Abstract
Some effects of cubic tensors in divergences, which characterize dually flat spaces, are investigated. Focusing on the small area around a specific point, how tangent space approximation is corrected by the cubic tensor at the point, is described. The cubic tensor in divergence is treated as a perturbation, supposing that the coordinates in tangent space are unperturbed solutions. A first order perturbation solution is presented as a main result. On the other hand, the unperturbed solution is yet another coordinate different from the usual affine coordinate. The divergences are shown to be written in a simple form if it is written in terms of the unperturbed solution. Furthermore, a relation which transforms similarities to the divergences is shown. Consequently, the asymmetry in the divergences between the tangent point and the other point is shown to be attributed to the difference between the squared norms of the two kinds of coordinates.
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Notes
In the following, Einstein notation is also used for Roman letters as \(\langle \theta , \eta \rangle = \theta ^i \eta _i\) and derivatives are also written as \(\theta ^i = \partial \phi / \partial \eta _i = \partial ^i \phi \), \(\eta _i = \partial \psi / \partial \theta ^i = \partial _i \psi \). Furthermore, we identify specific points by Greek letters \(\iota \), \(\kappa \) etc. and use them as indices like \(\theta _\iota \), \(\eta ^\iota \) etc., representing a point \(\iota \) as a vector \(\theta _\iota = (\theta ^1_\iota \cdots \theta ^n_\iota )^\prime \) or a dual vector \(\eta ^\iota = (\eta ^\iota _1 \cdots \eta ^\iota _n)^\prime \). Einstein notation is not applied for these Greek letters.
We assume that the diagonal components \(\hat{d}_{\iota \iota }\) equal to zero.
In the following, we denote the Kronecker delta by \(\delta ^{ij}\) or \(\delta _{ij}\).
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The author greatly appreciates the valuable comments of the reviewers.
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Kumagai, A. A perturbative picture of cubic tensors in dually flat spaces. Japan J. Indust. Appl. Math. 35, 107–115 (2018). https://doi.org/10.1007/s13160-017-0274-8
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DOI: https://doi.org/10.1007/s13160-017-0274-8