetric tensor field τ on Σ, we define (M, g) by M = T Σ, g = g + π τ. Here π : T Σ Σ is the bundle projection and g denotes the Patterson-Walker Riemann extension metric associated with. If is torsionfree and, at every point of Σ, the Ricci tensor of is skew-symmetric and nonzero, then, for a suitable orientation of M, this g is a type III SDNE Walker metric. andrzej/indianapolis.pdf p.33

35 SKEW-SYMMETRY OF THE RICCI TENSOR The local structure of torsionfree surface connections with skewsymmetric Ricci tensor was described by Wong (1964): A torsionfree connection on a surface Σ has skew-symmetric Ricci tensor if and only if, on some neighborhood of any point of Σ, there exist coordinates in which the component functions of are Γ11 1 = 1 ϕ, Γ 22 2 = 2ϕ for a function ϕ, and Γ i jk = 0 unless i = j = k. The Ricci tensor of then is given by R 12 = 1 2 ϕ. p.34

36 Wong s paper provided three coordinate expressions which, locally, represent all torsionfree surface connections such that Ric is skew-symmetric and nonzero everywhere. The above version is simplified: Ric is allowed to vanish, and the three coordinate forms are replaced with just one. p.35

37 STRICTLY NON-WALKER UNIVERSAL MODELS Let there be given a contractible open set U in IR 2, a real constant K, a real-plane vector bundle P over the surface U, a section Ω of [P ] 2 without zeros, a connection in P with Ω = 0, a twice-covariant symmetric tensor field τ on Σ, sections c, q of P such that c has no zeros and R ( 1, 2 ) = KΩ(c, )q +KΩ(q, )c, Ω(c, 1 1 c 2 2 q) = 1. Here R is the curvature tensor of, and j denotes the -covariant derivative in the direction of the coordinate vector field j on U. andrzej/indianapolis.pdf p.36

38 Next, we it trivialize P, in the sense of identifying it with the product bundle U Π, for a vector plane Π. We choose the identification so that both c and Ω become onstant functions on U (valued in Π and [Π ] 2 ). In general, sections of P are now treated as functions U Π. The symbol Π + denotes the open half-plane in Π on which Ω(, c) > 0. The four-dimensional product M = Σ Π + will be the underlying manifold of the metric g, defined below. p.37

39 The connection gives rise to functions λ, µ : U [Π ] 2, taking values in symmetric bilinear forms on Π, and defined by λ(v, u) = Ω( 1 v, u), 2µ(v, u) = Ω( 2 v, u) for any constant sections v, u of P. The inclusion mapping U Π is identified with the radial vector field on U and denoted by X. Vector fields on the factor manifolds U and Π + are treated as vector fields on M = Σ Π +, tangent to the factor distributions. This includes constant fields v on Π + (such as c), the radial field X on Π +, and the coordinate fields j on U. p.38

40 We now define a metric g on M = Σ Π + by declaring that, for constant vector fields v, u on Π +, g(v, u) = 0, g(v, 1 ) = Ω(X, v), g(v, 2 ) = Ω(c, v), and g( j, k ) = g jk with g 11 = 2λ(X, X ) 4Ω(X, q), g 22 = 2λ(c, c) 4µ(c, X ) + Kφ 2, g 12 = λ(c, X ) µ(x, X ) 2Ω(q, c) + 4rφ 2. Here φ = Ω(X, c), and q is viewed as a vector field on M = Σ Π + tangent to the Π + direction. For a suitable orientation of M, this g is a strictly non-walker type III SDNE metric on Σ Π +. andrzej/indianapolis.pdf p.39

41 PROOF THAT (an outline) (Σ Π +, g) IS A UNIVERSAL MODEL Let (M, g) be any type III SDNE manifold with the scalar curvature 12K. Algebraically, this means that R = 1 2 (ζ η + η ζ) + K g g. for some (unique) C global trivialization (ζ, η, θ) of Λ + M satisfying the equalities ζ, ζ = ζ, η = η, θ = θ, θ = 0, ζ, θ = 2 = η, η, plus a sign-fixing condition on η (stating that V is the eigendistributions of η for the eigenvalue 1). We then have W + ζ = 0, W + η = ζ, W + θ = η. In addition to V = Ker ζ, another null two-dimensional distribution on (M, g) is H = Ker θ, and TM = H V. p.40

42 A two-plane system is a sextuple (Σ, ξ, σ, Π, c, Ω) consisting of a real affine plane Σ, a real vector plane Π, two linearly independent constant 1-forms ξ, σ on Σ, a nonzero constant vector field c on Π, and a nonzero constant 2-form Ω on Π. This gives rise to a septuple (M, V, h, α, β, θ, ζ), in which M = Σ Π +, for half-plane Π + Π on which Ω(, c) > 0, V is the vertical distribution on M, tangent to the Π + factor, h is a partial metric on M, with h(v, w) = Ω(Y w, v) and h(v, u) = 0 for vertical vectors u, v and vectors w tangent to the Σ factor, where Y w = ξ(w)x + σ(w)c and X denotes the radial vector field on Π, while, with φ = Ω(X, c) : M IR, α = d log φ on V, β = φ 2 ξ, θ = φ 2 Ω, ζ = 2φ 1 ξ σ. p.41

43 On the other hand, any type III SDNE manifold (M, g) leads to a similar septuple (M, V, h, α, β, θ, ζ), with V, θ, ζ defined earlier, h obtained by restricting g, and the 1-forms α, β characterized by ζ = 2α ζ + 2β η. The requirement that g be strictly non-walker amounts to β 0 everywhere. One shows that the septuple (M, V, h, α, β, θ, ζ) associated with any strictly non-walker type III SDNE manifold (M, g) also arises as described here from some two-plane system. andrzej/indianapolis.pdf p.42

44 A horizontal distribution for a two-plane system (Σ, ξ, σ, Π, c, Ω) is a vector subbundle H of TM with TM = H V. Any such H gives rise to a neutral-signature pseudo-riemannian metric g on M. Namely, g is the unique total-metric extension of h such that H is g-null. QUESTION. Which choices of H lead, for a suitable orientation of M, to type III SDNE metrics with the scalar curvature 12K, such that H = Ker θ? p.43

45 FOUR CURVATURE CONDITIONS: R(w, u)v = Kh(v, w)u, dβ + 2β α = Kζ/2, [ w θ](u, v) = 2α(w)θ(u, v), (dγ + 2α γ)(, w) = g(, w)/2 for all sections u, v of V, and w of H. p.44

46 DEFORMATIONS OF HORIZONTAL DISTRIBUTIONS Horizontal distributions H for a given two-plane system (Σ, ξ, σ, Π, c, Ω) form an affine space (the space of sections of a specific affine bundle over M = Σ Π + ). Namely, bundle morphisms F : TM TM valued in V and vanishing on V are sections of a specific vector bundle. We define the sum H + F to be a new horizontal distribution, with the sections w + Fw, where w runs through sections of H + F. However, (Σ, ξ, σ, Π, c, Ω) has a distinguished horizontal distributions H, tangent to the Σ factor. The bijective correspondence F H = H + F allows us to rewrite the four curvature conditions as a system of second-order nonlinear partial differential equations, imposed on F. andrzej/indianapolis.pdf p.45