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Classification of Left Invariant Riemannian Metrics on Complex Hyperbolic Space

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Abstract

It is well known that \({\mathbb {C}}H^n\) has the structure of a solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that each of those metrics is of constant negative scalar curvature, only one of them being Einstein (up to isometry and scaling).

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References

  1. Alekseevsky, D.V.: Conjugacy of polar factorizations of Lie groups. Mat. Sbornik 84 13, 14–26 (1971)

    Google Scholar 

  2. Alekseevsky, D.V.: Homogeneous Riemannian spaces of negative curvature. Mat. Sbornik 96 25, 87–109 (1975)

    Article  Google Scholar 

  3. Bérard-Bérgery, L.: Homogeneous Riemannian spaces of dimension four. Seminar A. Besse, Four-dimensional Riemannian geometry (1985)

  4. Bokan, N., Šukilović, T., Vukmirović, S.: Lorentz geometry of 4-dimensional nilpotent Lie groups. Geom. Dedic. 177, 83–102 (2015)

    Article  MathSciNet  Google Scholar 

  5. Calvaruso, G., Zaeim, A.: Four-dimensional Lorentzian Lie groups. Differ. Geom. Appl. 31, 496–509 (2013)

    Article  MathSciNet  Google Scholar 

  6. Christodoulakis, T., Papadopoulos, G.O., Dimakis, A.: Automorphisms of real four-dimensional Lie algebras and the invariant characterization of homogeneous 4-spaces. J. Phys. A Math. Gen. 36, 427–441 (2002)

    Article  MathSciNet  Google Scholar 

  7. Cordero, L.A., Parker, P.E.: Left-invariant Lorentz metrics on 3-dimensional Lie groups. Rend. Mat. Appl. 17, 129–155 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Domínguez-Vázquez, M., Sanmartín-López, V., Tamaru, H.: Codimension one Ricci soliton subgroups of solvable Iwasawa groups. Journal de Mathématiques Pures et Appliquées 152, 69–93 (2021)

  9. de Gosson, M.: Symplectic Geometry and Quantum Mechanics. Birkhäuser, Boston (2006)

    Book  Google Scholar 

  10. Hashinaga, T., Tamaru, H., Terada, K.: Milnor-type theorems for left-invariant Riemannian metrics on Lie groups. J. Math. Soc. Jpn. 68, 669–684 (2016)

    Article  MathSciNet  Google Scholar 

  11. Heintze, E.: On homogeneous manifolds of negative curvature. Math. Ann. 211, 23–34 (1974)

    Article  MathSciNet  Google Scholar 

  12. Heintze, E.: Riemannsche Solvmannigfaltigkeiten. Geom. Dedic. 1, 141–147 (1973)

    Article  MathSciNet  Google Scholar 

  13. Heber, J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133, 279–352 (1998)

  14. Kodama, H., Takahara, A., Tamaru, H.: The space of left-invariant metrics on a Lie group up to isometry and scaling. Manuscr. Math. 135, 229–243 (2011)

    Article  MathSciNet  Google Scholar 

  15. Kondo, Y., Tamaru, H.: A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups. arXiv:2011.09118 [math.DG] (2020)

  16. Kubo, A., Onda, K., Taketomi, Y., Tamaru, H.: On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups. Hiroshima Math. J. 46, 357–374 (2016)

    Article  MathSciNet  Google Scholar 

  17. Lauret, J.: Degenerations of Lie algebras and geometry of Lie groups. Differ. Geom. Appl. 18, 177–194 (2003)

    Article  MathSciNet  Google Scholar 

  18. Lauret, J.: Homogeneous nilmanifolds of dimension 3 and 4. Geom. Dedic. 68, 145–155 (1997)

    Article  MathSciNet  Google Scholar 

  19. Lauret, J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319, 715–733 (2001)

    Article  MathSciNet  Google Scholar 

  20. Lauret, J.: Ricci soliton solvmanifolds. Journal für die reine und angewandte Mathematik 2011, 1–21 (2011)

    Article  MathSciNet  Google Scholar 

  21. Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)

    Article  MathSciNet  Google Scholar 

  22. Nasehi, M.: On the geometry of higher dimensional Heisenberg groups. Mediterr. J. Math. 16, 29 (2019)

    Article  MathSciNet  Google Scholar 

  23. Rahmani, S.: Metriques de Lorentz sur les groupes de Lie unimodulaires de dimension 3. J. Geom. Phys. 9, 295–302 (1992)

    Article  MathSciNet  Google Scholar 

  24. Šukilović, T.: Classification of left invariant metrics on 4-dimensional solvable Lie groups. Theor. Appl. Mech. 47, 181–204 (2020)

    Article  Google Scholar 

  25. Šukilović, T.: Geometric properties of neutral signature metrics on 4-dimensional nilpotent Lie groups. Rev. Un. Mat. Argentina 57, 23–47 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Vukmirović, S.: Classification of left-invariant metrics on the Heisenberg group. J. Geom. Phys. 94, 72–80 (2015)

    Article  MathSciNet  Google Scholar 

  27. Vukmirović, S., Šukilović, T.: Geodesic completeness of the left-invariant metrics on \({{\mathbb{R}}H^{n}}\). Ukr. Math. J. 72, 702–711 (2020)

    Article  Google Scholar 

  28. Williamson, J.: On the algebraic problem concerning the normal forms of linear dynamical systems. Am. J. Math. 58, 141–163 (1936)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research has been supported by the Project no. 7744592 MEGIC “Integrability and Extremal Problems in Mechanics, Geometry and Combinatorics” of the Science Fund of Serbia and Ministry of Education, Science and Technological Development, Republic of Serbia, Grant no. 451-03-68/2022-14/200104.

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Authors and Affiliations

  1. Mathematical Institute, SANU, Belgrade, Serbia

    Andrijana Dekić & Marijana Babić

  2. Faculty of Mathematics, University of Belgrade, Belgrade, Serbia

    Srdjan Vukmirović

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  1. Andrijana Dekić
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  2. Marijana Babić
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  3. Srdjan Vukmirović
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Appendix A: Explicit Expression of Riemann Curvature Operator

Appendix A: Explicit Expression of Riemann Curvature Operator

$$\begin{aligned} R(X,Y_i)X&=\frac{1}{4z}( -x_iX+pY_i) \\ R(X,Y_i)Y_j&=\frac{1}{4z}(x_jY_i-\delta _{ij}\sigma _iX)\\ R(X,Y_i)Z_j&=\frac{\delta _{ij}}{4}W\\ R(X,Y_i)W&=-\frac{\beta }{4 \sigma _i}Z_i\\ \\ R(X, Z_i)X&=\frac{p}{4z}\left( Z_i+\frac{x_i}{\sigma _i}W\right) \\ R(X, Z_i)Y_j&=\frac{x_i}{4z}\left( Z_i+\frac{x_i}{\sigma _i}W-\delta _{ij}\frac{z}{x_j}W\right) \\ R(X, Z_i)Z_j&=-\frac{\sigma _i}{4z}\delta _{ij}X\\ R(X, Z_i)W&=\frac{\beta }{4z\sigma _i}\left( -2x_iX+x_i\sum _l\frac{x_l}{\sigma _l}Y_l+zY_i\right) \\ \\ R(X,W)X&=\frac{p\beta }{4z}\sum _l\frac{x_l}{\sigma _l^2}Z_l+ \frac{z+p}{2z}W\\ R(X,W)Y_i&=\frac{x_i\beta }{4z}\sum _l\frac{x_l}{\sigma _l^2}Z_l-\frac{\beta }{2\sigma _i}Z_i+\frac{x_i}{2z}W\\ R(X,W)Z_i&=\frac{\beta }{4z\sigma _i}\left( -3x_iX+2x_i\sum _l\frac{x_l}{\sigma _l}Y_l+2zY_i \right) \\ R(X,W)W&=\frac{\beta }{z}\left( -X+\frac{1}{2}\sum _l\frac{x_l}{\sigma _l}Y_l\right) \\ \\ R(Y_i,Y_j)X&=\frac{1}{4z}(x_iY_j-x_jY_i)\\ R(Y_i,Y_j)Y_k&=\frac{1}{4z}(\delta _{ik}\sigma _i Y_j-\delta _{jk}\sigma _j Y_i)\\ R(Y_i,Y_j)Z_k&=\frac{\beta }{4}\left( \frac{\delta _{ik}}{\sigma _j}Z_j-\frac{\delta _{jk}}{\sigma _i}Z_i\right) \\ R(Y_i,Y_j)W&=0\\ \\ \end{aligned}$$
$$\begin{aligned} R(Y_i,Z_j)X&=\frac{1}{4z}\left[ x_iZ_j+\left( \frac{x_ix_j}{\sigma _j}-2\delta _{ij}z\right) W \right] \\ R(Y_i,Z_j)Y_k&=\frac{1}{4} \left( \delta _{jk}\frac{\beta }{\sigma _i}Z_i +\delta _{ik}\frac{ \sigma _i}{z}Z_j +2\,\delta _{ij}\frac{\beta }{\sigma _k}Z_k +\delta _{ik}\frac{x_j \sigma _i}{z\sigma _j}W \right) \\ R(Y_i,Z_j)Z_k&=\frac{1}{4z}\left[ 2\,\delta _{ij}\frac{\beta }{\sigma _k}\left( x_kX-zY_k-x_k\sum _l\frac{x_l}{\sigma _l}Y_l\right) \right. \\&\quad \left. +\delta _{ik}\frac{\beta }{\sigma _j}\left( x_jX-zY_j-x_j\sum _l\frac{x_l}{\sigma _l}Y_l\right) -\delta _{jk}\sigma _jY_i \right] \\ R(Y_i,Z_j)W&=\frac{\beta }{4z}\left[ -\frac{x_j}{\sigma _j}Y_i+2\,\delta _{ij}(X-\sum _l\frac{x_l}{\sigma _l}Y_l) \right] \\ R(Y_i,W)X&=\frac{1}{4z}\left( \beta x_i\sum _l\frac{x_l}{\sigma _l^2}Z_l-\frac{\beta z}{\sigma _i}Z_i+2\,x_iW \right) \\ R(Y_i,W)Y_j&=\frac{\beta }{4z}\delta _{ij}\left[ \sigma _i\sum _l\frac{x_l}{\sigma _l^2}Z_l+(2\,\frac{\sigma _i}{\beta }-\frac{z}{\sigma _i})W \right] \\ R(Y_i,W)Z_j&=\frac{\beta }{4z}\left[ \delta _{ij}(X-\sum _l\frac{x_l}{\sigma _l}Y_l)-\frac{x_j}{\sigma _j}Y_i \right] \\ R(Y_i,W)W&=\frac{\beta ^2}{4z\sigma _i^2}\left[ -x_iX+x_i\sum _l\frac{x_l}{\sigma _l}Y_l+(z-2\frac{\sigma _i^2}{\beta })Y_i \right] \\ \\ R(Z_i,Z_j)X&=0 \\ R(Z_i,Z_j)Y_k&=\frac{\beta }{4z}\left[ \frac{\delta _{ik}}{\sigma _j}(-x_jX+x_j\sum _l\frac{x_l}{\sigma _l}Y_l+zY_j)\right. \\&\quad \quad \left. -\frac{\delta _{jk}}{\sigma _i}(-x_iX+x_i\sum _l\frac{x_l}{\sigma _l}Y_l+zY_i) \right] \\ R(Z_i,Z_j)Z_k&=\frac{1}{4z}\left[ \delta _{ik}\sigma _i(Z_j+\frac{x_j}{\sigma _j}W) -\delta _{jk}\sigma _j(Z_i+\frac{x_i}{\sigma _i}W) \right] \\ R(Z_i,Z_j)W&=\frac{\beta }{4z}\left( \frac{x_i}{\sigma _i}Z_j-\frac{x_j}{\sigma _j}Z_i \right) \\ \\ R(Z_i,W)X&=\frac{\beta }{4z\sigma _i}\left( -x_iX+x_i\sum _l\frac{x_l}{\sigma _l}Y_l+zY_i) \right) \\ R(Z_i,W)Y_j&=\frac{\beta }{4z}\delta _{ij}\left( -X+\sum _l\frac{x_l}{\sigma _l}Y_l \right) \\ R(Z_i,W)Z_j&=\frac{\beta }{4z}\left[ -\frac{x_j}{\sigma _j}Z_i+\delta _{ij}\sigma _i\sum _l\frac{x_l}{\sigma _l}^2Z_l +\left( \delta _{ij}\left( 2\frac{\sigma _i}{\beta }-\frac{z}{\sigma _j}\right) -\frac{x_ix_j}{\sigma _i\sigma _j} \right) W \right] \\ R(Z_i,W)W&=\frac{\beta ^2}{4z}\left[ \left( \frac{z}{\sigma _i^2}-\frac{2}{\beta }\right) Z_i + \frac{x_i}{\sigma _i}\sum _l\frac{x_l}{\sigma _l^2}Z_l \right] \\ \end{aligned}$$

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Dekić, A., Babić, M. & Vukmirović, S. Classification of Left Invariant Riemannian Metrics on Complex Hyperbolic Space. Mediterr. J. Math. 19, 232 (2022). https://doi.org/10.1007/s00009-022-02152-w

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