By Vincent Rivasseau (Chief Editor)

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Here φ−1 T N is the pullback of T N to M along φ. We work this out in local coordinates for k = 0, 1, 2. 4) where if the left-hand side is evaluated at x ∈ M , all objects on N are evaluated at φ(x). On the bundles ⊗k T ∗ M ⊗φ−1 T N , there is a natural inner product ·, · which in local coordinates is given by v, w = v α wβ hαβ ξ, η = ξiα ηjβ hαβ g ij and similarly for higher order tensors. t. ·, · . Further, letting ∆D v α = g ij Di Dj v α the identity ∆D v, w µg = − M Dv, Dw µg M holds, and thus the operator ∆D is self-adjoint with respect to the L2 pairing v, w µg .

The advantages of our results in [12] are as follows. (i) We can prove our assumptions for several mixing states of one-dimensional systems. The assumption of [12] is valid for Gibbs states for finite range interactions, ground states of the (massive) XY model, quasifree states of CAR algebras. (ii) Our limit theorem is valid for (not strictly local) certain quasi-local observables, which we named exponentially localized observables. 1), we can show CLT under the following condition: |ϕ(Q1 Q2 ) − ϕ(Q1 )ϕ(Q2 )| ≤ C(Q1 ) Q2 r−d−1− .

Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. 1, 19–44 (1994). [22] S. Klainerman and F. Nicol` o, On local and global aspects of the Cauchy problem in General Relativity, Class. Q. , 16, R73-R157 (1999). D. gov/gr-qc/0203012 . [24] G. D. Rendall, Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data, Comm. Math. Phys. 150, 561–583 (1992). [25] M. Struwe, Globally regular solutions to the u5 Klein-Gordon equation, Ann.

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Annales Henri Poincaré - Volume 4 by Vincent Rivasseau (Chief Editor)

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