Information Geometry
# Dual connections

## Properties

## Examples

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

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 \(M\).

- The dual is unique.
- \(\nabla^{**} = \nabla\).
- \(\overline\nabla = \frac{1}{2} (\nabla+\nabla^*)\) is a metric connection. In particular, if \(\nabla,\nabla^*\) are torsion-free, it is the Levi-Civita connection.
- Let \(\nabla\) be torsion-free. Then \(\nabla^*\) is torsion-free iff \(\nabla g\) is symmetric.
- 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 \(\langle X,Y \rangle_p = \langle \Pi_\gamma X, \Pi^*_\gamma Y \rangle_q\)
- Dual Riemann curvature tensors satisfy \(\langle R(X,Y)Z, W \rangle = - \langle R^* (X,Y)W, Z \rangle\).
- (Important!) \(\nabla\) is flat iff \(\nabla^*\) is flat.

** 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 \(\nabla^A\):

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

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.

**See also:** Dually flat manifolds