Riv. Mat. Univ. Parma, Vol. 13, No. 2, 2022
Federico A. Rossi [a]
New special Einstein pseudo-Riemannian metrics on solvable Lie algebras
Pages: 449-479
Received: 1 December 2021
Accepted in revised form: 28 March 2022
Mathematics Subject Classification: 53C50, 53C25, 53C30, 22E25, 32M10.
Keywords: Einstein metrics, nilsolitons, solvable Lie algebras, pseudo-Riemannian homogeneous metrics,
complex structures, para-complex structures.
Author address:
[a]: Università degli Studi di Perugia, Dipartimento di Matematica e Informatica, Perugia, Italy
This research was partially supported by GNSAGA of INdAM and the Young Talents Award of
Università degli Studi di Milano-Bicocca joint with Accademia Nazionale dei Lincei
Full Text (PDF)
Abstract:
We exhibit a concrete procedure to construct Einstein pseudo-Kähler and para-Kähler
metrics on solvable Lie algebras. We apply this method to classify all the rank-one pseudo-Iwasawa
extensions of type-(Nil4) nilsoliton in low dimension. We prove that such metrics exist on
the rank-one pseudo-Iwasawa extension of the generalized Heisenberg Lie algebra.
Further ideas and suggestions to produce more special Einstein pseudo-Riemannian metrics are exposed.
References
- [1]
-
D. V. Alekseevskii, C. Medori and A. Tomassini,
Homogeneous para-Kählerian Einstein manifolds,
Uspekhi Mat. Nauk 64 (2009), no. 1(385), 3-50.
MR250309
- [2]
-
A. Aubert and A. Medina,
Groupes de Lie pseudo-riemanniens plats,
Tohoku Math. J. (2) 55 (2003), no. 4, 487-506.
MR2017221
- [3]
-
W. Batat and K. Onda,
Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups,
J. Geom. Phys. 114 (2017), 138-152.
MR3610038
- [4]
-
S. Benayadi and M. Boucetta,
On para-Kähler and hyper-para-Kähler Lie algebras,
J. Algebra 436 (2015), 61-101.
MR3348469
- [5]
-
C. Böhm and R. A. Lafuente,
Non-compact Einstein manifolds with symmetry,
J. Amer. Math. Soc., to appear,
DOI: 10.1090/jams/1022.
- [6]
-
D. Conti, V. del Barco and F. A. Rossi,
Ad-invariant metrics on nonnice nilpotent Lie algebras,
arXiv:2111.11274 [math.DG], preprint, 2021.
DOI
- [7]
-
D. Conti, V. del Barco and F. A. Rossi,
Diagram involutions and homogeneous Ricci-flat metrics,
Manuscripta Math. 165 (2021), no. 3-4, 381-413.
MR4280489
- [8]
-
D. Conti, V. del Barco and F. A. Rossi,
Uniqueness of ad-invariant metrics,
Tohoku Math. J., to appear.
- [9]
-
D. Conti and F. A. Rossi,
Construction of nice nilpotent Lie groups,
J. Algebra 525 (2019), 311-340.
MR3911646
- [10]
-
D. Conti and F. A. Rossi,
Einstein nilpotent Lie groups,
J. Pure Appl. Algebra 223 (2019), no. 3, 976-997.
MR3862660
- [11]
-
D. Conti and F. A. Rossi,
Indefinite Einstein metrics on nice Lie groups,
Forum Math. 32 (2020), no. 6, 1599-1619.
MR4168706
- [12]
-
D. Conti and F. A. Rossi,
Indefinite nilsolitons and Einstein solvmanifolds,
J. Geom. Anal. 32 (2022), no. 3, Paper No. 88, 34 pp.
MR4363761
- [13]
-
D. Conti and F. A. Rossi,
Nice pseudo-Riemannian nilsolitons,
J. Geom. Phys. 173 (2022), Paper No. 104433, 20 pp.
MR4358603
- [14]
-
D. Conti and F. A. Rossi,
Ricci-flat and Einstein pseudoriemannian nilmanifolds,
Complex Manifolds 6 (2019), no. 1, 170-193.
MR3954004
- [15]
-
D. Conti and F. A. Rossi,
The Ricci tensor of almost parahermitian manifolds,
Ann. Global Anal. Geom. 53 (2018), no. 4, 467-501.
MR3803336
- [16]
-
D. Conti, F. A. Rossi and R. Segnan Dalmasso,
Pseudo-Riemannian Sasaki solvmanifolds,
J. Korean Math. Soc. 60 (2023), no. 1, 115-141.
MR4527958
%%doi: 10.4134/JKMS.j220232
- [17]
-
L. A. Cordero, M. Fernández and L. Ugarte,
Pseudo-Kähler metrics on six-dimensional nilpotent Lie algebras,
J. Geom. Phys. 50 (2004), no. 1-4, 115-137.
MR2078222
- [18]
-
I. Dotti Miatello,
Ricci curvature of left invariant metrics on solvable unimodular Lie groups,
Math. Z. 180 (1982), no. 2, 257-263.
MR0661702
- [19]
-
M. Fernández, A. Fino and V. Manero,
\(G_2\)-structures on Einstein solvmanifolds,
Asian J. Math. 19 (2015), no. 2, 321-342.
MR3337790
- [20]
-
M. Fernández, M. Freibert and J. Sánchez,
A non Ricci-flat Einstein pseudo-Riemannian metric on a \(7\)-dimensional nilmanifold,
Bull. Belg. Math. Soc. Simon Stevin 28 (2022), no. 4, 487-511.
MR4420432
- [21]
-
A. Fino, M. Parton and S. Salamon,
Families of strong KT structures in six dimensions,
Comment. Math. Helv. 79 (2004), no. 2, 317-340.
MR2059435
- [22]
-
M.-P. Gong,
Classification of nilpotent Lie algebras of dimension 7 (over algebraically closed fields and R),
ProQuest LLC, Ann Arbor, MI, 1998,
Thesis (Ph.D.)-University of Waterloo, Canada.
MR2698220
- [23]
-
J. Heber,
Noncompact homogeneous Einstein spaces,
Invent. Math. 133 (1998), no. 2, 279-352.
MR1632782
- [24]
-
M. Jablonski,
Survey: homogeneous Einstein manifolds,
arXiv:2111.09782 [math.DG], preprint, 2021.
DOI
- [25]
-
Y. Kondo and H. Tamaru,
A classification of left-invariant Lorentzian metrics on some nilpotent lie groups,
Tohoku Math. J. (2) 75 (2023), no. 1, 89-117.
MR4564844
- [26]
-
J. Lauret,
Einstein solvmanifolds and nilsolitons,
in ''New developments in Lie theory and geometry'', 491,
Contemp. Math., Amer. Math. Soc., Providence, RI, 2009, 1-35.
MR2537049
- [27]
-
J. Lauret,
Einstein solvmanifolds are standard,
Ann. of Math. (2) 172 (2010), no. 3, 1859-1877.
MR2726101
- [28]
-
J. Lauret,
Finding Einstein solvmanifolds by a variational method,
Math. Z. 241 (2002), no. 1, 83-99.
MR1930986
- [29]
-
J. Lauret,
Ricci soliton homogeneous nilmanifolds,
Math. Ann. 319 (2001), no. 4, 715-733.
- [30]
-
J. Lauret and C. Will,
Einstein solvmanifolds: existence and non-existence questions,
Math. Ann. 350 (2011), no. 1, 199-225.
MR2785768
- [31]
-
J. Lauret and C. Will,
On the diagonalization of the Ricci flow on Lie groups,
Proc. Amer. Math. Soc. 141 (2013), no. 10, 3651-3663.
MR3080187
- [32]
-
A. I. Mal'cev,
On a class of homogeneous spaces, (Russian),
Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949), 9-32.
MR0028842
- [33]
-
V. Manero,
Closed \(G_2\) forms and special metrics,
PhD thesis, Universidad del país Basco (ESP), 2015.
- [34]
-
J. Milnor,
Curvatures of left invariant metrics on Lie groups,
Advances in Math. 21 (1976), no. 3, 293-329.
MR0425012
- [35]
-
Y. Nikolayevsky,
Einstein solvmanifolds and the pre-Einstein derivation,
Trans. Amer. Math. Soc. 363 (2011), no. 8, 3935-3958.
MR2792974
- [36]
-
K. Onda,
Example of algebraic Ricci solitons in the pseudo-Riemannian case,
Acta Math. Hungar. 144 (2014), no. 1, 247-265.
MR3267185
- [37]
-
G. P. Ovando,
Invariant pseudo-Kähler metrics in dimension four,
J. Lie Theory 16 (2006), no. 2, 371-391.
MR2197598
- [38]
-
T. L. Payne,
The existence of soliton metrics for nilpotent Lie groups,
Geom. Dedicata 145 (2010), 71-88.
MR2600946
- [39]
-
T. L. Payne,
Applications of index sets and Nikolayevsky derivations to positive rank nilpotent Lie algebras,
J. Lie Theory 24 (2014), no. 1, 1-27.
MR3186326
- [40]
-
L. Schäfer and F. Schulte-Hengesbach,
Nearly pseudo-Kähler and nearly para-Kähler six-manifolds,
In '' Handbook of pseudo-Riemannian geometry and supersymmetry'',
IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zürich, 2010, 425-453.
MR2681597
- [41]
-
S. Weinberg,
Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,
John Wiley & Sons., New York, NY, 1972.
- [42]
-
C. Will,
Rank-one Einstein solvmanifolds of dimension 7,
Differential Geom. Appl. 19 (2003), no. 3, 307-318.
MR2013098
- [43]
-
Z. Yan,
Pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces,
J. Geom. 111 (2020), no. 1, Paper No. 4, 18 pp.
MR4048326
Home Riv.Mat.Univ.Parma