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 \(M\).


  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.


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.

See also: Dually flat manifolds