Dual connections are paired through the metric. They let information geometry carry two compatible notions of straightness, typically mixture and exponential.
Metric duality
The definition of duality says that differentiating an inner product can be split between two connections.
Definition. Two affine connections \(\nabla\) and \(\nabla^*\) are dual with respect to \(g\) if \[ Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_X^*Z) \] for all vector fields \(X,Y,Z\).
Proposition. If \(\nabla\) is torsion-free, then the dual connection \(\nabla^*\), when it exists, is uniquely determined by \(g\) and \(\nabla\).
Proof. For fixed \(X,Z\), the defining identity determines the covector \(Y\mapsto g(Y,\nabla_X^*Z)\). Nondegeneracy of \(g\) identifies this covector with a unique vector \(\nabla_X^*Z\).
Average connection
The Levi-Civita connection is the midpoint of a dual pair.
Theorem. If \(\nabla\) and \(\nabla^*\) are torsion-free and dual, then \[ \nabla^{(0)}=\frac12(\nabla+\nabla^*) \] is the Levi-Civita connection of \(g\).
Proof. The average connection is torsion-free. Adding the duality identity to the same identity with \(\nabla\) and \(\nabla^*\) exchanged shows that \(\nabla^{(0)}g=0\). By uniqueness of the torsion-free metric connection, it is Levi-Civita.
Dual geodesic coordinates
Dual flatness supplies primal and dual affine coordinate systems.
Definition. A pair \((\nabla,\nabla^*)\) is dually flat if both connections are flat and dual with respect to \(g\). Coordinates affine for \(\nabla\) and \(\nabla^*\) are called dual affine coordinates.
Remark. Exponential-family natural parameters and expectation parameters are the basic statistical example.
Pythagorean geometry
Duality is the reason orthogonal projection identities involve one geodesic of each type.
Remark. The generalized Pythagorean theorem uses a \(\nabla\)-geodesic and a \(\nabla^*\)-geodesic meeting orthogonally under \(g\).
Metric duality
Definition. Connections \(\nabla\) and \(\nabla^*\) are dual with respect to \(g\) if \[ Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_X^*Z) \] for all vector fields \(X,Y,Z\).
Proposition. Given \(g\) and \(\nabla\), there is a unique dual connection. In coordinates with lowered last index \(\Gamma_{ijk}=g_{k\ell}\Gamma^\ell{}_{ij}\), \[ \Gamma^*_{ikj}=\partial_i g_{jk}-\Gamma_{ijk}. \] Moreover \((\nabla^*)^*=\nabla\).
Proof. Apply the defining identity to \(X=\partial_i,Y=\partial_j,Z=\partial_k\): \[ \partial_i g_{jk}=\Gamma_{ijk}+\Gamma^*_{ikj}. \] Nondegeneracy of \(g\) uniquely determines the upper-index coefficients. Solving the same equation for \(\Gamma\) proves involution.
The Levi--Civita midpoint
Theorem. If \(\nabla\) and \(\nabla^*\) are torsion free, then \[ \nabla^{(0)}=\frac12(\nabla+\nabla^*) \] is the Levi--Civita connection. Their difference \(K=\tfrac12(\nabla-\nabla^*)\) satisfies that \[ C(X,Y,Z)=2g(K_XY,Z) \] is totally symmetric.
Proof. The average is torsion free. Adding the two duality identities with \(Y,Z\) interchanged shows \(\nabla^{(0)}g=0\), so uniqueness identifies the Levi--Civita connection. Subtracting them and using both torsion-free conditions gives symmetry in \(X,Y\); metric duality gives symmetry in \(Y,Z\), hence total symmetry.
Curvature duality
Proposition. Curvature tensors of dual connections satisfy \[ g(R^\nabla(X,Y)Z,W) =-g(Z,R^{\nabla^*}(X,Y)W). \] Therefore \(\nabla\) is flat if and only if \(\nabla^*\) is flat.
This identity follows by differentiating the duality relation twice and antisymmetrizing in \(X,Y\). It explains why dual flatness is a paired phenomenon rather than two unrelated assumptions.
Dual parallel transport
If \(V\) is \(\nabla\)-parallel and \(W\) is \(\nabla^*\)-parallel along the same curve, then \[ \frac{\mathrm{d}}{\mathrm{d}t}g(V,W)=0. \] Neither transport need preserve lengths separately; what is preserved is their dual pairing. This is the transport-level counterpart of primal and dual affine coordinates.