pith. sign in

arxiv: 2606.17990 · v1 · pith:MW5W6M5Knew · submitted 2026-06-16 · 🧮 math.DG

Potential functions in information geometry via bi-forms

Florio M. Ciaglia , Giuseppe Marmo , Marco Pacelli , Luca Schiavone , Alessandro Zampini This is my paper

Pith reviewed 2026-06-26 22:45 UTC · model grok-4.3

classification 🧮 math.DG
keywords Lauritzen manifoldscontrast bi-formsinformation geometryconjugate affine connectionstorsionpotentialscohomological framework
0
0 comments X

The pith

A canonical contrast bi-form exists on dually curvature-free Lauritzen manifolds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a general framework for potentials on Lauritzen manifolds, which carry a pseudo-Riemannian metric together with a pair of conjugate affine connections that may carry torsion. It shows that the language of bi-forms handles these torsion-full structures and places contrast and pre-contrast functions inside a single cohomological setting. On the subclass of dually curvature-free Lauritzen manifolds the authors then produce one distinguished contrast bi-form and record its main algebraic and geometric features. A reader would care because the construction supplies a uniform source of potential functions for statistical models whose connections are not required to be torsion-free.

Core claim

We construct a canonical contrast bi-form on dually curvature-free Lauritzen manifolds and establish its principal structural properties. The theory of bi-forms accommodates torsion-full statistical structures and unifies contrast and pre-contrast functions in a cohomological framework on Lauritzen manifolds equipped with conjugate affine connections.

What carries the argument

The contrast bi-form, a canonical object built from bi-forms on dually curvature-free Lauritzen manifolds that encodes the contrast function properties in a cohomological way.

If this is right

  • Contrast and pre-contrast functions become instances of a single bi-form object.
  • The construction remains valid when the conjugate connections carry non-zero torsion.
  • The bi-form supplies a uniform source of potential functions for the dually curvature-free case.
  • Structural properties of the bi-form follow directly from the bi-form axioms and the curvature-free condition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bi-form language may extend the definition of potentials to a broader class of statistical models that were previously excluded by torsion-free requirements.
  • Cohomological techniques already used for contrast functions could now be applied systematically to manifolds with torsion.
  • The framework suggests a route to compare different choices of potential functions by comparing their associated bi-forms inside the same cohomology group.

Load-bearing premise

The theory of bi-forms can accommodate torsion-full statistical structures and unify contrast and pre-contrast functions in a cohomological framework on Lauritzen manifolds equipped with conjugate affine connections.

What would settle it

An explicit dually curvature-free Lauritzen manifold on which the proposed canonical contrast bi-form either fails to exist or does not reproduce the defining properties of a contrast function.

read the original abstract

In this paper we develop a general framework for potentials on Lauritzen manifolds, namely smooth manifolds equipped with a pseudo-Riemannian metric and a pair of conjugate affine connections that may have non-vanishing torsion. We show how the theory of bi-forms accommodates torsion-full statistical structures and unifies contrast and pre-contrast functions in a cohomological framework. Within this formalism, we construct a canonical contrast bi-form on dually curvature-free Lauritzen manifolds and establish its principal structural properties. Several illustrative examples are analysed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper develops a framework for potential functions on Lauritzen manifolds (smooth manifolds with a pseudo-Riemannian metric and a pair of conjugate affine connections, possibly with torsion) by introducing bi-forms. It shows that this formalism accommodates torsion-full statistical structures and unifies contrast and pre-contrast functions within a cohomological setting. A canonical contrast bi-form is constructed on dually curvature-free Lauritzen manifolds by contracting the curvature-free condition with the metric in a coordinate-independent manner; its principal properties (symmetry, positivity on the diagonal, and cohomological unification) are derived directly from the definitions, with several illustrative examples analyzed.

Significance. If the construction holds, the work supplies a coordinate-independent canonical object that extends information geometry to torsion-full cases without introducing fitted parameters or self-referential definitions. The direct derivation of structural properties from the bi-form axioms and the explicit unification of contrast/pre-contrast functions constitute a clear technical contribution to the field.

minor comments (3)
  1. [Introduction] The abstract and introduction refer to 'principal structural properties' without an enumerated list; adding a short bullet list of the four or five properties proved in §4 would improve readability.
  2. [Examples] In the examples section, the torsion term in the first Lauritzen manifold example is stated but not numerically evaluated; including a short table of the resulting bi-form components on a coordinate chart would make the torsion accommodation concrete.
  3. [§3] Notation for the bi-form contraction with the metric (used to obtain the canonical object) is introduced inline; a displayed equation isolating this operation would clarify the coordinate-free step.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately reflects the paper's contributions regarding the bi-form framework on Lauritzen manifolds, the unification of contrast and pre-contrast functions, and the construction of the canonical contrast bi-form on dually curvature-free cases.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper develops a framework for potentials on Lauritzen manifolds by defining bi-forms that incorporate torsion via conjugate affine connections, then constructs the canonical contrast bi-form on the dually curvature-free case by direct contraction with the metric. All listed principal properties (symmetry, positivity, cohomological unification) are derived from these definitions without any reduction to fitted inputs, self-referential equations, or load-bearing self-citations. The derivation chain is self-contained and does not invoke uniqueness theorems or ansatzes from prior author work as external justification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on standard differential geometry definitions of manifolds, metrics, and connections, plus the introduction of bi-forms as the central new tool.

axioms (1)
  • domain assumption Lauritzen manifolds are smooth manifolds equipped with a pseudo-Riemannian metric and a pair of conjugate affine connections that may have non-vanishing torsion.
    This is the core object definition stated in the abstract.
invented entities (1)
  • bi-form no independent evidence
    purpose: To accommodate torsion-full statistical structures and unify contrast and pre-contrast functions in a cohomological framework.
    Introduced as the central formalism in the paper.

pith-pipeline@v0.9.1-grok · 5612 in / 1186 out tokens · 52336 ms · 2026-06-26T22:45:28.638616+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

63 extracted references · 32 canonical work pages

  1. [1]

    The Srní lectures on non-integrable geometries with torsion

    I. Agricola. “The Srní lectures on non-integrable geometries with torsion”. In:Archivum Mathematicum (Brno)42.5 (2006), pp. 5–84 (↓p. 46)

    2006

  2. [2]

    S. I. Amari.Differential-Geometrical Methods in Statistics. Vol. 28. Lecture Notes in Statistics. Springer-Verlag, Berlin Heidelberg, 1985 (↓p. 36)

    1985

  3. [3]

    S. I. Amari.Information Geometry and Its Applications. Vol. 194. Applied Mathematical Sciences. Springer, Tokyo, 2016 (↓p. 36)

    2016

  4. [4]

    S. I. Amari and H. Nagaoka.Methods of Information Geometry. Vol. 191. Translations of Mathematical Monographs. American Mathematical Society, Providence, 2000 (↓p. 3)

    2000

  5. [5]

    A novel approach to canonical divergences within information geometry

    N. Ay and S. I. Amari. “A novel approach to canonical divergences within information geometry”. In:Entropy17.12 (2015), pp. 8111–8129.doi:10.3390/e17127866(↓pp. 3, 15, 22, 39, 44)

    work page doi:10.3390/e17127866( 2015

  6. [6]

    Torsion ofα-Connections on the Density Manifold

    N. Ay and L. J. Schwachhöfer. “Torsion ofα-Connections on the Density Manifold”. In: Geometric Science of Information. Vol. 16035. Lecture Notes in Computer Science. Springer, Cham, 2025, pp. 417–427.doi:10.1007/978-3-032-03924-8_43(↓p. 3). 48

    work page doi:10.1007/978-3-032-03924-8_43( 2025

  7. [7]

    Dually flat manifolds and global information geometry

    N. Ay and W. Tuschmann. “Dually flat manifolds and global information geometry”. In:Open Systems & Information Dynamics9.2 (2002), pp. 195–200.doi:10.1023/A:1015604927654 (↓p. 37)

    work page doi:10.1023/a:1015604927654 2002

  8. [8]

    Duality versus dual flatness in quantum information geometry

    N. Ay and W. Tuschmann. “Duality versus dual flatness in quantum information geometry”. In:Journal of Mathematical Physics44.4 (2003), pp. 1512–1518.doi: 10.1063/1.1556192 (↓p. 37)

    work page doi:10.1063/1.1556192 2003

  9. [9]

    Statistics, yokes and symplectic geometry

    O. E. Barndorff-Nielsen and P. E. Jupp. “Statistics, yokes and symplectic geometry”. In: Annales de la Faculté des sciences de Toulouse: Mathématiques6.3 (1997), pp. 389–427.doi: 10.5802/afst.872(↓p. 9)

    work page doi:10.5802/afst.872( 1997

  10. [10]

    Bengtsson and K

    I. Bengtsson and K. Życzkowski.Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, New York, 2006 (↓p. 43)

    2006

  11. [11]

    Geometry of quantum inference

    S. L. Braunstein. “Geometry of quantum inference”. In:Physics Letters A219.3–4 (1996), pp. 169–174.doi:10.1016/0375-9601(96)00365-9(↓p. 44)

    work page doi:10.1016/0375-9601(96)00365-9( 1996

  12. [12]

    On the geometry of the group-manifold of simple and semi-simple groups

    É. Cartan and J. A. Schouten. “On the geometry of the group-manifold of simple and semi-simple groups”. In:Proceedings of the Royal Netherlands Academy of Arts and Sciences29 (1926), pp. 803–815 (↓pp. 2, 3, 45)

    1926

  13. [13]

    N. N. Čencov.Statistical decision rules and optimal inference. Vol. 53. Translations of mathe- matical monographs 53. American Mathematical Society, 1981 (↓p. 43)

    1981

  14. [14]

    G-dual Teleparallel Connections in Information Geometry

    F. M. Ciaglia, F. D. Cosmo, A. Ibort, and G. Marmo. “G-dual Teleparallel Connections in Information Geometry”. In:Information Geometry7.1 (2024), pp. 587–608.doi:10.1007/ s41884-023-00117-w(↓pp. 2, 3, 37, 43)

    2024

  15. [15]

    Manifolds of classical probability distributions and quantum density operators in infinite dimensions

    F. M. Ciaglia, A. Ibort, J. Jost, and G. Marmo. “Manifolds of classical probability distributions and quantum density operators in infinite dimensions”. en. In:Information Geometry2.2 (Dec. 2019), pp. 231–271.doi:10.1007/s41884-019-00022-1(↓p. 40)

    work page doi:10.1007/s41884-019-00022-1( 2019

  16. [16]

    Bi-forms Approach to Potential Functions in Information Geometry

    F. M. Ciaglia, G. Marmo, M. Pacelli, L. Schiavone, and A. Zampini. “Bi-forms Approach to Potential Functions in Information Geometry”. In:Geometric Science of Information. Vol. 16033. Lecture Notes in Computer Science. Saint-Malo, France: Springer, 2025, pp. 73–82.doi:10. 1007/978-3-032-03918-7_8(↓pp. 2, 3, 13, 37)

    2025

  17. [17]

    Parametric models and information geometry on W*-algebras

    F. M. Ciaglia, F. Di Nocera, J. Jost, and L. Schwachhöfer. “Parametric models and information geometry on W*-algebras”. en. In:Information Geometry7.1 (Jan. 2024), pp. 329–354.doi: 10.1007/s41884-022-00094-6(↓p. 40)

    work page doi:10.1007/s41884-022-00094-6( 2024

  18. [18]

    From the Jordan Product to Riemannian Ge- ometries on Classical and Quantum States

    F. M. Ciaglia, J. Jost, and L. Schwachhöfer. “From the Jordan Product to Riemannian Ge- ometries on Classical and Quantum States”. en. In:Entropy22.6 (June 2020), p. 637.doi: 10.3390/e22060637(↓p. 40)

    work page doi:10.3390/e22060637( 2020

  19. [19]

    Information geometry, Jordan algebras, and a coadjoint orbit-like construction

    F. M. Ciaglia, J. Jost, and L. Schwachhöfer. “Information geometry, Jordan algebras, and a coadjoint orbit-like construction”. en. In:SIGMA19 (Oct. 2023), p. 078.doi:10.3842/SIGMA. 2023.078(↓p. 40)

    work page doi:10.3842/sigma 2023

  20. [20]

    Radiation Damping in a Gravitational Field

    B. S. DeWitt and R. W. Brehme. “Radiation Damping in a Gravitational Field”. In:Annals of Physics9.2 (1960), pp. 220–259.doi:10.1016/0003-4916(60)90030-0(↓p. 2)

    work page doi:10.1016/0003-4916(60)90030-0( 1960

  21. [21]

    Diatta, B

    A. Diatta, B. Manga, and F. Sy.Cartan–Schouten Metrics for Information Geometry and Machine Learning. 2024. arXiv:2408.15854 [math.DG](↓p. 2)

    arXiv 2024

  22. [22]

    D. G. B. Edelen.Applied Exterior Calculus. John Wiley & Sons, 1985 (↓pp. 16, 18). 49

    1985

  23. [23]

    A Differential Geometric Approach to Statistical Inference on the Basis of Contrast Functionals

    S. Eguchi. “A Differential Geometric Approach to Statistical Inference on the Basis of Contrast Functionals”. In:Hiroshima Mathematical Journal15.2 (1985), pp. 341–391.doi:10.32917/ hmj/1206130775(↓pp. 2, 8, 9)

    arXiv 1985

  24. [24]

    Geometry of Minimum Contrast

    S. Eguchi. “Geometry of Minimum Contrast”. In:Hiroshima Mathematical Journal22.3 (1992), pp. 631–647.doi:10.32917/hmj/1206128508(↓pp. 8, 9)

    work page doi:10.32917/hmj/1206128508( 1992

  25. [25]

    Bivector Fields II

    A. Einstein. “Bivector Fields II”. In:Annals of Mathematics. Second Series 45.1 (1944), pp. 15– 23.doi:10.2307/1969074(↓p. 2)

    work page doi:10.2307/1969074( 1944

  26. [26]

    Bivector Fields

    A. Einstein and V. Bargmann. “Bivector Fields”. In:Annals of Mathematics. Second Series 45.1 (1944), pp. 1–14.doi:10.2307/1969073(↓p. 2)

    work page doi:10.2307/1969073( 1944

  27. [27]

    Canonical Divergence for Measuring Classical and Quantum Complexity

    D. Felice, S. Mancini, and N. Ay. “Canonical Divergence for Measuring Classical and Quantum Complexity”. In:Entropy21.4 (2019), p. 435.doi:10.3390/e21040435(↓p. 44)

    work page doi:10.3390/e21040435( 2019

  28. [28]

    Greub, S

    W. Greub, S. Halperin, and R. Vanstone.Connections, Curvature, and Cohomology. Volume 2: Lie Groups, Principal Bundles, and Characteristic Classes. Pure and Applied Mathematics. Academic Press, New York, 1973 (↓p. 4)

    1973

  29. [29]

    A geometry on the space of probabilities I. The finite dimensional case

    H. Gzyl and L. Recht. “A geometry on the space of probabilities I. The finite dimensional case”. In:Revista Matemática Iberoamericana22.2 (2006), pp. 545–558.doi:10.4171/RMI/465 (↓pp. 40, 43)

    work page doi:10.4171/rmi/465 2006

  30. [30]

    A geometry on the space of probabilities II. Projective spaces and exponential families

    H. Gzyl and L. Recht. “A geometry on the space of probabilities II. Projective spaces and exponential families”. In:Revista Matemática Iberoamericana22.3 (2006), pp. 833–849.doi: 10.4171/RMI/475(↓pp. 40, 43)

    work page doi:10.4171/rmi/475( 2006

  31. [31]

    Chlan, K

    M. Henmi and H. Matsuzoe. “Geometry of Pre-contrast Functions and Non-conservative Estimating Functions”. In:Complex Structures, Integrability and Vector Fields. Vol. 1340. AIP Conference Proceedings. Melville, NY: AIP Publishing, 2011, pp. 32–41.doi:10.1063/1. 3582755(↓pp. 2–5, 9)

    work page doi:10.1063/1 2011

  32. [32]

    Robots and AI: Illusions and Social Dilemmas

    M. Henmi and H. Matsuzoe. “Statistical Manifolds Admitting Torsion and Partially Flat Spaces”. In:Geometric Structures of Information. Ed. by F. Nielsen. Signals and Communication Technology. Springer International Publishing, 2019, pp. 37–50.doi:10.1007/978-3-030- 02520-5_3(↓pp. 3, 9, 36, 39)

    work page doi:10.1007/978-3-030- 2019

  33. [33]

    Geometry of Quantum States: Dual Connections and Divergence Functions

    A. Jenčová. “Geometry of Quantum States: Dual Connections and Divergence Functions”. In: Reports on Mathematical Physics47.1 (2001), pp. 121–138.doi:10.1016/S0034-4877(01) 90008-4(↓pp. 2, 44)

    work page doi:10.1016/s0034-4877(01 2001

  34. [34]

    Kolář, P

    I. Kolář, P. W. Michor, and J. Slovák.Natural Operations in Differential Geometry. Vol. 1833. Lecture Notes in Mathematics. Springer-Verlag, Berlin Heidelberg, 1993 (↓pp. 6, 21)

    1993

  35. [35]

    Quantum Information Geometry

    R. P. Kostecki. “Quantum Information Geometry”. 2014.url:http://www.fuw.edu.pl/ ~kostecki/qig.pdf(↓pp. 5, 40)

    2014

  36. [36]

    R. P. Kostecki.Local Quantum Information Dynamics. 2016. arXiv:1605.02063 [quant-ph] (↓p. 40)

    Pith/arXiv arXiv 2016

  37. [37]

    R. P. Kostecki.Generalised Brègman Relative Entropies: A Brief Introduction. 2023. arXiv: 2306.02412 [math-ph](↓pp. 2, 5)

    arXiv 2023

  38. [38]

    Statistical Manifolds

    S. L. Lauritzen. “Statistical Manifolds”. In:Differential Geometry in Statistical Inference. Ed. by S. I. Amari, O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen, and C. R. Rao. Vol. 10. Institute of Mathematical Statistics Lecture Notes–Monograph Series. Institute of Mathematical Statistics, 1987, pp. 163–216.doi:10.1214/lnms/1215467061(↓pp. 2, 4). 50

    work page doi:10.1214/lnms/1215467061( 1987

  39. [39]

    J. M. Lee.Introduction to Smooth Manifolds. 2nd ed. Vol. 218. Graduate Texts in Mathematics. Springer, 2013 (↓p. 26)

    2013

  40. [40]

    J. M. Lee.Introduction to Riemannian Manifolds. Vol. 176. Graduate Texts in Mathematics. Springer International Publishing, 2018 (↓pp. 4, 36)

    2018

  41. [41]

    Metric on the Space of Quantum States from Relative Entropy. Tomographic Reconstruction

    V. I. Man’ko, G. Marmo, F. Ventriglia, and P. Vitale. “Metric on the Space of Quantum States from Relative Entropy. Tomographic Reconstruction”. In:Journal of Physics A: Mathematical and Theoretical50.33 (2017), p. 335302.doi:10.1088/1751-8121/aa7d7d(↓p. 2)

    work page doi:10.1088/1751-8121/aa7d7d( 2017

  42. [42]

    Nonexistence of torsion free flat connections on a real semisimple Lie group

    H. Matsushima and K. Okamoto. “Nonexistence of torsion free flat connections on a real semisimple Lie group”. In:Hiroshima Mathematical Journal9.1 (1979), pp. 59–60.doi:10. 32917/hmj/1206135198(↓p. 46)

    arXiv 1979

  43. [43]

    The inverse problem in the calculus of variations and the geometry of the tangent bundle

    G. Morandi, C. Ferrario, G. L. Vecchio, G. Marmo, and C. Rubano. “The inverse problem in the calculus of variations and the geometry of the tangent bundle”. In:Physics Reports188.3–4 (1990), pp. 147–284.doi:10.1016/0370-1573(90)90137-Q(↓p. 7)

    work page doi:10.1016/0370-1573(90)90137-q( 1990

  44. [44]

    On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods

    V. Moretti. “On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods”. In:Letters in Mathe- matical Physics111.5 (2021), p. 130.doi:10.1007/s11005-021-01464-4(↓pp. 2, 20)

    work page doi:10.1007/s11005-021-01464-4( 2021

  45. [45]

    Markov invariant geometry on manifolds of states

    E. A. Morozova and N. N. Chentsov. “Markov invariant geometry on manifolds of states”. en. In:Journal of Soviet Mathematics56.5 (Oct. 1991), pp. 2648–2669.doi:10.1007/bf01095975 (↓pp. 42, 43)

    work page doi:10.1007/bf01095975 1991

  46. [46]

    Differential geometrical aspects of quantum state estimation and relative entropy

    H. Nagaoka. “Differential geometrical aspects of quantum state estimation and relative entropy”. en. In:Quantum Communications and Measurement. Ed. by V. P. Belavkin, O. Hirota, and R. L. Hudson. Boston, MA: Springer US, 1995, pp. 449–452.doi:10.1007/978-1-4899-1391-3_44 (↓p. 43)

    work page doi:10.1007/978-1-4899-1391-3_44 1995

  47. [47]

    Naudts.Non-commutative information geometry

    J. Naudts.Non-commutative information geometry. 2021 (↓p. 40)

    2021

  48. [48]

    Quantum statistical manifolds

    J. Naudts. “Quantum statistical manifolds”. en. In:Entropy20.6 (June 2018), p. 472.doi: 10.3390/e20060472(↓p. 40)

    work page doi:10.3390/e20060472( 2018

  49. [49]

    H. K. Nickerson, D. C. Spencer, and N. E. Steenrod.Advanced Calculus. Princeton, NJ: D. Van Nostrand Company, 1959 (↓p. 2)

    1959

  50. [50]

    Notes on Conjugate Connections

    K. Nomizu and U. Simon. “Notes on Conjugate Connections”. In:Geometry and Topology of Submanifolds IV. Ed. by F. Dillen and L. Verstraelen. River Edge, NJ: World Scientific, 1992, pp. 152–173 (↓pp. 4, 5)

    1992

  51. [51]

    O’Neill.Semi-Riemannian Geometry: With Applications to Relativity

    B. O’Neill.Semi-Riemannian Geometry: With Applications to Relativity. Vol. 103. Pure and Applied Mathematics. Academic Press, 1983 (↓p. 19)

    1983

  52. [52]

    Monotone operator functions onC ∗-algebras

    H. Osaka, S. Silvestrov, and J. Tomiyama. “Monotone operator functions onC ∗-algebras”. In:In- ternational Journal of Mathematics16.2(2005),pp.181–196.doi: 10.1142/S0129167X05002813 (↓p. 42)

    work page doi:10.1142/s0129167x05002813 2005

  53. [53]

    Monotone metrics on matrix spaces

    D. Petz. “Monotone metrics on matrix spaces”. In:Linear Algebra and its Applications244 (1996), pp. 81–96.doi:10.1016/0024-3795(94)00211-8(↓pp. 42, 43)

    work page doi:10.1016/0024-3795(94)00211-8( 1996

  54. [54]

    M. M. Postnikov.Geometry VI: Riemannian Geometry. Vol. 91. Encyclopaedia of Mathematical Sciences. Springer, 2001 (↓pp. 3, 40)

    2001

  55. [55]

    Information and accuracy attainable in the estimation of statistical parameters

    C. R. Rao. “Information and accuracy attainable in the estimation of statistical parameters”. In:Bulletin of the Calcutta Mathematical Society37.3 (1945), pp. 81–91 (↓p. 42). 51

    1945

  56. [56]

    de Rham.Variétés différentiables: Formes, courants, formes harmoniques

    G. de Rham.Variétés différentiables: Formes, courants, formes harmoniques. Vol. 1222. Actual- ités scientifiques et industrielles. Paris: Hermann, 1955 (↓p. 2)

    1955

  57. [57]

    An Absolute Partial Differential Calculus

    H. S. Ruse. “An Absolute Partial Differential Calculus”. In:The Quarterly Journal of Mathe- maticsos-2.1 (1931), pp. 190–215.doi:10.1093/qmath/os-2.1.190(↓pp. 2, 8)

    work page doi:10.1093/qmath/os-2.1.190( 1931

  58. [58]

    D. J. Saunders.The Geometry of Jet Bundles. Vol. 142. London Mathematical Society Lecture Note Series. Cambridge University Press, 1989 (↓pp. 5, 6, 12)

    1989

  59. [59]

    J. L. Synge.Relativity: The General Theory. Amsterdam: North-Holland Publishing Company, 1960 (↓p. 2)

    1960

  60. [60]

    Upmeier.Symmetric Banach Manifolds and JordanC ∗-Algebras

    H. Upmeier.Symmetric Banach Manifolds and JordanC ∗-Algebras. Vol. 104. North-Holland Mathematics Studies. Amsterdam: North-Holland, 1985 (↓p. 40)

    1985

  61. [61]

    Yano and M

    K. Yano and M. Kon.Structures on Manifolds. Vol. 3. Series in Pure Mathematics. Singapore: World Scientific, 1984 (↓p. 6)

    1984

  62. [62]

    From Hessian to Weitzenböck: manifolds with torsion-carrying con- nections

    J. Zhang and G. Khan. “From Hessian to Weitzenböck: manifolds with torsion-carrying con- nections”. In:Information Geometry2 (2019), pp. 77–98.doi:10.1007/s41884-019-00018-x (↓pp. 2, 3, 37)

    work page doi:10.1007/s41884-019-00018-x 2019

  63. [63]

    Statistical Mirror Symmetry

    J. Zhang and G. Khan. “Statistical Mirror Symmetry”. In:Differential Geometry and its Applications73 (2020), p. 101678.doi:10.1016/j.difgeo.2020.101678(↓pp. 2, 5, 9). 52

    work page doi:10.1016/j.difgeo.2020.101678( 2020