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.
Similar content being viewed by others
References
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).
L. Bianchi, Lezioni di Geometria Differenziale, Vol. 2, Pt. 2, Zanichelli, Bologna (1924).
L. Bianchi, Opere, vol. 3 : Sistemi Tripli Orthogonali, Cremonese, Roma (1955).
É. Cartan, Les systèmes differentials extérieurs et leur applications géométriques, Actual. Sci. Ind., No. 994, Hermann et Cie., Paris (1945).
G. Darboux, Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes, Gauthier-Villars, Paris (1910).
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).
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).
E. V. Ferapontov, “Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type,” Funct. Anal. Its Appl., 25, No. 3, 195–204 (1991).
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).
J. Haantjes, “On forming sets of eigenvectors,” Indag. Math., 17, 158–162 (1955).
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).
A. E. Mironov and I. A. Taimanov, “Orthogonal curvilinear coordinate systems corresponding to singular spectral curves,” Proc. Steklov Inst. Math., 255, 169–184 (2006).
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).
O. I. Mokhov, “Compatible and almost compatible metrics,” Russ. Math. Surv., 55, No. 4, 819–821 (2000).
O. I. Mokhov, “Compatible and almost compatible pseudo-Riemannian metrics,” Funct. Anal. Its Appl., 35, No. 2, 100–110 (2001).
O. I. Mokhov, “Flat pencils of metrics and integrable reductions of Lamé’s equations,” Russ. Math. Surv., 56, No. 2, 416–418 (2001).
O. I. Mokhov, “Integrability of the equations for nonsingular pairs of compatible flat metrics,” Theor. Math. Phys., 130, No. 2, 198–212 (2002).
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).
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).
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).
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).
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).
A. Nijenhuis, “Xn−1-forming sets of eigenvectors,” Indag. Math., 13, 200–212 (1951).
S. P. Tsarev, “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method,” Math. USSR Izv., 54, No. 5, 397–419 (1990).
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).
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.
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
In memory of Yuri Petrovich Solovyov
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 6, pp. 171–182, 2016.
Rights and permissions
About this article
Cite this article
Mokhov, O.I. On Metrics of Diagonal Curvature. J Math Sci 248, 780–787 (2020). https://doi.org/10.1007/s10958-020-04912-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04912-z