Dually flat manifolds are the cleanest setting in information geometry: two dual affine coordinate systems, a Hessian metric, and a canonical divergence all fit together.
Dual affine coordinates
Flatness gives coordinate systems in which the two dual connections have zero coefficients.
Definition. A statistical manifold \((M,g,\nabla,\nabla^*)\) is dually flat if \(\nabla\) and \(\nabla^*\) are flat, torsion-free, and dual with respect to \(g\). Coordinates \(\theta\) affine for \(\nabla\) and \(\eta\) affine for \(\nabla^*\) are dual affine coordinates.
Remark. Exponential-family natural parameters and expectation parameters are the main example.
Hessian metric
In affine coordinates, the metric is the Hessian of a convex potential.
Theorem. On a dually flat manifold, locally there is a potential \(\psi(\theta)\) such that \[ g_{ij}=\partial_i\partial_j\psi(\theta), \qquad \eta_i=\partial_i\psi(\theta). \]
Proof. In \(\nabla\)-affine coordinates, dual flatness and metric duality imply that the metric tensor has symmetric first derivatives: \(\partial_kg_{ij}=\partial_ig_{kj}\). The Poincar? lemma gives a local potential whose Hessian is \(g\). Defining \(\eta_i=\partial_i\psi\) yields dual affine coordinates.
Canonical divergence
The convex potential produces the divergence compatible with the dual geometry.
Definition. The canonical divergence in dual coordinates is \[ D(p,q)=\psi(\theta(p))+\psi^*(\eta(q))-\theta^i(p)\eta_i(q), \] where \(\psi^*\) is the Legendre dual of \(\psi\).
Proposition. This canonical divergence is the Bregman divergence generated by \(\psi\) in \(\theta\)-coordinates.
Proof. Since \(\psi^*(\eta(q))=\theta^i(q)\eta_i(q)-\psi(\theta(q))\), substitution gives \[ D(p,q)=\psi(\theta_p)-\psi(\theta_q)-\langle\nabla\psi(\theta_q),\theta_p-\theta_q\rangle. \]
Flatness and projections
Projection theorems become exact because primal and dual geodesics are affine lines.
Remark. This is the setting where generalized Pythagorean theorems, alternating projections, and maximum-entropy characterizations are most transparent.
Equivalent local descriptions
Theorem. For a simply connected statistical manifold \((\mathcal M,g,\nabla,\nabla^*)\) with torsion-free dual connections, the following local descriptions are equivalent:
- \(\nabla\) and \(\nabla^*\) are flat;
- there are \(\nabla\)-affine coordinates \(\theta\) and a strictly convex potential \(\psi\) such that \(g_{ij}=\partial_i\partial_j\psi\);
- there are dual coordinates \(\eta_i=\partial_i\psi(\theta)\) satisfying \(g(\partial/\partial\theta^i,\partial/\partial\eta_j)=\delta_i^j\).
Proof. Flatness and torsion freeness of \(\nabla\) provide affine coordinates with \(\Gamma_{ij}^k=0\). Duality and torsion freeness of \(\nabla^*\) imply \(\partial_k g_{ij}=\partial_i g_{kj}\). Poincaré's lemma first gives functions \(\eta_j\) with \(g_{ij}=\partial_i\eta_j\), and symmetry gives a potential \(\psi\) with \(\eta_j=\partial_j\psi\). Positive definiteness makes \(\psi\) strictly convex. Its Legendre coordinates have vanishing dual coefficients. The converse follows by direct differentiation of a Hessian metric.
Primal and dual affine coordinates
Let \[ \eta=\nabla\psi(\theta),\qquad \varphi(\eta)=\sup_\theta\{\theta^\top\eta-\psi(\theta)\}. \] Then \[ \theta=\nabla\varphi(\eta),\qquad \nabla_\eta^2\varphi=(\nabla_\theta^2\psi)^{-1}. \] The metric line element can be written in three equivalent forms: \[ \mathrm{d}s^2 =\mathrm{d}\theta^\top\nabla^2\psi(\theta)\,\mathrm{d}\theta =\mathrm{d}\eta^\top\nabla^2\varphi(\eta)\,\mathrm{d}\eta =\mathrm{d}\theta^\top\mathrm{d}\eta. \]
Canonical divergence and orthogonality
The dual affine structure determines \[ \mathcal D(p,q) =\psi(\theta(p))+\varphi(\eta(q)) -\theta(p)^\top\eta(q) =B_\psi(\theta(p),\theta(q)). \] If a primal geodesic from \(q\) to \(p\) and a dual geodesic from \(q\) to \(r\) are orthogonal at \(q\), the three-point identity reduces to \[ \mathcal D(p,r)=\mathcal D(p,q)+\mathcal D(q,r). \] This exact Pythagorean relation is stronger than the infinitesimal Pythagorean theorem of a general Riemannian manifold.
Global qualifications
Flat connections guarantee affine coordinates only locally unless topology and completeness cooperate. A punctured affine domain may have nontrivial holonomy; a convex potential may fail to be steep at its boundary; geodesics may leave the coordinate domain. “Dually flat” is therefore a statement about curvature, while global projection and optimization theorems also require convexity, closedness, and coercivity.