Information Geometry

# Dual connections

Let $$(M,g)$$ be a Riemannian manifold, and $$\nabla$$ be an any affine connection. The dual connection of $$\nabla$$ with respect to $$g$$ is a connection $$\nabla^*$$ such that

$X \langle Y,Z \rangle = \langle \nabla_X Y,Z \rangle + \langle Y, \nabla^*_X Z \rangle$

for $$X,Y,Z$$ vector fields.

The quadruple $$(M,g,\nabla,\nabla^*)$$ is called a dually connected manifold, and the triple $$(g,\nabla,\nabla^*)$$ is called a dualistic structure for $$M$$.

## Properties

1. The dual is unique.
2. $$\nabla^{**} = \nabla$$.
3. $$\overline\nabla = \frac{1}{2} (\nabla+\nabla^*)$$ is a metric connection. In particular, if $$\nabla,\nabla^*$$ are torsion-free, it is the Levi-Civita connection.
4. Let $$\nabla$$ be torsion-free. Then $$\nabla^*$$ is torsion-free iff $$\nabla g$$ is symmetric.
5. If $$X,Y \in T_p M$$ and $$\Pi_\gamma, \Pi^*_\gamma$$ denote parallel transport along a curve $$\gamma$$, with respect to $$\nabla$$ and $$\nabla^*$$ respectively, then $$\langle X,Y \rangle_p = \langle \Pi_\gamma X, \Pi^*_\gamma Y \rangle_q$$
6. Dual Riemann curvature tensors satisfy $$\langle R(X,Y)Z, W \rangle = - \langle R^* (X,Y)W, Z \rangle$$.
7. (Important!) $$\nabla$$ is flat iff $$\nabla^*$$ is flat.

## Examples

Example 1. Let $$\nabla^g$$ be the Levi-Civita connection of $$(M,g)$$, and $$A$$ be any symmetric 3-tensor. The following expression defines a torsion-free affine connection $$\nabla^A$$:

$\langle \nabla^A_X Y,Z \rangle = \langle \nabla^g_X Y,Z \rangle + A(X,Y,Z),$

The dual is given by $$(\nabla^A)^* = \nabla^{-A}$$.

Example 2 ($$\alpha$$-connections). Given two fixed torsion-free dual connections $$(\nabla,\nabla^*)$$, they can be extended to a family of $$\alpha$$-connections

$\nabla^{(\alpha)} = \frac{1+\alpha}{2}\nabla + \frac{1-\alpha}{2}\nabla^*,$

such that $$\nabla^{(\alpha)}$$ and $$\nabla^{(-\alpha)}$$ are duals, $$\nabla^{(0)}$$, is the Levi-Civita connection, $$\nabla^{(1)} = \nabla$$ and $$\nabla^{(-1)}=\nabla^*$$. In particular, if we define the Amari-Chentsov tensor as $$C(X,Y,Z) = \langle \nabla_X Y - \nabla^*_X Y,Z \rangle = \nabla g$$, then we can construct $$\nabla^{(\alpha)}$$ using the previous example for $$A= \frac{\alpha}{2}C$$.

An important case of a dualistic structure in information geometry is the pair of exponential and mixture connections.