3-Sasakian manifolds and intrinsic torsion Bogdan Alexandrov 1 - PDF document

3-Sasakian manifolds and intrinsic torsion Bogdan Alexandrov 1 Intrinsic torsion Let T := R n and G ⊂ GL ( n, R ) be a subgroup. The following map is clearly G -invariant. δ : T ∗ ⊗ g ֒ → T ∗ ⊗ gl ( n, R ) = T ∗ ⊗ T ∗ ⊗ T → Λ 2 T ∗ ⊗ T β ⊗ γ ⊗ x �→ ( β ∧ γ ) ⊗ x Therefore Ker δ , Im δ and Λ 2 T ∗ ⊗ T/ Im δ are also representations of G and the projection π : Λ 2 T ∗ ⊗ T → Λ 2 T ∗ ⊗ T/ Im δ is G -invariant. So, if P G M ⊂ P GL ( n, R ) M is a G -structure on a manifold M , then all the above spaces de- fine corresponding associated with P G M bun- dles TM , T ∗ M , g ( M ) . . . , and δ and π induce correctly defined bundle maps. 1

Let ∇ and ∇ ′ be connections in P G . Then ∇ ′ − ∇ ∈ Γ( T ∗ M ⊗ g ( M )) , δ ( ∇ ′ − ∇ ) = T ∇ ′ − T ∇ ⇒ T ∇ ′ − T ∇ ∈ Γ(Im δ ) � T ∇ ′ − T ∇ � ⇒ π = 0 . This shows that the following definition is inde- pendent of the choice of ∇ . � T ∇ � ∈ Γ(Λ 2 T ∗ ⊗ T/ Im δ ) is Def. T P G M := π the intrinsic torsion of the G -structure P G M . We have furthermore T ∇ ′ = T ∇ ⇔ δ ( ∇ ′ −∇ ) = 0 ⇔ ∇ ′ −∇ ∈ Γ(Ker δ ) ⇔ ∇ ′ = ∇ + A for some A ∈ Γ(Ker δ ). Thus, given ∇ , the connections ∇ ′ satisfying T ∇ ′ = T ∇ are parametrized by Γ(Ker δ ). Def. Let W ⊂ Λ 2 T ∗ ⊗ T/ Im δ be a G -invariant subspace. P G M is said to be of (Gray-Hervella) type W if T P G M ∈ Γ( W ( M )). 2

E.g., T P G M = 0 iff there exists a connection ∇ in P G M with T ∇ = 0 (1 -integrable G -structure ). Suppose now that N is a G -invariant comple- ment of Im δ in Λ 2 T ∗ ⊗ T . Then there exists a connection with Torsion in Γ( N ( M )). If further- more δ is injective, then this connection ∇ 0 ,N is unique and is called the canonical connection of P G M with respect to N . Examples: 1. G = SO ( n ) or O ( n ). δ so ( n ) : T ∗ ⊗ so ( n ) → Λ 2 T ∗ ⊗ T ���� � �� � ∼ = T ∗ ∼ =Λ 2 T ∗ is an isomorphism. Therefore • Im δ so ( n ) = Λ 2 T ∗ ⊗ T ⇒ Λ 2 T ∗ ⊗ T/ Im δ so ( n ) = 0 ⇒ T P SO ( n ) M = 0 and thus there exists a connection ∇ in P SO ( n ) M with T ∇ = 0. • Ker δ so ( n ) = 0. Therefore ∇ is unique. ∇ is the Levi-Civita connection. 3

2. G ⊂ SO ( n ) or O ( n ). Then g ⊂ so ( n ) = g ⊕ g ⊥ Thus δ so ( n ) i δ g : T ∗ ⊗ g → T ∗ ⊗ ( g ⊕ g ⊥ ) = T ∗ ⊗ so ( n ) ֒ → Λ 2 T ∗ ⊗ T = δ so ( n ) ( T ∗ ⊗ g ) ⊕ δ so ( n ) ( T ∗ ⊗ g ⊥ ) So δ g is injective, Im δ g = δ so ( n ) ( T ∗ ⊗ g ) and δ so ( n ) ( T ∗ ⊗ g ⊥ ) is a G -invariant complement of Im δ g in Λ 2 T ∗ ⊗ T . Thus there exists a unique connection ∇ 0 with Torsion T ∇ 0 ∈ Γ( δ so ( n ) ( T ∗ M ⊗ g ⊥ ( M ))). Equiva- lently, ∇ 0 is characterized by ∇ 0 = ∇ + A 0 with A 0 ∈ Γ( T ∗ M ⊗ g ⊥ ( M )). Because of the isomorphisms Λ 2 T ∗ ⊗ T/ Im δ g ∼ = δ so ( n ) ( T ∗ ⊗ g ⊥ ) ∼ = T ∗ ⊗ g ⊥ T P G M ↔ T ∇ 0 ↔ A 0 we have and the Gray-Hervella-type classification is usually done in terms of a decomposition T ∗ ⊗ g ⊥ = W 1 ⊕ · · · ⊕ W k of T ∗ ⊗ g ⊥ into irreducible G -invariant summands. 4

2 3 -Sasakian manifolds Def. ( M, g ) is a 3 -Sasakian manifold if the g = r 2 g + d r 2 ) is hyper- cone ( � M = M × R + , � K¨ ahler. In this case there exist orthogonal I, � � J, � K ∈ Γ( End ( T � M )) which satisfy the quaternionic identities. Let ξ I = − � I∂ r | r =1 , ξ J = − � J∂ r | r =1 , ξ K = − � K∂ r | r =1 , V = span { ξ I , ξ J , ξ K } , I = � J = � K = � I | V ⊥ , J | V ⊥ , K | V ⊥ , I | V = 0 , J | V = 0 , K | V = 0 . Then TM = V ⊥ ⊕ V , V is trivialised by the or- thonormal frame ξ I , ξ J , ξ K , and I, J, K satisfy the quaternionic identities and are orthogonal on V ⊥ . Thus we obtain an Sp ( n )-structure on M , where the action of Sp ( n ) ⊂ SO (4 n + 3) on R 4 n +3 = R 4 n ⊕ R 3 is given by the standard representation on R 4 n and the trivial one on R 3 . 5

3 Sp ( n ) Sp (1) -structures on (4 n + 3) -dimensional manifolds If we consider an Sp ( n )-structure as above, we have T ∗ ⊗ sp ( n ) ⊥ = 15 R ⊕ other summands . � �� � 57 54 33 n ≥ 3 or n =2 or n =1 Since the dimension of the trivial representation is too big, we shall consider a more general G - structure. Let G := Sp ( n ) Sp (1) ⊂ SO (4 n + 3) acting on R 4 n +3 = R 4 n ⊕ R 3 by the standard rep- resentation of Sp ( n ) Sp (1) on R 4 n ∼ = H n and through the projection Sp ( n ) Sp (1) → SO (3) on R 3 . (Then Sp ( n ) ⊂ G acts on R 4 n +3 as above.) We have T ∗ ⊗ g ⊥ = 2 R ⊕ other summands , � �� � 31 for n> 1 or 18 for n =1 T ∗ ⊗ g = R ⊕ other summands . � �� � 9 for n> 1 or 8 for n =1 6

One basis of 2 R ⊂ T ∗ ⊗ g ⊥ is given by A ( P, Q ) = S ( g ( IP, Q ) ξ I − η I ( Q ) IP ) , B ( P, Q ) = S ( η I ( P ) IQ − nη J ∧ η K ( P, Q ) ξ I ) and R ⊂ T ∗ ⊗ g is spanned by C ( P, Q ) = S η I ( P )( IQ +2 η J ( Q ) ξ K − 2 η K ( Q ) ξ J ) . Here η I , η J , η K are dual to ξ I , ξ J , ξ K and S denotes the cyclic sum with respect to I, J, K . Let T A , T B , T C be the corresponding torsions. Then all invariant complements of R = span { T C } = Im δ ∩ span { T A , T B , T C } � �� � 3 R ⊂ Λ 2 T ∗ ⊗ T are of the form N x,y = span { T A + xT C , T B + yT C } , x, y ∈ R . For the canonical connections ∇ 0 ,N x,y we have T ∇ 0 ,Nx,y = λ ( T A + xT C ) + µ ( T B + yT C ) , ∇ 0 ,N x,y = ∇ + λ ( A + xC ) + µ ( B + yC ) , where in the first instance λ and µ are functions. Notice that they are the same for all x, y . 7

Thm 1 If the the torsion of ∇ 0 = ∇ 0 ,N 0 , 0 is T 0 = λT A + µT B , then λ and µ are constants and the curvature tensors of ∇ 0 and ∇ satisfy R 0 = � R 0 + R hyper , R = � R + R hyper , R 0 and � where � R are explicit G -invariant ten- sors (which depend on λ, µ ) and R hyper is ahler curvature tensor on V ⊥ . In a hyper-K¨ particular, Ric has two eigenvalues: Ric | V = 2( n + 2)(2 λ 2 + 4 λµ + ( n + 2) µ 2 ) , Ric | V ⊥ = 2 λ ((4 n + 5) λ + 2( n + 2) 2 µ ) . R 0 ∈ Λ 2 ⊗ g , ∇ 0 T 0 ∈ 2 T ∗ ⊗ R Proof: and T 0 ( T 0 ( · , · ) , · ) is an explicit G -invariant ten- Then decompose the spaces Λ 2 ⊗ g and sor. 2 T ∗ into G -irreducible components and use the Bianchi identity b ( R 0 − ∇ 0 T 0 − T 0 ( T 0 ( · , · ) , · )) = 0 and Schur’s lemma. � 8

General constructions: Let ( M, g ) have a G -structure, so that the po- tential of ∇ 0 is λA + µB . 1. Then for g c,d = d 2 ( g | V ⊥ + c 2 g | V ) we obtain a G -structure, where the potential of ∇ 0 ,g c,d is λ c,d A g c,d + µ c,d B g c,d with � � µ − 2( c 2 − 1) λ c,d = c µ c,d = 1 dλ, n + 2 λ . cd 2. If we change the sign of ξ I , ξ J , ξ K , then we obtain a G -structure, where the signs of λ and µ are also changed. 9

Examples: 1. Let ( M, g ) be 3-Sasakian. Then 1 λ = − 1 , µ = for g, n + 2 2 − c 2 λ = − c d, µ = for g c,d . ( n + 2) cd In all cases λ < 0 , 2 λ + ( n + 2) µ < 0. 2. Let ( M, g ) be 3-Sasakian with signature (3 , 4 n ). Then for the metric d 2 ( − g | V ⊥ + c 2 g | V ) µ = − 1 + 2 c 2 λ = c d, ( n + 2) cd. In all cases λ > 0 , 2 λ + ( n + 2) µ < 0. 3. Let M ′ be hyper-K¨ ahler, M = M ′ × SO (3) with the product metric. On M we have a G - structure with λ = 0 , µ < 0 (depending on the scaling of the metric on SO (3)) and we have 2 λ + ( n + 2) µ < 0. 10

4. Let ( M ′ , g ′ , I ′ , J ′ , K ′ ) be hyper-K¨ ahler. Then dΩ I ′ = 0 , dΩ J ′ = 0 , dΩ K ′ = 0 . Hence locally there exist α I ′ , α J ′ , α K ′ such that Ω I ′ = d α I ′ , Ω J ′ = d α J ′ , Ω K ′ = d α K ′ . Let M = M ′ × R 3 and u, v, w be the coor- dinates on R 3 . Fix ν < 0 and define ξ I = ∂ u , η I = d u − να I ′ , ξ J = ∂ v , η J = d v − να J ′ , ξ K = ∂ w , η K = d w − να K ′ , V = span { ξ I , ξ J , ξ K } = T R 3 , V ⊥ = { X : η I ( X ) = η J ( X ) = η K ( X ) = 0 } (notice that V ⊥ � = TM ′ ), g = g ′ + η 2 I + η 2 J + η 2 K , I | V = 0 , J | V = 0 , K | V = 0 , IX ′ = hI ′ X ′ , X ′ = hJ ′ X ′ , KX ′ = hK ′ X ′ for X ′ ∈ TM ′ . Thus we obtain a G -structure λ = − ν ν on M with 2 > 0 , µ = n +2 < 0 and 2 λ + ( n + 2) µ = 0. 11

5. We obtain further examples if we apply the second ”general construction” on the above ones. 6. Let M ′ be hyper-K¨ ahler. Then M = M ′ × R 3 with the product metric has a G -structure with λ = 0 , µ = 0. Thm 2 1. Every pair ( λ, µ ) appears exactly once in the above list of examples. 2. A manifold with a G -structure of the con- sidered type with torsion T ∇ 0 = λT A + µT B is locally equivalent to the corresponding example. Rem. For each ( λ, µ ) there exists a unique con- nection with totally skew-symmetric torsion: ∇ a = ∇ + λA + µB + ( λ − µ ) C. Consider a 3-Sasakian Wolf space, written in the form H · Sp (1) /L · Sp (1). Then the second Ein- stein metric is one of the normal metrics and ∇ a is the corresponding canonical connection. 12