Parallel spinors on Riemannian and Lorentzian manifolds Bernd Ammann1 1Universität Regensburg, Germany 1st Geometry Conference for German-Japanese friendship Tokyo, Sept. 2019 Images/150JapanGermany.jpg Talk I: Parallel spinors on Riemannian manifolds joint work with Klaus Kröncke, Hartmut Weiß, and Frederik Witt Arxiv 1512.07390, Web-Link with further info Math. Z. 291 303-311 (2019) A) Basic facts about Ricci-flat manifolds, special holonomy and spin geometry B) Our results C) Applications to the space of metrics g with scal g ≥ 0 (D. Wraith, T. Schick) Images/zahnrad-germany-and-japan.jpg Talk II: A) Parallel spinors on Lorentzian manifolds (joint work with Klaus Kröncke and Olaf Müller) Arxiv 1903.02064 B) Topology of initial data sets with strict dominant energy condition (work by Jonathan Glöckle) Arxiv 1906.00099 Images/zahnrad-germany-and-japan.jpg Goal for Talk I Study the space of all Ricci-flat metrics on a given compact manifold M. Main interest: (Pre-)Moduli space fRicci-flat metricsg= DiffId(M) Talk I: Parallel spinors on Riemannian manifolds A) Basic facts about Ricci-flat manifolds, special holonomy and spin geometry B) Our results C) Applications to the space of metrics g with scal g ≥ 0 (D. Wraith, T. Schick) Images/zahnrad-germany-and-japan.jpg Cheeger-Gromoll splitting theorem Theorem (Cheeger-Gromoll splitting theorem) Let (M; g) be a complete Riemannian manifold with RicM ≥ 0. Suppose γ : R ! M is a line, i.e. a geodesic satisfying 8t; s 2 R : d(γ(t); γ(s)) = jt − sj : Then we have (M; g) = (M0; g0) × R : My favorite reference: Takashi Sakai, Riemannian Geometry, AMS, Theorem V.3.9 Following work by J. Eschenburg and E. Heintze (Augsburg in Bavaria within Germany) Images/zahnrad-germany-and-japan.jpg Consequence of Cheeger-Gromoll splitting theorem Lemma If (M; g) is a complete Riemannian manifold with infinite diameter. Let Γ act cocompactly and isometrically on (M; g). Then (M; g) carries a line. The Cheeger-Gromoll splitting theorem implies: Theorem If (M; g) is a compact Riemannian manifold with RicM ≥ 0. Then the universal covering Me is isometric to k N × R where N is a simply-connected compact manifold with RicN ≥ 0. Note that k is a topological invariant, namely the growth rate of π1(M). Non-positive Ricci curvature Lemma Let K be a Killing vector field on a Riemannian manifold, then r∗rK = Ric(K ) Proof Let K be a Killing field on a Riemannian manifold. Then 2 Y 7! rY K is skew-symmetric, and thus Y 7! rX;Y K is also skew-symmetric for fixed X. 0 = g(R(X; Y )K ; W ) + g(R(Y ; W )K ; X) + g(R(W ; X)K ; Y ) 2 2 = g(rX;Y K ; W ) − g(rY ;X K ; W ) + g(R(Y ; W )K ; X) 2 2 + g(rW ;X K ; Y ) − g(rX;W K ; Y ) 2 = 2g(rX;Y K ; W ) + 2g(R(Y ; W )K ; X) Thus 2 rX;Y K + R(K ; X)Y = 0: 2 rX;Y K + R(K ; X)Y = 0: Taking the trace (= metric of X- and Y -component) we get r∗rK = Ric(K ) which is the lemma. Corollary If (N; g) is a compact Riemannian manifold with Ric ≤ 0, then every Killing vector field is parallel. If such an N is simply connected then its isometry group is finite. Proof. Z Z krK k2 = hRic(K ); K i ≤ 0: N N N Thus: Ric ≤ 0 and π1(N) = f1g implies # Isom(N) < 1. Ricci-flat manifolds Combining Ric ≥ 0 and Ric ≤ 0 one gets: Corollary If (M; g) is a compact Riemannian manifold with Ric = 0. Then Me is isometric to k N × R where N is a simply-connected compact manifold with RicN = 0, and Isom(N) is finite. By carefully studying the action of the Deck transformations one obtains: (Web-Link) Theorem (Cheeger-Gromoll, Fischer-Wolf) Every (closed) Ricci-flat manifold has a finite normal covering of the form k N × (R =Γ) where Γ is a lattice in Rk , and where N is a closed Riemannian manifold with π1(N) = f1g. The restricted holonomy group Hol0(M; g) := fPγ j γ contract. path from p to pg = Hol(Me ; g). (M; g) is irreducible if Hol0(M; g) ! GL(TpM) is an irred. repr. Theorem (de Rham splitting theorem) Let (Me ; g~) be a complete simply-connected Riemannian manifold. Then as a Riemannian product k Me = M1 × · · · × Mr × R where each Mi is a complete non-flat irreducible Riemannian manifold. For Ric ≥ 0 on a univ. covering Me of a closed M this refines the Cheeger-Gromoll splitting. Holonomy Definition The holonomy group Hol(M; g) := fPγ j γ path from p to p in Mg ⊂ GL(TpM): (M; g) is irreducible if Hol0(M; g) ! GL(TpM) is an irred. repr. Theorem (de Rham splitting theorem) Let (Me ; g~) be a complete simply-connected Riemannian manifold. Then as a Riemannian product k Me = M1 × · · · × Mr × R where each Mi is a complete non-flat irreducible Riemannian manifold. For Ric ≥ 0 on a univ. covering Me of a closed M this refines the Cheeger-Gromoll splitting. Holonomy Definition The holonomy group Hol(M; g) := fPγ j γ path from p to p in Mg ⊂ GL(TpM): The restricted holonomy group Hol0(M; g) := fPγ j γ contract. path from p to pg = Hol(Me ; g). Theorem (de Rham splitting theorem) Let (Me ; g~) be a complete simply-connected Riemannian manifold. Then as a Riemannian product k Me = M1 × · · · × Mr × R where each Mi is a complete non-flat irreducible Riemannian manifold. For Ric ≥ 0 on a univ. covering Me of a closed M this refines the Cheeger-Gromoll splitting. Holonomy Definition The holonomy group Hol(M; g) := fPγ j γ path from p to p in Mg ⊂ GL(TpM): The restricted holonomy group Hol0(M; g) := fPγ j γ contract. path from p to pg = Hol(Me ; g). (M; g) is irreducible if Hol0(M; g) ! GL(TpM) is an irred. repr. Holonomy Definition The holonomy group Hol(M; g) := fPγ j γ path from p to p in Mg ⊂ GL(TpM): The restricted holonomy group Hol0(M; g) := fPγ j γ contract. path from p to pg = Hol(Me ; g). (M; g) is irreducible if Hol0(M; g) ! GL(TpM) is an irred. repr. Theorem (de Rham splitting theorem) Let (Me ; g~) be a complete simply-connected Riemannian manifold. Then as a Riemannian product k Me = M1 × · · · × Mr × R where each Mi is a complete non-flat irreducible Riemannian manifold. For Ric ≥ 0 on a univ. covering Me of a closed M this refines the Cheeger-Gromoll splitting. Berger’s holonomy list Theorem If (M; g) is an irreducible Riemannian manifold, n = dim M, then (M; g) is locally symmetric or Hol0(M; g) is one of the following: (1) Hol0(M; g) = SO(n) (generic) (2) Hol0(M; g) = U(n=2) (Kähler) (3) Hol0(M; g) = SU(n=2) (Ricci-flat Kähler= Calabi-Yau) (4) Hol0(M; g) = Sp(n=4) (hyper-Kähler) (5) Hol0(M; g) = Sp(n=4) · Sp(1) (quaternionic-Kähler) (6) Hol0(M; g) = G2 and n = 7 (7) Hol0(M; g) = Spin(7) and n = 8 There are compact examples in each case. In cases (3), (4), (6) and (7) we have Ric = 0. If M is compact with Ric = 0, then we are in case (1), (3), (4), (6) or (7). The locally symmetric factors are Ricci flat only if they are flat. Summary Let (M; g) be a compact Ricci-flat manifold. After passing to a finite covering Mb ! M, we can decompose k Mb = M1 × · · · × Mr × T ; where all Mi have irreducible holonomy. Spin geometry Let (N; h) be a time- and space-oriented semi-Riemannian manifold. We assume that we have a fixed spin structure, i.e. a choice of a complex vector bundle $hN, called the spinor bundle, with h h ^•=even ∗ $ N ⊗C $ N = T N ⊗R C: This bundle carries (fiberwise over p 2 M) I a non-degenerate hermitian product (positive definit in the Riemannian case) I a compatible connection h h I a compatible Clifford multiplication cl : TN ⊗ $ N ! $ N, cl(X ⊗ ') =: X · ' such that X · Y · ' + Y · X · ' + 2h(X; Y ) ' = 0: Spinors and holonomy Let (N; h) be a Riemannian or Lorentzian spin manifold. Parallel transport along a loop c :[0; 1] ! N, p = c(0) = c(1) gives a map $hN h h P (c) 2 Spin($pN) ⊂ U($pN) 2 : 1 TN P (c) 2 SO(TpN; h) Assume that ' 6= 0 is a parallel spinor, i.e. a parallel section of Γ($hN). Then the holonomy group TN Hol(N; h; p) := fP (c) j c loop based in pg ( SO(TpN; h): is special, i.e. dim Hol(N; h; p) < dim SO(TpN; h). In the Riemannian case: Ric = 0 In the Lorentzian case: Ric(Y ) is lightlike for all Y ) Ric = f α ⊗ α for a lightlike 1-form α. Remark A product M1 × · · · × Mk of (semi-)Riemannian spin manifolds carries a parallel spinor if and only if each factor carries a parallel spinor. Parallel spinors Let (N; h) be a Riemannian or Lorentzian spin manifold. Assume that ' 6= 0 is a parallel spinor, ) RX;Y ' = 0 P ! 1 ) 0 = ±ei · Rei ;Y ' = − 2 Ric(Y ) · ' ) h(Ric(Y ); Ric(Y ))' = − Ric(Y ) · Ric(Y ) · ' = 0 In the Lorentzian case: Ric(Y ) is lightlike for all Y ) Ric = f α ⊗ α for a lightlike 1-form α. Remark A product M1 × · · · × Mk of (semi-)Riemannian spin manifolds carries a parallel spinor if and only if each factor carries a parallel spinor.