Abstract
In this section we will discuss some basic facts and background notions. We will start by recalling some well known definitions in Hamiltonian dynamics [10, 12, 30, 109] and in Kahler geometry [18, 30, 95, 106, 121, 123, 133] (which will be of use in view of their generalization later on), and will then pass to collect some basic notions on Kahler and hyperkahler manifolds and structures [13, 14, 74, 92, 100, 104, 115, 133]. Finally we will restrict to the special (but relevant!) case of Euclidean such manifolds, recalling that in this case there are some “standard” hyperkahler structures.
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Notes
- 1.
By “smooth” we will always mean \(\mathcal {C}^\infty \); note also that all objects (manifolds, vector fields, forms, etc.) we consider are real unless explicitly stated otherwise.
- 2.
Many excellent discussions of Symplectic Geometry are available in the literature, see e.g. [10, 30, 82]; the reader is referred to these for details.
- 3.
Vector fields are always assumed to be smooth.
- 4.
Note that \(A_{ij}\) are the elements of the matrix A (defined in equation (1)) while \(A^{ij}\) are the elements of the inverse matrix \(A^{-1}\). If we think of A as a (symplectic) metric this notation is quite natural. It should be stressed that here we have used the notation with upper and lower indices—in order to prepare for the situation where a Riemannian metric is defined on M—but we do not need assuming such a metric is defined. In fact, as mentioned above, a symplectic manifold does not require the introduction of a metric. Note however that we have assumed some metric g is defined in M and hence in \(U \subseteq M\); we can thus dispense with such worries and just use this to raise and lower indices when needed.
- 5.
We are showing the proof of this well known fact in order to emphasize, later on, the differences with the hyperhamiltonian case.
- 6.
It should be stressed that not all symplectic manifolds admit a Kahler structure; see e.g. [116].
- 7.
In our earlier works, this was denoted as the quaternionic structure; however this notation is potentially confusing, as the notion of quaternionic structure is routinely used in the literature in a different sense [74, 115]; thus we prefer to use this (less suggestive but also less likely to cause misunderstandings) denomination.
- 8.
In particular, for \(M = \mathbf{R}^{4 n} = \mathbf{R}^4 \oplus \ldots \oplus \mathbf{R}^4\), with Euclidean metric and a standard hyperkahler structure in each \(\mathbf{R}^4\) component, we have a set of \(2^n\) mutually dual hyperkahler structures.
- 9.
If M is hyperkahler and of dimension \(m = 4 n\), and it can be decomposed as the product of n hyperkahler manifolds, we say it is fully decomposable. This is notably the case of \(\mathbf{R}^{4 n}\) with standard structures, i.e. seen as the product of standard \(\mathbf{R}^4\) hyperkahler manifolds.
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Gaeta, G., Rodríguez, M.A. (2017). Background Material. In: Lectures on Hyperhamiltonian Dynamics and Physical Applications. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-54358-1_1
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DOI: https://doi.org/10.1007/978-3-319-54358-1_1
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