On Metrics of Diagonal Curvature

Abstract

In this paper, the theory of spaces of diagonal curvature is developed. An efficient necessary condition for metrics of diagonal curvature, namely, the vanishing of the Haantjes tensor for the Ricci affinor, is obtained. Examples are constructed.

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References

  1. 1.

    D. A. Berdinsky and I. P. Rybnikov, “On orthogonal curvilinear coordinate systems in constant curvature spaces,” Sib. Math. J., 52, No. 3, 394–401 (2011).

    MathSciNet  Article  Google Scholar 

  2. 2.

    L. Bianchi, Lezioni di Geometria Differenziale, Vol. 2, Pt. 2, Zanichelli, Bologna (1924).

  3. 3.

    L. Bianchi, Opere, vol. 3 : Sistemi Tripli Orthogonali, Cremonese, Roma (1955).

  4. 4.

    É. Cartan, Les systèmes differentials extérieurs et leur applications géométriques, Actual. Sci. Ind., No. 994, Hermann et Cie., Paris (1945).

  5. 5.

    G. Darboux, Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes, Gauthier-Villars, Paris (1910).

    Google Scholar 

  6. 6.

    D. M. Deturck and D. Yang, “Existence of elastic deformations with prescribed principal strains and triply orthogonal systems,” Duke Math. J., 51, No. 2, 243–260 (1984).

    MathSciNet  Article  Google Scholar 

  7. 7.

    B. A. Dubrovin and S. P. Novikov, “Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov–Whitham averaging method,” Sov. Math. Dokl., 27, 665–669 (1983).

    MATH  Google Scholar 

  8. 8.

    E. V. Ferapontov, “Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type,” Funct. Anal. Its Appl., 25, No. 3, 195–204 (1991).

    MathSciNet  Article  Google Scholar 

  9. 9.

    E. V. Glukhov, Algebraic-Geometric Methods of Constructing Orthogonal Coordinates in Riemannian Spaces, Graduate work, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University (2015).

    Google Scholar 

  10. 10.

    J. Haantjes, “On forming sets of eigenvectors,” Indag. Math., 17, 158–162 (1955).

    MathSciNet  Article  Google Scholar 

  11. 11.

    I. M. Krichever, “Algebraic-geometric n-orthogonal curvilinear coordinate systems and solutions of the associativity equations,” Funct. Anal. Its Appl., 31, No. 1, 25–39 (1997).

    MathSciNet  Article  Google Scholar 

  12. 12.

    A. E. Mironov and I. A. Taimanov, “Orthogonal curvilinear coordinate systems corresponding to singular spectral curves,” Proc. Steklov Inst. Math., 255, 169–184 (2006).

    Article  Google Scholar 

  13. 13.

    O. I. Mokhov, “Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems,” Russ. Math. Surv., 53, No. 3, 515–622 (1998).

    MathSciNet  Article  Google Scholar 

  14. 14.

    O. I. Mokhov, “Compatible and almost compatible metrics,” Russ. Math. Surv., 55, No. 4, 819–821 (2000).

    MathSciNet  Article  Google Scholar 

  15. 15.

    O. I. Mokhov, “Compatible and almost compatible pseudo-Riemannian metrics,” Funct. Anal. Its Appl., 35, No. 2, 100–110 (2001).

    MathSciNet  Article  Google Scholar 

  16. 16.

    O. I. Mokhov, “Flat pencils of metrics and integrable reductions of Lamé’s equations,” Russ. Math. Surv., 56, No. 2, 416–418 (2001).

    Article  Google Scholar 

  17. 17.

    O. I. Mokhov, “Integrability of the equations for nonsingular pairs of compatible flat metrics,” Theor. Math. Phys., 130, No. 2, 198–212 (2002).

    MathSciNet  Article  Google Scholar 

  18. 18.

    O. I. Mokhov, “Compatible metrics of constant Riemannian curvature: local geometry, nonlinear equations, and integrability,” Funct. Anal. Its Appl., 36, No. 3, 196–204 (2002).

    MathSciNet  Article  Google Scholar 

  19. 19.

    O. I. Mokhov, “Lax pairs for equations describing compatible nonlocal Poisson brackets of hydrodynamic type and integrable reductions of the Lamé equations,” Theor. Math. Phys., 138, No. 2, 238–249 (2004).

    Article  Google Scholar 

  20. 20.

    O. I. Mokhov, “Riemann invariants of semisimple non-locally bi-Hamiltonian systems of hydrodynamic type and compatible metrics,” Russ. Math. Surv., 65, No. 6, 1183–1185 (2010).

    Article  Google Scholar 

  21. 21.

    O. I. Mokhov, “Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type,” Theor. Math. Phys., 167, No. 1, 403–420 (2011).

    MathSciNet  Article  Google Scholar 

  22. 22.

    O. I. Mokhov and E. V. Ferapontov, “Non-local Hamiltonian operators of hydrodynamic type related to metrics of constant curvature,” Russ. Math. Surv., 45, No. 3, 218–219 (1990).

    Article  Google Scholar 

  23. 23.

    A. Nijenhuis, “Xn−1-forming sets of eigenvectors,” Indag. Math., 13, 200–212 (1951).

    MathSciNet  Article  Google Scholar 

  24. 24.

    S. P. Tsarev, “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method,” Math. USSR Izv., 54, No. 5, 397–419 (1990).

    MathSciNet  MATH  Google Scholar 

  25. 25.

    V. E. Zakharov, “Description of the n-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type. I: Integration of the Lamé equations,” Duke Math. J., 94, 103–139 (1998).

    MathSciNet  Article  Google Scholar 

  26. 26.

    V. E. Zakharov, “Application of the inverse scattering transform to classical problems of differential geometry and general relativity,” in: J. Bona, R. Choudhury, and D. Kaup, eds., The Legacy of the Inverse Scattering Transform in Applied Mathematics, Contemp. Math., Vol. 301, Amer. Math. Soc., Providence (2002), pp. 15–34.

    Google Scholar 

  27. 27.

    V. E. Zakharov and A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I,” Funct. Anal. Its Appl., 8, No. 3, 226–235 (1974).

    Article  Google Scholar 

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Correspondence to O. I. Mokhov.

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In memory of Yuri Petrovich Solovyov

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 6, pp. 171–182, 2016.

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Mokhov, O.I. On Metrics of Diagonal Curvature. J Math Sci (2020). https://doi.org/10.1007/s10958-020-04912-z

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