Hesse Manifolds: Optimization, Geometry & Cosmology

• Hesse manifold is a term covering three principal definitions: weak-Riemannian manifolds with metric sprays for optimization, affine Hessian manifolds in differential geometry, and Riemannian manifolds defined via the Hesse equation for cosmology.
• In optimization, a weak-Riemannian Hesse manifold provides a metric spray that enables unique covariant derivatives and a second-order Taylor framework despite the limitations of infinite-dimensional settings.
• In affine differential and information geometry, Hessian manifolds feature flat, torsion-free connections with locally potential-derived metrics, while the Hesse-equation approach reveals hidden symmetries in multifield cosmological models.

“Hesse manifold” is not a single universally fixed term. In recent infinite-dimensional optimization, it denotes a weak Riemannian $C^\infty$-manifold admitting a metric spray, so that a Levi–Civita connection and a Riemannian Hessian become available even when the manifold is modeled on locally convex, possibly non-Banach spaces (Zalbertus et al., 26 Mar 2026). In affine differential and information geometry, the same label is often used interchangeably with “Hessian manifold,” meaning an affine manifold $(M,\nabla)$ with $\nabla$ flat and torsion-free and a metric locally of the form $g=\nabla(df)$ (Liu, 1 Sep 2025). A third usage, developed in the study of hidden symmetries of multifield cosmological models, calls a Riemannian manifold $(M,G)$ a Hesse manifold when it admits nontrivial solutions of the PDE $\mathrm{Hess}_G(A)=GA$ (Lazaroiu, 2020). These notions are explicitly distinguished in the literature and are not equivalent.

1. Terminological scope

The present literature supports three principal meanings.

Usage	Defining condition	Typical context
Weak-Riemannian Hesse manifold	$(M,g)$ admits a metric spray $S_g$	Optimization on weak Riemannian manifolds
Affine Hesse/Hessian manifold	$(M,\nabla)$ is affine and $g=\nabla(df)$ locally	Affine differential geometry, information geometry
Hesse-equation manifold	$(M,\nabla)$0 has nontrivial solutions	Hyperbolic geometry, cosmology

The ambiguity is substantive rather than stylistic. The weak-Riemannian notion is designed to recover second-order optimization on manifolds where the musical morphism may fail to be surjective and where Levi–Civita connections need not exist a priori. The affine notion fixes a flat torsion-free connection and studies metrics that are Hessians of local potentials. The Hesse-equation notion does not assume an affine flat structure at all; it is instead defined by the existence of special eigenfunctions of the Riemannian Hessian.

Two explicit warnings recur in the literature. The optimization-theoretic Hesse manifold is stated to be unrelated to the classical Hessian manifold of information geometry (Zalbertus et al., 26 Mar 2026). Conversely, the cosmological Hesse manifold is stated not to be confused with a Hessian manifold, because the defining equation $(M,\nabla)$1 does not assert that the metric itself is Hessian in the affine sense (Lazaroiu, 2020).

2. Weak-Riemannian Hesse manifolds in optimization

A weak Riemannian manifold in the sense of Bastiani calculus is a $(M,\nabla)$2-manifold modeled on locally convex spaces, equipped with a smooth positive-definite bilinear form $(M,\nabla)$3 whose induced norm need not reproduce the model topology. This weak setting creates three obstructions: the fiberwise map $(M,\nabla)$4 may fail to surject onto $(M,\nabla)$5, so the Riemannian gradient of a functional may not exist intrinsically; there is no natural Hilbert completion compatible with the manifold topology; and Levi–Civita connections or metric sprays may fail to exist. Definition 3.4 of “Optimization on Weak Riemannian Manifolds” therefore calls a weak Riemannian $(M,\nabla)$6-manifold $(M,\nabla)$7 a Hesse manifold precisely when it admits a metric spray $(M,\nabla)$8 (Zalbertus et al., 26 Mar 2026).

The existence of a metric spray is the minimal additional structure needed to recover second-order geometry. On a Hesse manifold, for every smooth curve $(M,\nabla)$9, there is a unique covariant derivative operator

$\nabla$0

satisfying linearity, Leibniz, chain, and metric product rules. If $\nabla$1, then

$\nabla$2

and

$\nabla$3

This yields the second-order Taylor formula

$\nabla$4

for $\nabla$5 and $\nabla$6 (Zalbertus et al., 26 Mar 2026).

The optimization theory differs sharply from finite-dimensional or Hilbert-manifold practice. First-order optimality retains the familiar condition $\nabla$7 for all $\nabla$8, or equivalently $\nabla$9 whenever the gradient exists. Second-order criticality requires $g=\nabla(df)$0 and positive semidefiniteness of $g=\nabla(df)$1, but local minimality requires more: positive definiteness is not sufficient in infinite dimensions, and coercivity

$g=\nabla(df)$2

for some $g=\nabla(df)$3 is the condition used to deduce strict local minimality. The paper also establishes the structural hierarchy

$g=\nabla(df)$4

This places Hesse manifolds strictly between arbitrary weak manifolds and settings where full Hilbert-space machinery is available (Zalbertus et al., 26 Mar 2026).

The principal examples come from shape analysis and shape optimization. They include $g=\nabla(df)$5 with the invariant $g=\nabla(df)$6 metric

$g=\nabla(df)$7

mapping spaces $g=\nabla(df)$8 with $g=\nabla(df)$9 metrics induced from $(M,G)$0, and right-invariant weak metrics on regular Lie groups such as diffeomorphism groups used in LDDMM-type registration. The same framework also explains algorithmic preferences: on weak manifolds, retractions may fail to be locally surjective, so gradient descent is formulated with normalized local additions rather than retractions (Zalbertus et al., 26 Mar 2026).

3. Affine Hesse or Hessian manifolds

In the classical affine-geometric sense, an affine manifold is a pair $(M,G)$1 where $(M,G)$2 is flat and torsion-free, equivalently $(M,G)$3 and $(M,G)$4. In affine local coordinates $(M,G)$5, the condition $(M,G)$6 holds. A Riemannian metric $(M,G)$7 on $(M,G)$8 is Hessian if locally there exists a smooth potential $(M,G)$9 such that

$\mathrm{Hess}_G(A)=GA$0

A Hessian manifold is then a triple $\mathrm{Hess}_G(A)=GA$1 (Liu, 1 Sep 2025).

A distinguished subclass is formed by Hessian manifolds of Koszul type. These are manifolds for which there exists a $\mathrm{Hess}_G(A)=GA$2-form $\mathrm{Hess}_G(A)=GA$3 with

$\mathrm{Hess}_G(A)=GA$4

Because $\mathrm{Hess}_G(A)=GA$5 is torsion-free, symmetry of $\mathrm{Hess}_G(A)=GA$6 forces $\mathrm{Hess}_G(A)=GA$7 to be closed. The recent topological study develops a flat-line-bundle formulation of this structure. The canonical flat line bundle is

$\mathrm{Hess}_G(A)=GA$8

and the obstruction bundle $\mathrm{Hess}_G(A)=GA$9 is constructed from local Hessian potentials $(M,g)$0 via transition functions $(M,g)$1. Triviality of $(M,g)$2 is equivalent to the existence of a global potential $(M,g)$3 with $(M,g)$4. For Koszul type, positive flat line bundles, space-like Lagrangian immersions into $(M,g)$5, and hyperbolic affine structures are proved equivalent (Liu, 1 Sep 2025).

The same work identifies a Legendre-type duality with radiant manifolds. If $(M,g)$6 is of Koszul type with $(M,g)$7-form potential $(M,g)$8, then the conjugate connection $(M,g)$9 makes $S_g$0 radiant, and $S_g$1 becomes an Euler vector field. Conversely, if $S_g$2 is radiant with Euler field $S_g$3, then $S_g$4 is of Koszul type with $S_g$5-form potential $S_g$6, satisfying $S_g$7. In Euclidean affine coordinates this reproduces the classical Legendre transform picture (Liu, 1 Sep 2025).

The curvature of a Hessian manifold is encoded by the totally symmetric Amari–Chentsov tensor

$S_g$8

The paper derives the identity

$S_g$9

and corresponding Ricci bounds

$(M,\nabla)$0

This places Hessian geometry simultaneously in affine differential geometry and information geometry, while retaining a genuinely Riemannian curvature theory (Liu, 1 Sep 2025).

4. Topology and cohomology of compact affine Hessian manifolds

Compact Hessian manifolds satisfy strong global restrictions. The Euler characteristic vanishes, $(M,\nabla)$1 is infinite and torsion-free, and if $(M,\nabla)$2 is abelian then $(M,\nabla)$3 is homeomorphic to a flat torus $(M,\nabla)$4. Compact Hessian manifolds are aspherical, all rational Pontryagin classes vanish, and for compact orientable manifolds the curved case forces mapping-torus structure: either $(M,\nabla)$5 is flat, or $(M,\nabla)$6 is a mapping torus. More specifically, if $(M,\nabla)$7 for an orientable compact Hessian manifold, then $(M,\nabla)$8 is flat and $(M,\nabla)$9 is the Levi–Civita connection of $g=\nabla(df)$0. For compact orientable curved Hessian manifolds, the universal cover splits as $g=\nabla(df)$1 (Liu, 1 Sep 2025).

In dimension three, the classification becomes explicit. Every compact, orientable Hessian $g=\nabla(df)$2-manifold is either the Hantzsche–Wendt manifold or admits the structure of a Kähler mapping torus. The refined list given in the three-dimensional classification includes the Hantzsche–Wendt manifold, the $g=\nabla(df)$3-torus $g=\nabla(df)$4, $g=\nabla(df)$5, three Euclidean Seifert fibered spaces

$g=\nabla(df)$6

and quotients $g=\nabla(df)$7 with $g=\nabla(df)$8 discrete, free, and properly discontinuous (Gnandi, 23 Oct 2025).

A complementary cohomological viewpoint is supplied by Dolbeault–Koszul cohomology for flat affine manifolds. For a Hessian manifold $g=\nabla(df)$9, the metric determines a class

$(M,\nabla)$00

and metrics in the same class differ by $(M,\nabla)$01 for a closed $(M,\nabla)$02-form $(M,\nabla)$03. The Künneth formula

$(M,\nabla)$04

holds for flat affine manifolds $(M,\nabla)$05 when at least one factor is compact. Applied to Hessian metrics on products, this yields the structural statement

$(M,\nabla)$06

with $(M,\nabla)$07 and $(M,\nabla)$08 Hessian metrics on the factors and $(M,\nabla)$09 a closed $(M,\nabla)$10-form on $(M,\nabla)$11. If one factor is compact hyperbolic flat affine, then $(M,\nabla)$12 on that factor, so all Hessian metrics on the product arise from the other factor up to a $(M,\nabla)$13 term (Osipov, 11 May 2026).

5. Hesse manifolds defined by the Hesse equation

Another line of work defines a Hesse manifold analytically rather than affinely. Let $(M,\nabla)$14 be a Riemannian manifold of positive dimension $(M,\nabla)$15. A Hesse function is a smooth function $(M,\nabla)$16 satisfying

$(M,\nabla)$17

The space of solutions is denoted $(M,\nabla)$18, and the Hesse index is

$(M,\nabla)$19

By definition, $(M,\nabla)$20 is a Hesse manifold when $(M,\nabla)$21 (Lazaroiu, 2020).

This definition leads to a rigid linear theory. The index satisfies

$(M,\nabla)$22

For $(M,\nabla)$23, the extended Hesse pairing is

$(M,\nabla)$24

When $(M,\nabla)$25, the quantity $(M,\nabla)$26 is constant, so the pairing restricts to a symmetric bilinear form on $(M,\nabla)$27. Any Hesse function satisfies the Helmholtz and Monge–Ampère consequences

$(M,\nabla)$28

A basic corollary is that Hesse manifolds are non-compact (Lazaroiu, 2020).

The complete maximally Hesse case is characterized exactly. A complete Hesse manifold with $(M,\nabla)$29 is isometric to the Poincaré ball $(M,\nabla)$30, whose metric is

$(M,\nabla)$31

Moreover,

$(M,\nabla)$32

is canonically identified with Minkowski space $(M,\nabla)$33 via the Weierstrass map, and the Hesse pairing has Lorentzian signature $(M,\nabla)$34. Local maximality also has a geometric meaning: a Riemannian manifold is locally maximally Hesse if and only if it is hyperbolic. In dimension two, complete Hesse surfaces are exactly the elementary hyperbolic surfaces: the Poincaré disk, the hyperbolic punctured disk, and hyperbolic annuli (Lazaroiu, 2020).

The individual Hesse functions admit a distance-theoretic description. Writing $(M,\nabla)$35, one distinguishes timelike, spacelike, and lightlike Hesse functions. If $(M,\nabla)$36 is the corresponding characteristic subset and $(M,\nabla)$37 the signed distance to $(M,\nabla)$38, then

$(M,\nabla)$39

for $(M,\nabla)$40,

$(M,\nabla)$41

for $(M,\nabla)$42, and

$(M,\nabla)$43

for $(M,\nabla)$44. In multifield cosmology these functions generate the hidden, or Hessian, Noether symmetries of the minisuperspace Lagrangian; the characteristic system includes both $(M,\nabla)$45 and the scalar-potential constraint $(M,\nabla)$46 (Lazaroiu, 2020).

6. Flows, selfsimilarity, and related constructions

Within affine Hessian geometry, one major development is the Hesse–Koszul flow on compact Hessian manifolds. In the normalization used in the convergence theory, the untwisted flow is

$(M,\nabla)$47

while the normalized flow for negative first affine Chern class is

$(M,\nabla)$48

If $(M,\nabla)$49, the normalized flow exists for all $(M,\nabla)$50 and converges in $(M,\nabla)$51 to the unique Hesse–Einstein metric $(M,\nabla)$52 satisfying

$(M,\nabla)$53

More generally, for any $(M,\nabla)$54, the twisted flow

$(M,\nabla)$55

exists for all time and converges to the unique metric solving $(M,\nabla)$56 (Puechmorel et al., 2020).

The corresponding self-similar solutions are Hesse solitons. On a Hessian manifold, a Hesse soliton satisfies

$(M,\nabla)$57

and in the gradient case

$(M,\nabla)$58

The compact theory is highly restrictive: there are no compact shrinking Hesse solitons, compact steady Hesse solitons are trivial and non-proper, and any non-trivial compact gradient Hesse soliton is proper. Duality is built into the theory: if $(M,\nabla)$59 is a Hesse soliton, then the dual Hessian structure $(M,\nabla)$60 is again a Hesse soliton, and the dual of a Hesse–Einstein manifold is a Hesse soliton generated by $(M,\nabla)$61 (Maeta, 2021).

A related but distinct selfsimilarity theory studies Hessian manifolds $(M,\nabla)$62 endowed with a homothetic vector field $(M,\nabla)$63. If the homothety is potential, then the manifold is locally isomorphic to a direct product of radiant selfsimilar Hessian manifolds. On a radiant factor with $(M,\nabla)$64 and $(M,\nabla)$65, there is a canonical Hessian potential

$(M,\nabla)$66

At $(M,\nabla)$67, positivity fails, which is the “extensive” case emphasized in thermodynamic geometry (Osipov, 2019).

This extensive case is developed separately through degenerate Hessian structures on radiant manifolds. If $(M,\nabla)$68 is radiant and $(M,\nabla)$69 with $(M,\nabla)$70 extensive, then $(M,\nabla)$71 is necessarily degenerate and $(M,\nabla)$72. That result is interpreted as the geometric form of the Gibbs–Duhem relation and motivates the use of embedded Hessian submanifolds as the non-degenerate arenas for Ruppeiner-type thermodynamic geometry (García-Ariza, 2015).

Finally, in special Kähler geometry and its deformations, the Hesse potential provides a real, symplectically covariant formulation in special real coordinates $(M,\nabla)$73, with

$(M,\nabla)$74

This realizes the scalar manifold as a Hesse or Hessian manifold in the affine sense and underlies both the local $(M,\nabla)$75-map and the Hessian reinterpretation of the holomorphic anomaly equation (Mohaupt et al., 2011, Cardoso et al., 2015).