# Dual connections for Information Geometry

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?

• 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! – Alex Degtyarev Dec 19 '14 at 8:41
• I think there is a typo here: the second summand in the display should be $g(Y,\nabla^*_X Z)$? – Fran Burstall Dec 21 '14 at 18:10
• Oh,yes. I made a mistake, thank you! – user64142 Dec 22 '14 at 7:48

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).
$\nabla$ is self-dual ($\nabla=\nabla^*$)$\Longleftrightarrow \nabla$ is compatible with $g$
and of course the right condition is dependent on $g$.
• 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. – Jjm Dec 19 '14 at 12:23