Dirac Operators in Riemannian Geometry Thomas Friedrich

1 Torino, February 2014 Dirac operators in Riemannian geometry Thomas Friedrich (Humboldt University Berlin) 2 , Graduate Twistor and Killing c operator up to the year nors up to the year 1991 , Teubner-Verlag Leipzig/Stuttgart , Lecture Notes No. 1976, Springer 2009. General References Dirac Operators in Riemannian Geometry The Dirac Spectrum 2009 Th. Friedrich, Studies in Mathematics No. 25, AMS 2000. This book contains 275 references up to theN. year Ginoux, 2000 This book contains 240 references on eigenvalues of the Dira H. Baum, Th.spinors Friedrich, on R.1991. Riemannian Grunewald, manifolds I. Kath, This book contains 107 references on Twistor and Killing spi 3 2 T and 2 S Content 1. Motivation 2. Clifford algebras 3. Spin structures 4. The Dirac operator 5. Conformal changes of the metric 6. The index of the Dirac7. operator Eigenvalue estimates for the Dirac operator 8. Riemannian manifolds with Killing spinors 9. The twistor operator 10. Intrinsic upper bounds for metrics on 4 via twistor operator via deformations 2 2 ) ) /D /D ( ( 16. Killing and twistor spinors with torsion 17. Remarks on the second Dirac eigenvalue 15. Eigenvalue estimates for 11. Surfaces, mean curvature and the12. Dirac operator Dirac operators depending on connections13. with torsion The Casimir operator 14. Eigenvalue estimates for 5 by g f ) 2 y ∂ C ; } 2 ) R y = 0 y ( γ {z i∂ 1 0 ∞ x 0 1 − γ + C | · x y ∂ → + ( γ ) 1 2 2 + g f C y ; = γ x 2 · ∂ ¯ R ∂ x } ( i 0 ∞ x , γ {z C ) , γ i 0 y : | Id i∂ Consider the Cauchy-Riemann operators P Motivation − − = x = ∂ ( Clifford relations 2 y 2 1 ¯ (Laplacian). ∂g ∂f γ i = = = ∆ ∂ 2 x = 2 γ ¯ ∂∂ satisfy the g f y = 4 , γ ¯ P ∂ x ∂ γ 4 a) From complex analysis: Then Define a differential operator and 6 , } with ) 1 X , ( 0 Λ iη − ⊕ 0 , 1 )= =Λ JX ( 1 η . Λ : : Energy. . η 4 c { E 2 ¯ ∂f ) = ∆ m = + ¯ ∂∂ + 1 , 2 0 + ∂f p Λ ¯ 2 ∂ c ∂ , p ) =: Consider a free classical particle with 2( } ) df – manifold, Kähler ( = X 1 , : momentum, ) ( 0 2 E Λ iη /c 2 Then: ,g,J v n )= vm − 2 1 )+pr M √ JX df ( ( ( 0 η = , 1 : p Does there exist a generalization of the Cauchy-Riemann Λ η { = pr = df 0 , : mass, 1 Λ More generally: operator on a more general class ofb) manifolds? From theoretical physics: m Then special relativity predicts the relation and Question: 7 ψ , closed manifold . can be realized as the 1 p smooth ”Dirac equation” 4 Z M . . is 4 Z 4 X 1 over 4 c 24 2 of a 4-dimensional, compact and M L m ) )= Z 4 ; 4 M ∆+ manifold ( 2 X ) = 0 mod 16 ( ~ σ 4 2 2 1 8 c M H , both acting on some state function ( √ σ ∇ = ~ = i L topological then ψ t →− ∂ ~ p i (Rochlin 1950): If What is the meaning of the square root? ) = 0 (Freedman 1982): (Hirzebruch) 4 ⇒ , t M ∂ ( ~ 2 i ω → s. t. intersection form According to the quantization rules of quantumE mechanics: Question: c) From topology: Theorem Theorem Theorem Any unimodular quadratic form simply connected 8             1 and an elliptic − 4 1 0 M ? − → 1 0 1 2 0 S − − . 1 0 0 2 1 01 0 2 0 = 8 − − − 8 = 0 mod 2 s. t. E ) D 1 2 1 0 0 0 0 S − − Γ( 1 2 100000 → ) = dim − − 8 ) dim coker E S ( 1 2 1000000 − σ − − : Γ( D , ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 2 2 D 4 − M (             σ 8 1 = Does there exist a vector bundle 8 ) = dimker )= E is of typ II and D D ( ( 0 ≥ 8 E a-index Example: Question: differential operator t-index 9 . of the } 1 − . 2] complexification = (2) 2 1 n/ [ C , e M = 2 : ∼ = n = 0 n i ) 2 e ∆ denotes its · R ) ( j n dim∆ e C R ( + Cl C , j ) e n Cl · i . e n { R / shows: n End(∆ of y R γ Clifford algebras by endomorphisms. More generally: −→ 7→ O 2 ) standard euclidean scalar product. 2 n C : an orthonormal basis. Then the (finite dimensional!) R , e g ( x on a complex vector space n C , ) := γ ) n n Cl R 7→ = 2 ( R acts on Clifford algebra 1 ( n ,...,e ) e C Cl 1 2 e R Cl , ( There exists a unique representation of smallest dimension ) C space of (Dirac) spinors Then : , g Cl n n R ∆ ( Example. ⇒ Thm. algebra associative algebra is called the 10 .

= 1 | and can be realized i x ) | n . ) . and µ SO( n n ”Clifford multiplication” ∆ R ker( ∈ splits into ⊕ i : −→ n n n n ∆ , x l ∆ by an endomorphism: ∆ 2 ⊗ n x ∈ = ∆ ⊗ n · ∆ n ψ R n · R .... ∆ x : · ⊗ 1 µ n x acts on 7−→ R is a two-fold covering of ) n R ) = x, ψ ∈ ( n group x -representation ∋ ) ) n n n Spin( ∆ , ) × n Spin( Spin( n R ( R Cl Every vector The The in • • • 11 Dirac er, the , ) . µ ψ . k · ∓ 2 x ∆ ker( · i e −→ ⊗ onto k i ± 2 e n representation splits into two ∆ n : ∆ =1 ) i X ⊗ n 1 n n R , x + . k Spin( − 2 ψ ∆ ⊗ ⊕ x k + 2 ) = ψ = ∆ k ⊗ twistor operator 2 x ∆ ( p is even, then the k and the = 2 n If This splitting yields two differential operators of first ord There is a universal projection of operator • • • irreducible pieces, 12 8 8 quaternionic mod mod structure. 3 5 , , 2 4 real ≡ ≡ n n -invariant : n quaternionic structures + 4) ∆ -invariant k ) 8 8 k mod mod Spin(8 7 1 , , Spin(8 6 0 ≡ ≡ real structures n n admits a admits a +4 k k ± 8 ± 8 -invariant structures in ) ∆ ∆ n Spin( n α commutes with Clifford multiplication anti-commutes with Clifford multiplication Additional structure. The representation • Proposition: The representation 13 n M ) to x ( n ∆ P→F admits a spin x n M n of a Riemannian manifold M x x T admits a reduction F . to every point ) n n Spin Structures. the oriented frame bundle. ∆ ) n x ) -principal bundle SO( ( ) M n , g n x Riemannian spin manifold n → T ∆ ) n n M SO( n ( M M S S ∈ ∈ F x x Spin( However: )= )= : notion of a n n ) Attach a copy of M M → , g ( : ( n Spinor bundle: S Tangent bundle: T Denote by M ( the group structure iff the Idea • 14 . The P

) p . ( ) 1 n π ) = the projection onto the Spin( ) m n ∈ ( ˜ ˜ π . M ˜ m, p : / ) · admits a spin structure , g P ( γ n ) ∗ g := Γ ×P M · /π n n ˜ ) = ( M M acts properly discontinuous on a manifold = Γ m, p ∈ Γ → ) via m, p P n ( ˜ ) = ( ˜ · ˜ M P m,p γ g ( ( ˜ : ∗ · π ) π ) = m, p P ( ˜ ( ∗ acts on -principal bundle with the action of the spin group π ) Γ n and denote by Spin( n ˜ Moreover, and the fixed spin bundle can be reconstructed, is a orbit space. Moreover, suppose that induced bundle M Different spin structures: Suppose that a discrete group 15 is isomorphic Γ and introduce a . ) ) n 2 . Z ; )) , a new spin structure of γ n Spin( ( n ǫ M · ) ( M 1 } ⊂ P . In particular we proved 1 ( H n ∗ − , then the group ˜ m, p , M · /π 1 n ǫ γ of ) = M 2 ) → { Z n := Γ , M ) :Γ ) = ( ǫ ( n 1 ǫ P π M ( m, p 1 ( ˜ π · admits at least one spin structure, then all spin γ n Hom( -principal fiber bundle over ) M n If Spin( -action via the formula is the universal covering of ǫ n Γ ˜ M structures correspond to the set Consider a homomorphism Theorem: new is still a The space to the fundamental group the manifold. If 16 . ) n under ) 2 Spin( Z B )) = 0 ); n n cture, namely SO( Spin( . Consequently, the B B ( ( 2 1 of the tangent bundle. is zero. This argument π H ) ) = 0 ) 2 2 n ∈ Z Z 2 . ); ); ω SO( n n B ) = 0 → n Spin( Spin( n M B B ( ( ( M 2 lifts into the classifying space 1 )) = 0 and 2 : ω n f H H f → )= ) 2 2 (Spin( Z 1 Z π ); ); n n )) = SO( Spin( n B . Indeed , the condition is sufficient, too. B ( An oriented manifold admits a spin structure iff its second ( 2 2 H H Spin( ) = 0 B n ( admits a spin structure iff 2 M π n ( 2 we have Existence of a spin structure Consider the classifying map Theorem: Stiefel-Whitney class vanishes, M Since image of the second Stiefel-Whitney class the map yields a necessary conditionω for the existence of a spin stru 17 . n ∆ ) n Spin( P× can be lifted from the tangent bundle to are not spin manifolds. ∇ := ,... S . (3) are spin manifolds with a unique spin structure. in a unique way. S S /SO different spin structures. ,... be a Riemannian spin manifold with a fixed spin structure. n (3) +1 ) 2 n P 2 ) ,SU P ,g, n ( 2 n ) C spinor bundle admits , M P ( n ( n S T C Let the spinor bundle Examples: • • • The Levi-Civita connection The associated bundle is the 18 . ψ. i e . · ∇ i e n =1 i X , Hodge - de Rham ) n = M ( q Λ ψ, Dψ not a topological invariant ◦ ∇ is a topological invariant is µ ) ) n n = D M ( q b ,...,e 1 Dirac operator , Dψ e ) dim ker( S on differential forms in Sh¨dne 1932,(Schrödinger Lichnerowicz 1962) ) =: Γ( q q ∆ : −→ Scal D 4 1 ) S + S dim ker(∆ : Γ( Dirac operator, Laplacian = ∆ D is an elliptic differential operator of first order 2 D D For the In an orthonormal frame Properties of • • theory implies that For the 19 ,... 1 , . = 0 2 of its Lie j . e as 3 · s p 6 = 2 , e i 2 , ...,q , e 2 if 1 ] = 2 e , 1 , e 3 = 1 = 0 e [ i j , , p , e 1 2 i e ) with multiplicity e · q h and the basis − . 3 ... p ) S 2 q λ, ] = 2 , 3 + ( + = , e 2 p | ( 2 = 1 3 e e [ | pqλ 4 Spin(3) = , , , p p 3 1 2 λ λ e · = 1 ± + (see Hitchin 1974) | 2 2 p λ λ e | ] = 2 2 = , e | with multiplicity 1 1 e e | [ ) = ) = λ λ ( ( p ± µ p,q ν algebra with the commutator relations Basic Example: Consider the Lie group We introduce a left invariant metric defined by the condition The eigenvalues of the Dirac operator are given by the formul 20 . . p 2 ± ∞ ast ) = λ , i.e. ( ± p,q = 0 ν is a solution. In this 0 . − p,q q → ν lim λ 2 = ) p q p, − 2 p ± , then + ( 4 2 ) = λ pqλ ( 4 ± p,p ν 0 = 4 → lim λ 4 is a transcendent number, then the kernel of the λ , λ ∞ is an integer divisible by ) = p λ ( p µ = 4 0 λ → lim If the parameter If λ Remark that The kernel of the Dirac operator corresponds to • • • Dirac operator is trivial . case the dimension of the kernel of the Dirac operator is at le 21 2 2 2 2 ) ) ) ) − 11 − − − 22 33 44 ν ν ν ν ( ( ( ( 16 12 l 8 0 4 0 20 60 50 40 30 10 22 2 2 2 2 2 2 ) ) ) ) ) ) 2 1 − − + 11 22 11 + 22 µ µ ν ν ν ν ( ( ( ( ( ( 10 8 6 l 4 2 0 80 60 40 20 23 ula . n M with positive scalar 1 . g ) ψ 1 4 − n σ ( D 4 +1 n , the kernel of the Dirac operator is − 2 σ S ) = be a compact Riemannian spin manifold. If ψ ( ) 1 , g D n M ( Conformal change of the metric – the corresponding Dirac operators. The dimension of the kernel of the Dirac operator is a – two conformally equivalent metrics on Let For any metric on g D · σ and = 1 1 g D After a suitable identification of spinors we obtain the form curvature, then the kernel of the Dirac operator is trivial. the metric is conformally equivalent to a metric • • • Theorem: conformal invariant. Corollary: Example: trivial. 24 . -genus, . ) ˆ ) splits, the A . 2 4 p ) − n 4 + M ( ∆ S − σ 2 1 ⊕ Γ( 8 1 p + n (7 −→ 1 ) = ) = ∆ 5760 4 − . is given by the n S = M ) ∆ ( + n ˆ ˆ A A D M : Γ( ( ˆ − A , then ) = ,D = 8 ) + − n D S Γ( . If splits and the Dirac operator splits, too, 1 − -manifold we have index( p S 4 −→ 1 24 ) ⊕ + = + S ˆ S A = The index of the Dirac operator : Γ( is a complete Riemannian manifold, then the Dirac operator is even, then the representation S + ) be compact. Then the index of k , then D , g n n = 2 = 4 M M n n ( If Let If In case a a compact If is essentially self-adjoint. • • • • • spin bundle 25 is positive 2 -dimensional 4 CP -genus vanishes, ˆ -genus is an even A ˆ A the + 4 k 8 is not spin. 2 is not spin. 2 . CP 16 CP . 8 . But / be compact and spin. If it admits a Riemannian 6 = 0 -genus of any compact spin manifold is an integer, n ) = 1 ˆ 8 A 2 / M CP ( . Moreover, in dimensions The Let . ˆ ) = 1 (Rochlin) The signature of a smooth, compact, The scalar curvature of the metric Kähler of A Z 2 ∈ CP ) = 0 ) ( n n ˆ A M M ( ( ˆ ˆ A metric with positive the scalar curvature, then the Theorem: Example: Corollary: Theorem: Example: and A spin manifold is divisible by number. 26 4 / 0 and R 1 ≥ R 2 λ → ly n M : f ψ . · . 0 X R · · f 1) + − n ψ n X 4( ∇ ≥ := 2 λ ψ Fix a real-valued function f X ∇ (The Riemannian case – Friedrich 1980): : compact, Riemannian spin manifold ) Eigenvalue estimates for the Dirac operator , g n not optimal. : minimum of the scalar curvature The formula Schrödinger-Lichnerowicz implies immediate 0 : eigenvalues of the Dirac operator M introduce a new spinorial connection R λ ( • → Theorem: Idea of the proof: 27 . ) ψ ( . D · 2 || ψ . and integrate. X · ψ 1 2 n · || f ) λ/n + R n = n ψ − f M g X Z ∇ 1 4 + (1 − ψ 2 ) := · 2 L X || R ) )( 1 4 ψ ψ ( ( + D || ψ 1 f ,P n − S) n = ∆ ⊗ = ψ 2 2 ) Γ(T 2 L f || ) , then use the latter formula with − → ψ Twistor Operator ( ψ D P · ( || λ : Γ(S) = P Dψ and prove the formula (A. Lichnerowicz 1987): Next generalize the Schrödinger-Lichnerowicz-formula Idea of a second proof: Consider the The estimate follows after some elementary computation. If 28 ere are satisfies a stronger ψ ψ . · ). X 1) 0 − R n ( n s Killing spinors 2 1 ± = ,then the spinor field ψ ψ X · ∇ 0 R 1) − n n 4( = ψ 2 D Remark: If Riemannian Killing spinors Example: The lower boundother is manifolds, realized too for (see example later – on all spheres. But th equation, namely 29 ard 1997): . . . 4 n/ is even is odd 2 := 2 k n/ n/ := := m where m if if 0 0 R 0 · R R · · 1) + 2) + 3 m − + 1 k · k m 4 4( m m 4( ≥ ≥ ≥ 2 2 λ λ 2 λ The case Kähler (Kirchberg 1986-1990): The case quaternionic-Kähler (Kramer, Semmelmann, Weing Further Results: • • 30 nstant such that ]) 0 g ([ C = C . . /n the inequality holds : ) 2 ) 2 , g S π , g 2 · n C S ( on 4 M ( g vol vol ≥ ≥ ) g ) D g ( 2 D ( λ 2 the inequality holds λ ] 0 g [ . Then there exists a constant (Hijazi 1986, 1991): Bär ∈ , i.e. for any metric 2 π g S · ) = 0 0 g = 4 D be a conformal structure on a Riemannian spin manifold such ( metric ] C 0 g [ has only one conformal structure. The corresponding Lott co any Conformal estimate (Lott 1986): The case of 2 • • equals S that ker Let for 31 e other W. See for 4. 02), 196 - 207. 0-716. Eigenvalue estimates for the Dirac operator depending on th • components of the curvatureexample tensor, i.e. depending on Ric or Th.Friedrich and K.-D.Kirchberg, Journ.Th.Friedrich Geom. and Phys. K.-D.Kirchberg, Math. 41 (20 Ann. 324 (2002), 70 Th. Friedrich and E.C. Kim, Journ. Geom. Phys. 37 (2001), 1-1 32 . 0 R> . 1 R ∈ λ and . ) 1) n − R M n ( ( T n ∈ s is given by the scalar curvature, 2 1 λ ± = ψ, X : · λ ) X , g · n λ M ( = on ψ X is an Einstein manifold with positive scalar curvature ∇ ) Riemannian manifolds with Killing spinors , g n M ( The so-called Killing number Killing Spinor Necessary conditions: • • 33 . n S = n M admits two Killing spinors. over manifolds Kähler-Einstein +1 k 2 k 2 X -manifold. 2 M G → +1 k 2 M has positive constant curvature, i.e. n M is an Einstein-Sasakian manifold. is a nearly manifold. Kähler is a nearly parallel -fibrations : 1 be a simply-connected spin manifold. Then is admits a 5 6 7 S 8 (Friedrich,Grunewald,Kath, Hijazi 1986-1989) ) , M M M 4 , g , : : : n M = 3 = 5 = 6 = 7 ( . k n n n n Any Einstein-Sasakian manifold 2 • • • • • Killing spinor if and only if Theorem: Let Examples: X 34 ⊗ ) n M ( ∗ T )) . µ ( )) ψ ker ( . Γ( D ) · i µ ( e −→ 1 n of the Clifford multiplication ) ker S + S ⊗ −→ ψ i : Γ( ) e S n ∇ P ( ⊗ M ) ( ⊗ n T i ψ of any spinor field is a section in e M ⊂ ( ψ n =1 ) ◦ ∇ T i X ∇ µ p : ( . In a local frame we obtain The twistor operator p ) = ψ ) := and we can apply the projection. The operator ( ψ S ( P P ⊗ ) n M ( twistor operator T = is the S The covariant derivative Consider the kernel ker as well as the projection onto this subbundle, 35 . = 0 ψ · λX . − ) n ψ . X M ) ( ∇ n T and then we obtain ∈ ) = n λ ψ ,X n λψ − − = 2dim(∆ ( · )= ψ X ( ) = 0 2]+1 1 n reads as ψ D ( n/ [ + 2 D · is connected, then it is bounded by ψ ≤ ) = 0 X n X ψ ) implies ∇ ( 1 M n P P ψ · + ) = ker( ψ ψ ( X λX The dimension of the kernel of the twistor operator is a Any Killing spinor is a solution of the twistor equation. D ∇ = · ψ X X 1 n ∇ + twistor equation ψ X ∇ conformal invariant. If The Proposition: Proof: Proposition: 36 ated. 2 ) such that ψ · i , e ) ψ ( D Re( . Then the functions n n =1 i X M − ) ψ ( 2 C , − ) 2 ψ | ( ) ψ ( D | ψ,D 2 | ψ | (a local result – Friedrich 1989) (Lichnerowicz 1987, Friedrich 1989) ) := Re ) := ψ ψ ( ( be a twistor spinor on a connected C Q ψ are constant. Theorem: Let Proposition: The zeros ofOutside the a zero twistor setthe there spinor twistor is spinor on a becomes conformal a the change sum of connected of the two metric manifold Killing spinors. are isol 37 . ) = 6 = 0 P ) miting P . ker( ) ∗ ker( , g n M ( such that the space ∗ g on twistor spinors with zeros: be a compact Riemannian spin manifold with (a global result – Lichnerowicz 1989) ) , g coincides with the space of Killing spinor on n ) ∗ M ( P K. Habermann, J. Geom. Phys. 1990W. and and Kühnel H.-B. 1992 Rademacher – several papers since 1994. ker( Then there exists an Einstein metric Proof: Use thecase solution in of the the estimate of Yamabe the problem Dirac as operator. well as the li Theorem: Let Further results • • 38 . 0 ψ · 2 f T . = . ) 0 ψ . ) 1 u and ( M,g R 0 ( g 2 2 such that 2 L → | S 0 2 ψ ψ grad · ) 0 M ) 2 1 u ( M,g + 0 Λ: ( g 2 0 2 L g | , D ψ 0 grad u ψ u 1 2 e · | − e ) + = = Λ ψ ( 0 g 0 g ψ D 0 | g there exists a spinor field ,D , then ) = 0 0 0 g g ,D u ) , g ∆ 2 2 1 u e ) 2 M,g M ( ( − = 2 M,g e 2 L ( g | 2 || ≡ ) 0 2 L = | ψ ψ ( ψ and || g g | D ∆ | = 2 n Intrinsic upper bounds for metrics on If The Rayleigh quotient • • Suppose that on Now apply the Rayleigh quotient with a family of test spinor . ) 0 g ( 2 39 dM .

→ 2 ) 2 0 || g ) ( M u and we 2 ( : 0 g f dM 2 . || grad ) = 0 ) ) f u u 0 ( ( 2 1 g 0 0 ( g g 2 )+ f dM ( grad grad 0 u f || g − 2 1 e and any function Z 2 2 1 4 Λ grad )+ || f M Z ( 0 + ) + g 2 on 0 ≤ f g 0 ( 2 g ) 2 Λ grad 0 u 2 g ( e dM 2 2 Z = Λ g dM ≤ u Z ) . Then e 0 g 2 ( ≤ Z 2 u/ ) ) − g e . Then , g dM D 1 2 2 ( = f 2 1 ≡ M u λ 2 f For any metric f e Z ) vol( g ) g D the following estimate holds: ( D 2 1 ( Consider Consider 1 2 1 λ obtain Theorem: • • R λ 40 0 2 ψ M Λ = . ) , where 0 0 g ψ D · ( 1 H λ ) = Λ= 0 ψ -parallel spinor to ( 3 D and the eigenspinors R spin structure and of a 0 ) = 0 0 ψ g D non-trivial ker( is the Killing spinor, 0 ψ with a ) and ) . The restriction flat 3 can , g R 2 , g ⊂ T 2 2 S M )=( )=( 0 . Then we know that 0 is the mean curvature. ) , g , g 0 2 2 g H M D M ( ( A surface ( 1 Λ= have constant length. has constant length and satisfies the equation These estimates can be used in the• following cases: • λ • 41 . ) a ( 2 2 1 , λ 2 ≃ →∞ a lim inf = 1 + ln(2) . ≤ 2 2 2 4 1 3 1 4 z a ≤ ≤ + 2 ) ) y a a ( ( 2 1 2 1 + λ λ 2 x 0 the first eigenvalue of the square of the Dirac → →∞ a a ) lim sup lim sup a (M. Kraus 1999) ( 2 1 ≤ λ 2 Application: Consider the ellipsoid A further result: Consequently, the upper bound is in the asymptotic optimal. and denote by operator. Then we obtain 42 0 ψ . 2 2 being u/ − 0 dT -parallel e 2 ψ o 2 || g ) := u/ u 3 . The kernel ∗ 0 0 − ψ g u f e 2 grad( . e f := , = − ψ . . Then g = 0 0 ) , = 0 f ψ 2 2 Γ / 2 fdT ) = 0 grad( fdT R g 2 || T R D u = Z ( 3 2 1 2 − λ e T = 2 T -parallel spinor Z ,g 0 ) = 2 g 0 T g ≤ . We use test spinors ( g L D 2 -dimensional and coincides with the ) ( D ∗ 2 0 2 1 λ dT is u . Fix a ) ) − g ψ , ψ g e with a trivial spin structure: 2 D | D ( 2 f 2 2 | T λ 2 ker( T For any function such that Z ≃ ) ) g o g D ( D 2 2 – the flat matric on the torus λ 0 We estimate ker( belongs to the kernel of spinors. Consequently The case of g Theorem: orthogonal to the kernel, . 43 2 . 2 dT || 2 ) || u ) u -348. grad( erator || grad( 99), 231-258. || + u 22. 3 2 − || ∗ e v 2 . || T ∗ Z Γ = + ∈ 2 2 ∗ || ) dT u u 3 , we obtain , v ∗ − i Γ e ,x 2 grad( ∈ ∗ T v f ∗ Z h i v ) − e 0 ) g 6 = f 0 D ( 2 2 ) = λ x ( grad( ∗ || v ≤ f 2 , ∗ dT u − i f v e 2 T Z ) = ) f g D ( 2 2 λ Then grad( We apply the inequality for eigenfunctions of the Laplace op I. Agricola, Th. Friedrich,I. Journ. Agricola, Geom. B. Phys. Ammann,M. Th. 30 Kraus, (1999), Friedrich, Journ. Manusc. 1- Geom. Math. Phys. 100 (19 31 and 32 (1999), 209-216 and 341 Minimizing with respect to 44 . n . M ψ . n · H dM 2 2 n H = n -spinor on M n Z M 2 | 4 0 n – second fundamental form. ψ n ≤ ψ, Dψ := , H – the mean curvature. · ) T M ) n ψ , g , X M n → ( +1 S n M n 1 2 R T M = : ) vol( S D ψ is compact and oriented, then ( X 2 1 n λ , ∇ M +1 If n R ⊂ – parallel spinor in n 0 Surfaces, mean curvature and the Dirac operator M ψ D – the Dirac operator on Then we have • • • Theorem: 45 , g es) by holds. 2 ψ · ) T M alent to the X ( → ∗ 2 S . 2 1 and a spinor field ) 1 ψ T M = · : R ψ a Riemannian metric ∗ of length one such that X S → n ψ ∇ 2 ψ,Y M X M ∇ : . Then there exits an isometric H H ψ = induces on ) = 2Re( -tuple consisting of a simply-connected +1 Dψ 4 n and a spinor field , Y R 1 ) R ⊂ X ( n ∗ be a . Define the endomorphism → 3 S n M ( R ) g M ⊂ . : -manifold, a function 2 2 H M H ψ (The spin formulation of the fundamental theorem for surfac ,g,H,ψ is a symmetric endomorphism and 2 n 2 ∗ S M ( )= ψ of length one such that Any immersion ( The construction of the immersion using the spinor a function The integrability conditionGauss- of and Codazzi-equation. the latter equation is equiv D • Theorem: Let Idea of the proof: Riemannian Then ψ immersion 46 05-2012. is given by 3 R -form. 1 = . – not only minimal ones) C 3 -form. ψ ⊕ R 1 Ω 1 = 0 , R -dimensional spin representation. 2 ψ ψ ω → Ω d 2 2 M M = : Z ψ f – a complex valued = – a real valued dω ) )) f ψ ( ψ , ψ · ψ , α · X X ):=( ) := Re( The Weierstrass representation of a surface X X ( ( – the quaternionic structure in the ψ ψ Ω ω α • • • Th. Friedrich, Journ. Geom.Generalization by Phys. B. 28 Morel, (1998), M.-A. 143-157. Lawn and J. Roth between 20 (Weierstrass representation of a surface in and the isometric immersion These forms are closed, 47 . ) ψ. − · ψ, k T) e X,Y, · ∇ X T( ( 2 1 1 4 T) + k + e ( ψ Y n g X =1 g X X k ∇ ∇ -form. := 3 := := ψ ψ Y is a D X X ∇ ∇ – Riemannian manifold, ∇ of totally skew-symmetric torsion T) T , ∇ ,g, n M ( The torsion linear metric connection covariant derivative on spinors a first order differential operator The Dirac operator depending on a connection with • • • • • 48 , g . 2 Scal || 1 4 T . || + 1 8 T) − k e g − D ( ∧ T ∇ δ Scal T) 1 2 4 1 . k + 3 e / ( T T + T σ d n =1 X k 2 1 1 4 1 2 − + , T T T := Slebarski (1987), Bismut (1989), Kostant (1999), Agricola d 3 4 T σ = ∆ + -shift: 2 T 3 /D / 1 = ∆ 2 D -form derived from – Dirac operator of the connection – Dirac operator related to 4 a D /D (2002). • • • First formula: Second formula: History of the 49 in 0 T = ≤ ) . ψ, ψ · T d T = 0 ( , 6 = 0 ψ be a compact, Riemannian spin T) ,g, n M ( -form and the scalar curvature vanish, 3 Let . If there exists a spinor can only admit parallel spinors if 0 ≤ is parallel with respect to the Levi-Civita connection. g , then the T ψ T = 0 d ∆ Scal On a Calabi-Yau or Joyce manifold, a metric connection with s.t. , and g T -form manifold s.t. A Vanishing Theorem. Corollary. 3 the kernel of 0 = Scal 50 R R . 8 1 – Riemannian Dirac operator. 4 1 D = Ω+ = ∆+ 2 2 D D The Casimir operator is a symmetric space, then (Parthasarathy formula): H / = G – Riemnannian spin manifold, n ) M , g n croigrLcnrwc formula: Schrödinger-Lichnerowicz If M • • ( 51 . ) g 188-204. (T) + Scal (T) δ δ 1 4 + 2 T ) + σ T 2 acting on spinor fields of the triple 2 σ || 2 − T || − T 1 d 16 T d (3 ( − 1 8 1 8 g + + T Scal 2 1 8 /D Casimir operator - Riemannian manifold with torsion. coincides with the usual Casimir operator (Parthasarathy, − = ∆ For a naturally reductive space and its canonic connection, T) Ω The , Ω := ∇ ,g, n M is defined by ( Definition: Motivation: the operator I. Agricola and Th. Friedrich, Journ. Geom. Phys. 50 (2004), 1972; Kostant, 1999; Agricola, 2002). 52 -parallel -parallel ∇ ∇ re no g yield that the kernel of , we obtain Scal 2 1 8 /D (T = 0) = ∆+ g Scal 8 1 − 2 D Ω = The kernel of the Casimir operator contains all Lower bounds for the eigenvalues of For the Levi-Civita connection spinor. Example: Proposition: Corollary: spinors. the Casimir operator is trivial. In particular, then there a 53 commute 2 /D 2 T . and 4 1 2 Ω /D − ◦ . 2 . 2 /D || || T g T || || T = T + + ◦ g g 2 -parallel, then ∇ T + Scal d 2Scal , /D 2Scal 2 Ω 1 ◦ 16 8 1 1 16 + + , − T preserves the kernel of T T 2 /D T T = T : ◦ = ∆ = ∆ Ω If the torsion form is Ω = T = 0 ∇ with the endomorphism In the compact case, The case Proposition: 54 . , , . 2 1 5 1 2 e 4) − − ∧ − g ) , g 0 34 e , Scal Scal 0 1 8 + , 1 8 , − 12 − 4 e 2 − 2 /D Ω /D : ⊕ = = 2( = 4 2 7 Ω 1 2 := T dη T = diag(4 c ⊕ − ∧ + 0 T g , η g Scal 1234 Scal e 1 8 1 8 T = 8 Ω = Ω + + − , T T = 8 = ∆ = ∆ 4 2 0 T = 0 -dimensional Sasakian manifold. ± T Ω 5 ∇ Ω – a 5 – the contact structure. the Casimir operator splits into M η The characteristic connection, -Dimensional Sasakian Manifolds 5 • • • ⇒ 55 , - ∇ 28 > g , 2 -dimensional Scal 5 dy . 1 2 or 2 = 4 dx 4 − · -Einstein manifolds of 2 η , e y < 2 exist on the − ) g dx 4 . 1 1 2 ± 28 dx Scal . Examples: Friedrich/Ivanov, coincides with the space of = · ≤ 1 0 3 y = 0 g Ω Ker(Ω , e . If − 1 ψ · (Friedrich/Kim, 2000). Scal and dy dz ) η T occur on Sasakian 1 2 0 ≤ 2 1 4 ⊗ ) = 0 4 = ± 0 η − · Ω 2 := Ker(Ω , e η The kernel of 1 + 6 Ker(Ω such that : g dx = , 4 · ψ 2 1 − 5 . 2 4 e − = − = 1 g = e 6 = ) = 0 g 4 g ± Scal Ric Scal 2002. Spinors in both kernels Spinors in the kernel of parallel spinors If The interesting cases: Case Heisenberg group Ker(Ω type 56 ψ . 4 . 3 − 2 − -parallel spinors /D 2 ∇ = /D -dimensional Sasakian- 5 T 1= = ∆ − : 4 T ± Ω = 20 = ∆ g , 4 4 ± Scal Ω − 2 /D , + 3 . Examples: Friedrich/Ivanov, 2002. coincides with the space of ψ +4= T 4 4 T ± ± Ω : = ∆ = 0 = ∆ ψ 0 Ω · = 28 The Casimir operator of a compact Ω g T Scal The kernel of Case • Sasakian-Einstein manifolds, Theorem: Einstein manifold has trivial kernel. such that 57 , a. } ag. . = 2 0 2 = 2 || T ∇ || a > · . Ric 0 : , Killing spinors { , ω = 2 a> := T ∧ g c ω ) = , T g ∇ a Scal · a 2 4T = N 15 · 2 5 , ≥ T = = d 2 g /D = -dimensional nearly manifold. Kähler Ric 6 T T = 0 σ ∇ Ker(Ω) = Ker( 2 – a ) J is compact, then 6 ,g, is Einstein, 6 6 M M ( M The characteristic connection, If -Dimensional nearly manifolds Kähler 6 • • • • 58 . 2 || T || . 1 2 . − 0 ψ 2 ) · ) ) , ω ω (T) = 0 ∗ = 0 ω satisfies dω, ∗ ( ω, δ = 2(T ω d 1 6 · ) g − ω ∗ = Scal 0 dω, , ψ ( -manifold ( · T 6 1 2 σ T G · + is constant. , ) corresponding to dω ω 0 6 = 2 ∗ ψ = 0 T -Manifolds − ∗ 2 0 dω, G ψ ( , d ∇ – cocalibrated T = ) 6 = 0 T ,g,ω 7 ∇ M ( Suppose that The characteristic connection: Main difference to the previous examples: The parallel spinor -Dimensional 7 • • • • • 59 -parallel . -Sasakian ) ∇ 3 ∇ -manifold and 2 G = Ker(

ψ · . ) a . ω 7 6 2 ∗ a ( = a 49 144 − ψ · = − T 2 coincides with the space of , dω /D ) ∇ . = 0 ,g, ψ be a compact, nearly parallel Ω = 7 = 7 ) ∇ n M : ( ψ ,g,ω -structures: 7 2 G M ( its characteristic connection. The kernel of the Casimir Let ∇ This case includes Sasakian-Einstein manifolds and Ker(Ω) = spinors, operator of the triple denote by Nearly parallel Theorem: Remark: manifolds in dimension 60 . . = 0 ) 2 0 || ψ T · operators). || T 2 − . , T with parallel spinors σ 2 = 0 ) = 0 1) ω , 0 − ∗ T ψ (1 T ∆ ∇ N d dω, − ( (3 , , 1 8 2 -structures such that Ω 2 || + G T = 0 , || 2 T ω 2 1 /D -forms on ∗ 3 − d − : ) = ∆ = Ω 3 T σ g W 2 − Scal T d ( 8 1 dω, The metrics and + − ∗ 2 /D -structure of type Torsion and parallel spinor: Casimir operator: 2 T = Ω= described before yield examples of are negative or positive (no general relation between these G • • Results: 61 persymmetry, 72. 36. EMS Publishing House 2010. Th. Friedrich, Asian Journ.Th. Math. Friedrich, 5 S. (2001), Ivanov,I. 129-160. Asian Agricola Journ. , Math. Th.I. 6 Friedrich, Agricola, (2002), Math. Arch.Math. 303-3 Ann. 42I. 328 (2006), (2004), 5-84. Agricola, 711-748. Handbook of pseudo-Riemannian Geometry and Su Some references Th. Friedrich, E.C. Kim , Journ. Geom. Phys. 33 (2000), 128-1 62 [2004] polynomial in = ψ , · µ S ) ∀ X · µ symmetric and parallel Σ S via deformations ⊂ + be the splitting of the spinor µ µ µ 2 S . · Σ Σ Σ by a µ /D µ ∀ ∇ X ⊕ ( ∇ , for example µ 2 1 Σ − , i.e. Σ= ⊂ Σ Γ(Σ) ψ µ X . Then: Σ → T ∇ 2 /D und let := ψ = 0 , i.e. : Γ(Σ) 2 S X T /D ∇ S ∇ ◦ T = Deform the connection Estimate on every subbundle of T Assume Eigenvalue estimates for ◦ preserves the splitting of ⇒ 2 /D ∇ , b) bundle into eigenspaces of a) Thm. endomorphism T Idea: 63 + 2 k T ψ Appl. 26 ase to an k 1 4 i − 2 k ψ i } ) TSψ,ψ i e · −h {z ??? S 2 ”) k D,Sψ /Dψ, ψ h | + g Sψ S D k +2 · · i 2 } + e k X ( n k − {z Sψ k | r.h.s., o.k. = n dM =1 i X 2 + ψ k 4 1 i g X } ψ o.k. k − ∇ , g 2 2 n k ψ, ψ k {z 2 ψ ψ k /D 2 Scal S h | λ n k∇ M = Z = 1 4 i + 2 k ψ, ψ ψ 2 ) · k S 2 k + T /D k ( 8 1 The formula: h I. Agricola, Th.(2008), 613-624. Friedrich and M. Kassuba, Diff. Geom. and its For example l.h.s.: The last term needs to be estimated and leads in the equality c equation of twistor type (“ g min 64 Scal univ β 5) 80 √ 4(9 + 4 40 60 20 4 − 10 30 20 -dimensional Sasaki case univ 5 β 4 − , 3 =: 4 Σ The − ± 4 ⊕ , − 0 fixed 0 g min Σ > ⊕ = 8 Scal 4 g min 2 1 4 k has EV T ≥ T k Scal Universal estimate: 2 • • • • λ Σ = Σ g min 65 Scal univ β 5) 80 S √ β 4(9 + 4 40 60 20 4 − 10 30 20 S β =: -dimensional Sasaki case univ 2 5 β 4 + 1 − , 3 =: 4 Σ The − ± g min 4 ⊕ , − 0 fixed 0 g min Σ Scal > 1 4 ⊕ = 8 Scal 4 g min 1 2 1 4 16 k has EV -deformed estimate: T ≥ ≥ T k Scal Universal estimate: S 2 2 • • • • • λ λ Σ = Σ g min 66 Scal g univ ∗ 5) β β 5) 80 S √ √ β . 78 ) , 4 4(9 + 4 ± shows: 4(9 + 4 71 Σ 2 | T /D ≥ ∼ = ( 40 60 2 min g min λ 20 )= Scal 0 is an EV of Σ 2 | for 0 /D 4 ( − 10 30 20 g 2 min β λ =: g min S β Scal =: -dimensional Sasaki case univ 1) 2 5 β − n n 4 4( + 1 − , 3 =: 4 = Σ The − , we have in addition ± g min 4 ⊕ , ∗ − 0 g min fixed 0 g min Σ Scal > 1 4 ⊕ Scal = 8 Scal 4 g min 1 5 2 1 4 16 16 k has EV -deformed estimate: T ≥ ≥ ≥ T k Scal Universal estimate: S 2 2 2 • • • • • A subtle argument based on the fact that λ λ λ Σ = Σ In the region 67 = S -PFB β 1 S , ) ,g,η if it is Einstein 5 -Einstein-Sasaki η M ( is an ) . R -Einstein-Sasaki iff the base η ∈ ,g,η ? 5 a ∗ M ( dependence on the scalar curvature! iff -Einstein-Sasaki mnfd η 2 /D -dimensional Sasaki mnfds are for some 5 4) quadratic -dim. Sasaki mnfds were constructed by Boyer / 5 a, a, a, a, is an EV of 2 Examples in the region + 1 Regular compact Ric = ( On a simply connected Sasaki mnfd g min -dimensional mnfds; Kähler these are , i.e. A Sasaki mnfd is called an 4 Scal ⊥ 4 1 η 1 16 over is a mnfd. Kähler-Einstein Non regular compact First known estimate with [Sasaki condition is not scaling invariant] Dfn. Thm. Example. Open problem: on mnfd. Galicki. 68 de . Xψ = 0 i e ψ s ⊗ i e XD 1 n =1 2 n i k + char. conn.) ψ P s ψ 1 n might be: = ! s X s k∇ 4 + 3 ∇ / · 1 1 4 = ψ ∇ ⇔ 2 via twistor operator = ⊗ k s ) s 2 ψ X P s − /D /D D k ker )= 1 n s ψ ∈ X,Y, and ( + right value of ⊗ ψ ◦ ∇ 2 1 4 sT k p X ⇔ ( ψ = s = p + 2 s s P : : Clifford multiplication k Y P m ∇ M g X Σ ∇ ker → := Y M -twistor spinor s X Σ s , not clear what the is the ”standard” normalisation, ∇ ⊗ : 4 s Use possible improvements of an eigenvalue estimate as a gui / twistor operator: ∇ Eigenvalue estimates for projection on T M : = 1 is called A priori different scaling in = s Fundamental relation: ψ p ( m Idea: to the ‘right’ twistor spinor dM,

i 69 , dM ϕ,ϕ ) 2 2 2 k k T ϕ h k ) g M 3 Z ,...,µ 2 2 1 Scal . 4) µ 3) 2 n> M − T − Z n n ( max( 1) n 4( 2 − 4) n 3) − satisfies n satisfies ( − − 2 4( ϕ n n /D dM ( ( 2 + n 4 k of , and ”=” iff ϕ − λ k dM T 2 2 k , k Z T ϕ 3) 2 1 k s Any spinor k − − 2 n P T n 5) k k 3) 4( 2 − M − 5) = 3) Z n n ( n The first EV − ( 1 − s n 8 n n n − ( + n n 8( g min + = . are the eigenvalues of 3) k 1 Scal − corresponding to the largest eigenvalue of − n n 1) µ dM 4( i Σ − n = ,...,µ n is constant, 1 ( s ϕ,ϕ µ 4 g 2 is a twistor spinor for lies in /D ≥ h Scal ψ ψ M λ where Thm (twistor integral formula). Thm (twistor estimate). • • • where Z 70 ) 2 k 2 2 k , 296-329. k T k T T k k 8 3 7 12 − − abelian ) g min T g min 0 Scal Scal 4 1 Stab( → 3 10 ≥ T ≥ λ -parallel spinors λ c dominant (compared to ∇ g min (”balanced”), , no 3 k Scal T W k 2 √ ± , universal estimate: of class Sasaki: deformation technique yielded better estimates. twistor estimate: ) ) = 0 , g , g µ 6 5 M M On the other hand: ( ( reduces to Friedrich’s estimate for estimate is good for better than anything obtained by deformation • • Ex. • Ex. I.Agricola, J. Becker-Bender, H. Kim, Adv. Math. 243 (2013) Known: 71 . 3) 1 − 2 − n µ n . 2 4( 3) 4 = = 0 − − n s , n n ψ ) 4( ) iff for T n s = 0 − ψ ∧ = 0 · 2 k ψ X ψ · ( T n κX ) s k 3) 2 T P = − 3) 5 1 ∧ ψ n n − − X s X ( 2( n n ∇ 3) 8( + if − ψ + 1 · g n X 2( -twistor spinor ( Scal + n 3) s 1) − /Dψ − µ 1 · n n is an X 2( 4( 1 n ψ + = + κ 2 ψ c X − 3) ∇ Killing spinor with torsion ψ − c µ constant. Killing and Twistor Spinors with Torsion n is a = 2( g ψ ⇔ ∇ + is a twistor spinor with torsion for the same value satisfies the quadratic eq. κ ψ κ Scal In particular: Thm (twistor eq). Dfn. • • • n 72 torsion ing equation. . Then: 6 = 0 µ ψ 2 k µ T k 4 . s − µ is equivalent to the Killing equation 3 1 6 = 6 s = n -twistor spinor for some /Dψ depending on the geometry! 6 s is a for the same value of ψ ψ · eigenspinor with eigenvalue /D λX Assume = is a ψ the twistor equation for . . . with one exception: ψ s In general, this twistor equation cannot be reduced to a Kill Thm. • • Observation: The Riemannian Killingbehave / very twistor differently eq. and their analogue with ∇ + 73 ψ ) c T σ X )( 2 λ ψ. · 12 : ψ − dT X + (1 1 2 ψ. : ) + X T ∀ Xψ ψ 2 ) ) κ k k 1) be a Killing spinor with torsion X, e − X, e . Then ( ( ψ c n 3) T For any spinor field ( R − 1 k k n Let e e + 4( 2( ψ n =1 ) X X k := ) T 2 λ λ − + X ( 2 = , set λ sκ κ ψ · 16 ) +2(2 − X ( c = -dimensional Einstein-Sasaki mnfd with its characteristi 5 Ric ψ Integrability conditions & Einstein-Sasaki manifolds A ) X ( c Thm (curvature in spin bundle). Thm (integrability condition). Cor. with Killing number connection cannot have Killing spinors with torsion. Ric 74 . k T k 2 ± Σ g -parallel spinors c Scal 2 ∇ 15 Twistor spinors with torsion -parallel • • c ≥ ∇ in each of the subbundles λ , k ± T ϕ k 2 ± Σ ∈ g k ± T k ϕ Scal 2 2 15 ± , = 2 -dimensional nearly manifold Kähler = 0 k Killing spinors on nearly manifolds Kähler T µ ) 6 k The following classes of spinors coincide: ,g,J 6 its characteristic connection, torsion is parallel has EV 2 Riemannian KS c M ( Riemannian Killing spinors Killing spinors with torsion ∃ T ∇ - univ. estimate = twistor estimate, - - • Thm. • • There is exactly one such spinor - Einstein, - 75 . 4 m ) t m

⊕ 2 with t, 1 (2) − m so 2 t, − (4) = 2 Killing form t, so = , − 2 (2) , β t, 0 − /SO = 0 univ (4) β T tw ab, t> β · ∇ SO t ) =: = diag(2 = =: t g t − M 8 ) + 2 (irred. components of isotropy rep.), 25 Ric 1 − 2(1 , satisfying m t 5 2 X,Y 2 ≥ ( ⊕ ∇ ≥ β λ 4 − 2 1 λ m = = = 8 t i g : metric contact structure in direction ) m t Y,b Scal ( , , t ) : undeformed metric: 2 parallel spinors : Einstein-Sasaki with 2 Riemannian Killing spinors -dimensional example with Killing spinors with torsion 2 3 = 4 5 / / X, a 2 ( k h A = 1 = 2 T -dimensional Stiefel manifold For general Jensen metric: t t k Universal estimate: Twistor estimate: 5 • • • • • • • • characteristic connection 76 , and these 5 / univ tw EV β β = 2 t t 9 / 4 there exist 2 twistor spinors with torsion for Result: are even Killing spinors with torsion. 77 + 1 , again Sasaki n η 2 ⊗ η ) t − the Tanno deformation. 2 t t g has two Killing spinors with +( ) t tg , η t := , ξ t t g torsion : contact form, dimension M, g ) ( η ∗ R be Einstein-Sasaki, ∈ s.t. t ) ( t tη = t η M,g,ξ,η : Sasaki mnfd, , ( ) – establishes existence of examples in all odd dimensions – ξ t 1 Let = t ξ M,g,ξ,η Tanno deformation of metrics: ( If Einstein-Sasaki: admits two Riemannian Killing spinors Generalisation: deformed Sasaki mnfds with Killing spinors with torsion. Then there exists a • • • Thm. with 78 . ψ · cz/Obata R. η 1) + . ψ − 2 n · 012), 301-311. n na f 4( = = 4 ∗ ψ 1 ) = 2 : − R ) D . 2 ( n 1 D df µ ( > 2 0 1 – Killing spinor = µ = λ ψ η 2 a 2 : then , n ψ, n S · 6 = X n · a M = If ψ X Remarks on the second Dirac eigenvalue – Riemannian manifold, ∇ ) , g n – first eigenvalue of the Laplacian on functions. Lichnerowi 0 1 M λ : New test spinors for upper bounds of Th. Friedrich, Advances in Applied Clifford Algebras,( 22 , (2 A first family of test spinors 79 2 2 | | a a − | − | 2 2 ) ) n n − − 2 (1 | (1 2 2 a a a + − | + 0 1 0 1 2 λ ) λ n q q − . The numbers (1 ≤ ψ < 2 the eigenspinor, then the inner product · a ) ∗ ) 2 2 X ψ + · D D ( 0 1 ( a 2 be a compact Riemannian spin manifold with λ = n q < µ S ψ < µ , too. The second eigenvalue can be estimated by ± X ) 2 6 = ) 2 2 ∇ D n D , D ( ( ψ M 1 1 µ µ Let = = vanishes identically. 2 2 i n n ∗ 2 2 a a ψ , ψ is any “small” eigenvalue and h Theorem: Finally, if are eigenvalues of a Killing spinor 80 , ∗ m ψ . 2 )= a ∗ 1) ψ ( − . D | n . m = 8( = 0 | ≤ | ψ R a n . · 2 ψ o − | · ≥ dη η 2 1 is an eigenspinor, 3) = Λ + ψ ∗ − · η , ψ n η ( 2 m = a : ∗ = 0 − ) ψ + 2 a 1 D is a coclosed eigenform of the Laplace operator, 2) ( Λ 2 η − µ p n – eigenvalues of the problem η, δη ( ≤ -form n 1 na <... 2 ) = Λ The spinor field η Λ ( 1 < ∆ 1 Λ < In this case the if and only if and the eigenvalue can be estimated by Estimates for small Theorem: 0 . )

2 81 D ( is 2 ∗ µ ψ ≤ ), then 2 ) = 7 | a n . −| spinor ﬁeld ψ 2 · 3) η any − + n ( ψ 2 = 7 · a f n + = 1 ∗ Λ ψ , p η , 2 | a -form . In dimension 1 ψ −| 2 ) -dimensional Riemannian manifold ( n 7 and a − f is a (1 2 n a M + If The method applies also in some other small dimensions 0 1 . λ 8 , Fix a Killing spinor q 6 , = 5 min given by a function Corollary: Proof: Remark: n