By Morris, P.

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**Example text**

14). Fig. 14 Curled torus. The set of regular values R of EM is simply connected. This means that the monodromy is trivial for any closed path inside R. The question that we want to answer is what happens for a path Γ that encircles the origin and crosses C at a point p. Notice that EM−1 (Γ ) → Γ is not a T2 bundle since EM−1 (p) is not a T2 . In order to be more precise we have to give some details about how we deﬁne and compute ordinary monodromy. Recall that a system has monodromy if the regular T2 bundle is not trivial.

1 we describe the foliation of the phase space of the 1: − 2 resonance system by invariant tori. 2 we explain why we have to consider a sublattice of the period lattice and we compute its variation along a closed path that crosses the critical line. 3 we relate the period lattice description to the proof of fractional monodromy in [43]. 4 we give an explanation for the particular choice of the quadratic terms in H as related to the choice in [99]. 5. 6 we give two conjectures for the monodromy matrix in higher order resonances m: − n.

Direct computation shows that τ 2 = τ12 + τ22 + τ32 and σ 2 = σ12 + σ22 + σ32 are Td × T invariant and can be chosen as these two invariants. 9) indicates that there are no other principal or auxiliary invariants of this degree. 20. The 3-vector τ = (τ1 , τ2 , τ3 ) is the angular momentum vector. Both n and τ 2 = τ12 + τ22 + τ32 are totally symmetric with respect to the larger group O(3) × T . 8) which represents Td × T symmetry is σ 2 . We now come to a central result which provides the basis for the classiﬁcation of generic tetrahedral Hamiltonians.

### Metric projections onto subspaces of finite codimension by Morris, P.

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