Dually Flat Manifold

Updated 10 April 2026

Dually flat manifolds are smooth manifolds with a Hessian metric and two dual, flat affine connections that enable tractable analysis of geodesics and divergences.
They underpin information geometry by providing exponential and mixture coordinate systems through Legendre duality, simplifying the study of exponential and mixture families.
The canonical Bregman divergence derived from convex potentials supports generalized Pythagorean theorems, optimal projection properties, and efficient optimization in statistical models.

A dually flat manifold is a smooth manifold equipped with a Riemannian metric and two flat, torsion-free affine connections that are dual with respect to the metric. This structure underpins much of information geometry, providing the background for the geometry of exponential and mixture families, and supports canonical divergences satisfying generalized Pythagorean theorems. Dually flat manifolds admit coordinate systems and potentials that allow for tractable analysis of geodesics, divergences, and orthogonality, and can be interpreted both in terms of convex analysis and contact geometry. Their structure appears in statistics, statistical mechanics, Finsler geometry, optimization, and symplectic/Kähler geometry via torification.

1. Core Definition and Dual Connections

A dually flat manifold is a quadruple $(M, g, \nabla, \nabla^*)$ , where $M$ is a smooth manifold of dimension $n$ , $g$ is a Riemannian (or pseudo-Riemannian) metric, $\nabla$ is a torsion-free flat affine connection, and $\nabla^*$ is the $g$ -dual connection defined by

$X\,g(Y, Z) = g(\nabla_X Y, Z) + g(Y, \nabla^*_X Z), \quad \forall X,Y,Z \in \Gamma(TM)$

Both $\nabla$ and $\nabla^*$ must be flat (i.e., have vanishing curvature and torsion) (Fujita, 2023, Nielsen et al., 2018, Molitor, 2021, Felice et al., 2018). The Levi-Civita connection need not be flat unless the manifold is Euclidean.

The existence of this structure entails that there are coordinate systems—called $M$ 0-affine (or "exponential," $M$ 1-) and $M$ 2-affine (or "mixture," $M$ 3-) coordinates—that are flat for each connection, respectively. Moreover, in such coordinates, the metric has a Hessian form and is generated via a strictly convex potential. The key transformation linking these coordinates is Legendre duality: $M$ 4 where $M$ 5 and $M$ 6 are Legendre-conjugate convex functions (Fujita, 2023, Nielsen et al., 2018, Felice et al., 2018).

2. Convex Potentials, Affine Coordinates, and Bregman Divergence

On a dually flat manifold, a strictly convex $M$ 7 function $M$ 8 on a simply connected open set $M$ 9 defines the metric and dually flat structure: $n$ 0 The dual coordinate $n$ 1 parametrizes an alternative affine system for the dual connection $n$ 2. The Legendre transform gives a dual potential $n$ 3.

The canonical divergence is the Bregman divergence generated from the potential: $n$ 4 Or, equivalently, in mixed coordinates: $n$ 5 This divergence is central; it is nonnegative, vanishes only for coinciding points, and generates the metric by second derivatives. Geodesics for $n$ 6 and $n$ 7 correspond to straight lines in $n$ 8 and $n$ 9 coordinates, respectively (Fujita, 2023, Nielsen et al., 2018, Nishiyama, 2018, Nielsen, 2019, Felice et al., 2018).

3. Geometric Principles: Generalized Pythagorean Theorem and Projections

The geometry of dually flat spaces generalizes various classical geometric relationships:

Generalized Pythagorean theorem: For points $g$ 0 via $g$ 1-geodesic and $g$ 2 via $g$ 3-geodesic that are $g$ 4-orthogonal at $g$ 5, the canonical divergence satisfies

$g$ 6

This underpins the optimality properties of statistical projections and is tied to the convex structure of the underlying manifold (Fujita, 2023, Felice et al., 2018, Nakajima et al., 2020, Nielsen, 2019, Nishiyama, 2018).

Triangular relation: For any $g$ 7,

$g$ 8

Orthogonality of the connecting geodesics at $g$ 9 recovers the Pythagorean identity (Nishiyama, 2018, Nielsen, 2019).

Projection theorem: The $\nabla$ 0-projection of $\nabla$ 1 onto a submanifold $\nabla$ 2 is the point $\nabla$ 3 where the $\nabla$ 4-geodesic $\nabla$ 5 meets $\nabla$ 6 orthogonally; the canonical divergence is minimized at $\nabla$ 7 (Nakajima et al., 2020).

The theory extends to cases with boundary or singularities, using quasi-Hessian and coherent tangent bundle generalizations, maintaining the essential divergence and projection properties even in degenerate settings (Nakajima et al., 2020).

4. Statistical and Contact Geometric Realizations

Dually flat manifolds arise naturally in statistics:

Exponential families: The natural parameterization ( $\nabla$ 8) yields a $\nabla$ 9-flat geometry. The log-partition function is the potential generating the Fisher metric, and expectation coordinates are the dual ( $\nabla^*$ 0-affine) system. The Kullback-Leibler divergence is the canonical Bregman divergence (Nielsen et al., 2018, Nielsen, 2021, Omiya et al., 10 Dec 2025).
Mixture families and other parameterized probabilistic models similarly fit into the dual flatness structure, often using negative entropy as the potential.

In contact geometry, a dually flat structure arises from Legendre submanifolds: starting from a contact manifold $\nabla^*$ 1 and a strictly convex function $\nabla^*$ 2, the Legendre submanifold $\nabla^*$ 3 admits a natural dually flat structure, with induced metric and dual flat connections stemming from pullbacks of the contact form (Goto, 2015).

Dually flat structures are thus the geometric substrate for both exponential and mixture connections and divergences in information geometry, unifying statistical and contact/symplectic perspectives.

5. Advanced Structures: Toric Geometry, Kähler Correspondence, and Finsler Extensions

Dually flat geometry is compatible with more advanced symplectic and complex-geometric structures:

Toric correspondence: There is a bijective correspondence between regular toric Kähler manifolds and toric dually flat manifolds. Symplectic potentials on Delzant polytopes define dually flat metrics on open orbits, which compactify to Kähler toric manifolds via torification. Canonical divergences and Pythagorean formulas extend to boundary strata and facets (Fujita, 2023, Molitor, 2021, Figueirêdo et al., 2023).
Finsler generalizations: Dually flatness has been completely classified for general $\nabla^*$ 4-metrics in Finsler geometry, revealing a deep connection with projective flatness and providing explicit ODE/PDE characterizations of the allowable metric functions $\nabla^*$ 5 (Yu, 2013, Yu, 2013).
Twisted and warped product manifolds: Dual flatness can be characterized in higher-dimensional product manifolds under specific curvature and warping conditions, often reducing to combinations where the base manifold is dually flat and the fiber has constant sectional curvature (Diallo et al., 2014).

6. Divergences, Dynamics, and Algorithmic Applications

Several divergences arise canonically in dually flat spaces:

Canonical/Bregman divergence: Encodes the dually flat geometry, generates geodesics, and is critical in projection theorems (Nishiyama, 2018, Felice et al., 2018).
Affine divergence: Reduces to the Jeffreys divergence in probabilistic models (Nishiyama, 2018).
Jensen, Bhattacharyya, Jensen-Shannon: Associated with convex potential combinations, naturally consistent with mixture and exponential families (Nishiyama, 2018, Nielsen, 2021).

The canonical divergence also serves as a Hamilton principal function, allowing for variational and dynamical interpretations:

Gradient flows and geodesic generation: The gradient of the divergence with respect to one argument (time-rescaled) generates $\nabla^*$ 6- and $\nabla^*$ 7-geodesics. The dynamical systems induced by canonical divergence are integrable and coincide with geodesic equations in the appropriate coordinates (Felice et al., 2018).
Optimization: In statistical manifolds (especially exponential families), geodesic-based (e.g., $\nabla^*$ 8-geodesic and $\nabla^*$ 9-geodesic) optimization achieves one-step convergence to maximum-likelihood estimators or enables unconstrained optimization, leveraging the dually flat structure. This gives geometric backing to mirror descent, exponentiated gradient, and natural gradient algorithms (Omiya et al., 10 Dec 2025).

Practical computation of dually flat structure in high-dimensional spaces with intractable potentials is addressed by Monte Carlo Information Geometry, which replaces integrals by sample-based estimators of the potential. These estimators preserve the strict convexity needed for Bregman generators and thus maintain the dually flat structure almost surely, enabling efficient geometric algorithms (Nielsen et al., 2018).

7. Special Models, Classification, and Boundary Cases

Dually flat manifolds encompass important cases in statistics and geometry:

Simplex and categorical models: The simplex with KL divergence is a prototypical dually flat manifold, directly related to the Fubini-Study metric on $g$ 0 (Fujita, 2023, Molitor, 2021).
Escort probability geometry: By a projective conformal flattening of $g$ 1-geometry, the probability simplex with escort probabilities as affine coordinates admits a dually flat structure where the $g$ 2-divergence becomes the canonical Bregman divergence (Ohara et al., 2010).
Singular and quasi-Hessian structures: In statistical models with degenerate Fisher metrics or boundary singularities (e.g., mixture models on the simplex), the dually flat framework generalizes to quasi-Hessian manifolds, preserving key geometric properties (theorems on divergence and projection) via gluing and coherent tangent bundles (Nakajima et al., 2020).
1D toric models and complex space forms: The classification of toric dually flat manifolds in dimension one connects these objects to complex space forms (e.g., projective line, complex plane, unit disk) and yields explicit curvature and metric expressions (Figueirêdo et al., 2023).