Dual connections for Information Geometry

Question

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?

Answer 1

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.

Answer 2

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 $g$.