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In information Geometry, there is a definition of dual connection, which is: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfied $$X(g(Y,Z))=g(\nabla_XY,Z)+g(Y,\nabla^*_XZ)$$ It seems that this definition depends on the metric $g$. My question is could we find another way to define the dual connections above, without the metric involved?

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    $\begingroup$ One can always find a way to define something. E.g., one can probably speak about dual connections on dual vector bundles or such. The true question is what you want out of this definition! $\endgroup$ – Alex Degtyarev Dec 19 '14 at 8:41
  • $\begingroup$ I think there is a typo here: the second summand in the display should be $g(Y,\nabla^*_X Z)$? $\endgroup$ – Fran Burstall Dec 21 '14 at 18:10
  • $\begingroup$ Oh,yes. I made a mistake, thank you! $\endgroup$ – user64142 Dec 22 '14 at 7:48
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If you have any manifold $M$ and a vector bundle $\mathcal{V}$ over $M$, then associated to any connection $\nabla$ on $\mathcal{V}$, there is indeed a dual connection $\nabla^*$ on the dual bundle $\mathcal{V}^*$ satisfying $$ \partial_X[\xi(v)] = (\nabla^*_X \xi)(v) + \xi(\nabla_X v)$$ for sections $\xi$ of $\mathcal{V}^*$, $v$ of $\mathcal{V}$ and vectors $X \in TM$. This connection is unique, namely it is defined by the formula above (at least if the fiber space $V$ of $\mathcal{V}$ is reflexive, in particular if it is finite-dimensional).

There is no metric needed for this construction.

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No, the dual connection depends strictly on the metric (that is, a change of the metric would lead to a change of the dual connection), and therefore any definition makes sense only when a metric is defined. For example,

$\nabla$ is self-dual ($\nabla=\nabla^*$)$\Longleftrightarrow \nabla$ is compatible with $g$

and of course the right condition is dependent on $g$.

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  • $\begingroup$ Thanks for your answer, however, this is not what I want. Maybe because I didn't explain my question in a right way. Consider the dual connections over vector bundler and its dual space, I think they don not need the metrics. Since we talk about connections, I think the metric in the definition mentioned is also unnecessary. $\endgroup$ – user64142 Dec 19 '14 at 11:49
  • $\begingroup$ But if one connection is defined in a bundle and the other one in the dual bundle, then the description above makes no sense, since $X$, $Y$ and $Z$ are fields. I still do not understand the setting completely. $\endgroup$ – Jjm Dec 19 '14 at 12:23

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