An example The general procedure Conformally invariant operators Overdetermined systems, conformal differential geometry, and the BGG complex Andreas ˇ Cap University of Vienna Faculty of Mathematics IMA, July 2006 Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators Based on joint work with T. Branson, M. Eastwood, and A.R. Gover procedure for rewriting certain overdetermined systems in first order closed form works for symbols of geometric origin associated to various geometric structures we will restrict to the version for Riemannian manifolds comes from a method for constructing conformally invariant differential operators, which generalizes to parabolic geometries Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators Structure An example 1 The general procedure 2 Conformally invariant operators 3 Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators basic Riemannian geometry is closely related to representation theory of O ( n ) standard representation corresponds to the (co)tangent bundle use representation theory to organize symmetries strategy Embed O ( n ) into a larger group G ∼ = O ( n + 1 , 1) and analyze representations of G from the point of view of this subgroup. In the example, we will deal with the standard representation of G . Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators Consider V = R n +2 with the inner product � � x 0 y 0 n � . . . . , := x 0 y n +1 + x n +1 y 0 + x i y i . . i =1 x n +1 y n +1 Basis vectors e 1 , . . . , e n span a standard Euclidean R n The remaining two basis vectors span an R 2 with a (1 , 1)–metric and light-cone–coordinates � , � has signature ( n + 1 , 1) and hence G := O ( V ) is isomorphic to O ( n + 1 , 1). � 1 0 0 � Mapping A to defines an inclusion O ( n ) ֒ → G . 0 A 0 0 0 1 Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators As a representation of O ( n ), we have = R ⊕ R n ⊕ R V = V 0 ⊕ V 1 ⊕ V 2 ∼ with V 0 spanned by e n +1 and V 2 spanned by e 0 . Notation : column vectors with V 2 on top. For k = 0 , 1 , 2 consider Λ k R n ⊗ V . This splits as Λ k R n ⊗ V = ⊕ i (Λ k R n ⊗ V i ) but Λ k R n ⊗ V 1 admits a finer decomposition. As an example, for k = 1 we get R n ⊗ R n = R ⊕ S 2 0 R n ⊕ Λ 2 R n . Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators Doing the decompositions for k = 0 , 1 , 2 we notice coincidences: R n Λ 2 R n R 0 R n ⊕ Λ 2 R n R n ⊕ W 2 ⊕ Λ 3 R n R n R ⊕ S 2 R n Λ 2 R n R Assigning homogeneity k + i to elements of Λ k R n ⊗ V i we identify components of the same homogeneity. Use these identifications to define ∂ : V → R n ⊗ V as well as δ ∗ : Λ k R n ⊗ V → Λ k − 1 R n ⊗ V for k = 0 , 1 such that δ ∗ ◦ δ ∗ = 0. Explicit formulae in terms of bundles below. Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators Let ( M , g ) be a Riemannian manifold of dimension n . V gives rise to a vector bundle V = V 0 ⊕ V 1 ⊕ V 2 → M . Λ k R n ⊗ V corresponds to Λ k T ∗ M ⊗ V . sections of Λ k T ∗ M ⊗ V are triples consisting of two k –forms and one T ∗ M –valued k –form. We use subscripts 0 , 1 , 2 to indicate components. The maps ∂ and δ ∗ Using abstract indices, we define ∂ : V → T ∗ M ⊗ V , and δ ∗ : Λ k T ∗ M ⊗ V → Λ k − 1 T ∗ M ⊗ V for k = 0 , 1 by � − 1 � � � � � � h j � � 1 � � h ij � n − 1 φ ik k n φ j 0 h , δ ∗ , δ ∗ j ∂ = hg ij = = φ j φ jk φ ijk 1 − f i 2 f ij − φ i f f j f ij 0 0 Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators Let ∇ be the component–wise Levi–Civita connection on V . Note that while ∂ and δ ∗ preserve homogeneities, ∇ raises homogeneity by one. The basic system Define ˜ ∇ on V by ˜ ∇ Σ := ∇ Σ + ∂ Σ Choose a bundle map A : V 0 ⊕ V 1 → S 2 0 T ∗ M , and view it as A : V → T ∗ M ⊗ V . Consider the first order system ˜ for some ψ ∈ Ω 2 ( M , V ) ∇ Σ + A (Σ) = δ ∗ ψ The core of the method is to equivalently rewrite this in two ways, once as a higher order system on Σ 0 ∈ Γ( V 0 ) and once as a first order closed system. Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators First rewrite in terms of the V 0 –component Σ 0 . By construction A has values in ker( δ ∗ ) and δ ∗ ◦ δ ∗ = 0, so ˜ ∇ Σ + A (Σ) = δ ∗ ψ implies δ ∗ ˜ ∇ Σ = 0. The operator δ ∗ ˜ ∇ on Γ( V ) is given by 1 n ∇ j φ j + h h ∇ i h ˜ ∇ i δ ∗ φ j �→ ∇ i φ j + hg ij �→ −∇ j f + φ j f ∇ i f − φ i 0 It is evident, how to solve this: Choose f ∈ Γ( V 0 ) arbitrarily put φ i = ∇ i f , i.e. φ = df put h = − 1 n ∇ i ∇ i f = − 1 n ∆ f , where ∆ is the Laplacian Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators Proposition (splitting operator in degree zero) Given f ∈ Γ( V 0 ) there is a unique Σ ∈ Γ( V ) such that Σ 0 = f and δ ∗ ˜ ∇ Σ = 0. Mapping f to Σ defines a linear second order differential operator L : Γ( V 0 ) → Γ( V ), which is explicitly given by − 1 n ∆ f 0 2 � = ( − 1) ℓ ( δ ∗ ∇ ) ℓ L ( f ) = ∇ i f 0 f f ℓ =0 Observe that the components of L ( f ) in V 0 and V 1 are f and ∇ f , respectively. Hence For any A : V 0 ⊕ V 1 → S 2 0 T ∗ M , the map f �→ A ( L ( f )) is a first order operator Γ( V 0 ) → S 2 0 T ∗ M ⊂ T ∗ M ⊗ V , and all such operators can be obtained in that way. Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators We have seen that ˜ ∇ Σ + A (Σ) = δ ∗ ψ implies Σ = L ( f ), where f = Σ 0 . Now − 1 − 1 n ∆ f n ∇ i ∆ f ∇ L ( f ) = ˜ ˜ = ∇ i ∇ j f − 1 ∇ i ∇ j f n g ij ∆ f 0 f ∇ i ∇ j f − 1 n g ij f = ∇ ( i ∇ j ) 0 f , the tracefree part of ∇ 2 f . Adding A ( L ( f )) corresponds to adding D ( f ) to the middle component, where D : Γ( V 0 ) → Γ( S 2 0 T ∗ M ) is a first order operator. � h i � ∈ Ω 1 ( M , V ) is of the form δ ∗ ψ for some An element φ ij f i ψ ∈ Ω 2 ( M , V ) iff f i = 0 and φ ij is skew. Since our middle component is symmetric by construction it has to vanish and we get: Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators Proposition For any first order operator D : Γ( V 0 ) → Γ( S 2 0 T ∗ M ) there is a bundle map A : V → T ∗ M ⊗ V such that f �→ L ( f ) and Σ �→ Σ 0 induce inverse bijections between solutions of ∇ ( i ∇ j ) 0 f + D ( f ) = 0 and of ˜ ∇ Σ + A (Σ) = δ ∗ ψ for some ψ ∈ Ω 2 ( M , V ). � h � satisfies ˜ ∇ Σ + A (Σ) = δ ∗ ψ for some We also see that if Σ = φ i f ψ ∈ Ω 2 ( M , V ), then we actually have ∇ i h + τ i = 0 ∇ i φ j + hg ij + A ij ( f , φ ) = 0 ∇ i f − φ i = 0 for some one–form τ i . To rewrite in closed form, it remains to compute τ i . Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
An example The general procedure Conformally invariant operators Let d ˜ ∇ : Ω 1 ( M , V ) → Ω 2 ( M , V ) be the covariant exterior derivative associated to ˜ ∇ . Explicitly ˜ ∇ α ( ξ, η ) = ˜ ∇ ξ ( α ( η )) − ˜ ∇ η ( α ( ξ )) − α ([ ξ, η ]) . d ∇ ˜ d ˜ ∇ Σ = R • Σ, the action of the Riemann curvature on Σ. d ˜ ∇ ( A (Σ)) is concentrated in the middle component, and depends only on Σ 0 , Σ 1 and their first derivatives. Elements which are concentrated in the top component are reproduced by δ ∗ d ˜ ∇ . Hence τ i equals the top component of − δ ∗ ( R • Σ + d ˜ ∇ ( A (Σ))), and inserting the equations for ∇ Σ 0 and ∇ Σ 1 we already have, we obtain an expression in terms of the values of Σ. Andreas ˇ Cap Overdetermined systems, conformal geometry, and BGG
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