(T)-structures over two-dimensional F-manifolds: formal classification

  • Liana DavidEmail author
  • Claus Hertling


A (TE)-structure \(\nabla \) over a complex manifold M is a meromorphic connection defined on a holomorphic vector bundle over \({\mathbb {C}}\times M\), with poles of Poincaré rank one along \(\{ 0 \} \times M\). Under a mild additional condition (the so-called unfolding condition), \(\nabla \) induces a multiplication on TM and a vector field on M (the Euler field), which make M into an F-manifold with Euler field. By taking the pullbacks of \(\nabla \) under the inclusions \(\{ z\} \times M \rightarrow {\mathbb {C}}\times M\)\((z\in \mathbb {C}^*)\), we obtain a family of flat connections on vector bundles over M, parameterized by \(z\in {\mathbb {C}}^{*}\). The properties of such a family of connections give rise to the notion of (T)-structure. Therefore, any (TE)-structure underlies a (T)-structure, but the converse is not true. The unfolding condition can be defined also for (T)-structures. A (T)-structure with the unfolding condition induces on its parameter space the structure of an F-manifold (without Euler field). After a brief review on the theory of (T)- and (TE)-structures, we determine normal forms for the equivalence classes, under formal isomorphisms, of (T)-structures which induce a given irreducible germ of two-dimensional F-manifolds.


Meromorphic connections (T) and (TE)-structures F-manifolds Euler fields Frobenius manifolds Formal classifications 

Mathematics Subject Classification

53B15 35J99 32A20 53B50 



L.D. was supported by a grant of the Ministry of Research and Innovation, CNCS-UEFISCDI, Project No. PN-III-P4-ID-PCE-2016-0019 within PNCDI III. Part of this work was done during her visit at University of Mannheim (Germany) in October 2017. She thanks University of Mannheim for hospitality and great working conditions.


  1. 1.
    Anosov, D.V., Bolibruch, A.A.: The Riemann–Hilbert Problem, Aspects of Mathematics, vol. 22. Vieweg, Braunschweig (1994)CrossRefGoogle Scholar
  2. 2.
    Bourbaki, N.: Groupes et Algebres de Lie, Chapitres 4, 5 et 6. Hermann, Paris (1968)zbMATHGoogle Scholar
  3. 3.
    Coxeter, H.S.M.: Discrete groups generated by reflections. Ann. Math. 35, 588–621 (1934)CrossRefMathSciNetGoogle Scholar
  4. 4.
    David, L., Hertling, C.: Regular \(F\)-manifolds: initial conditions and Frobenius metrics. Ann. Sci. Norm. Super. Pisa Cl. Sci (5) XVII, 1121–1152 (2017)zbMATHGoogle Scholar
  5. 5.
    Dubrovin, B.: Geometry of \(2D\) topological field theories. In: Francoviglia, M., Greco, S. (eds.) Integrable Systems and Quantum Groups. Montecatini, Terme 1993, Lecture Notes in Mathematics, vol. 1620, pp. 120–348. Springer, Berlin (1996)Google Scholar
  6. 6.
    Dubrovin, B.: Differential geometry of the space of orbits of a Coxeter group. In: Terng, C.L., Uhlenbeck, K. (eds.) Survey of Differential Geometry [Integrable Systems]. Surveys in Differential Geometry, vol. IV, pp. 181–211. International Press, Boston, MA (1998)zbMATHGoogle Scholar
  7. 7.
    Givental, A.B.: Singular Lagrangian manifolds and their Lagrangian maps. J. Soviet. Math. 52(4), 3246–3278 (1998)CrossRefGoogle Scholar
  8. 8.
    Hertling, C.: \(tt^{*}\) geometry, Frobenius manifolds, their connections, and the construction for singularities. J. Reine Angew. Math. 555, 77–161 (2003)zbMATHGoogle Scholar
  9. 9.
    Hertling, C.: Frobenius Manifolds and Moduli Spaces for Singularities, Cambridge Tracts in Mathematics, vol. 151. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  10. 10.
    Hertling, C., Manin, Y.: Weak Frobenius manifolds. Int. Math. Res. Not. 6, 277–286 (1999)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Hertling, C., Manin, Y.: Unfoldings of meromorphic connections and a construction theorem for Frobenius manifolds. In: Hertling, C., Marcolli, M. (eds.) Frobenius Manifolds, Quantum Cohomology, and Singularities, pp. 113–144. Vieweg, Braunschweig (2004)zbMATHGoogle Scholar
  12. 12.
    Hertling, C., Manin, Y., Teleman, C.: An update on semisimple quantum cohomology and \(F\)-manifolds. Proc. Steklov Inst. Math. 264, 62–69 (2009)CrossRefGoogle Scholar
  13. 13.
    Hertling, C., Hoevenaars, L., Posthuma, H.: Frobenius manifolds, projective special geometry and Hitchin systems. J. Reine Angew. Math. 649, 117–165 (2010)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Lorenzoni, P., Pedroni, M., Raimondo, A.: \(F\)-manifolds and integrable systems of hydrodynamic type. Arch. Math. 47, 163–180 (2011)zbMATHGoogle Scholar
  15. 15.
    Malgrange, B.: Deformations de systemes differentielles et microdifferentielles. In: Seminaire E.N.S. Mathematique et Physique, pp. 351–379Google Scholar
  16. 16.
    Malgrange, B.: Deformations of differential systems II. J. Ramanjuan Math. Soc. 1, 3–15 (1986)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Malgrange, B.: Equations Differentielles a Coefficients Polynomiaux, Progress in Mathematics, vol. 96. Birkhauser, Basel (1991)zbMATHGoogle Scholar
  18. 18.
    Sabbah, C.: Isomonodromic Deformations and Frobenius Manifolds. Springer, Berlin (2007)zbMATHGoogle Scholar
  19. 19.
    Strachan, : Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures. Diff. Geom. Appl. 20, 67–99 (2004)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Dover Publications, Mineola (1965)zbMATHGoogle Scholar

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.“Simion Stoilow” Institute of Mathematics of the Romanian AcademyBucharestRomania
  2. 2.Lehrstuhl für MathematikUniversität Mannheim, B6, 2668131Germany

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