Abstract
We present new tools for determining whether two nilpotent metric Lie algebras are in the same isometry class. We exhibit new examples of continuous families of nilsoliton metric Lie algebras, including deformations of uniform metric Lie algebras. We show that all algebras of generalized Heisenberg type except for Heisenberg algebras and two others admit deformations by nilsoliton metric Lie algebras.
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References
Deloff, E.: Naturally reductive metrics and metrics with volume preserving geodesic symmetries. Thesis, Rutgers University (1979)
Eberlein P.: Geometry of 2-step nilpotent groups with a left invariant metric. Ann. Sci. École Norm. Sup. (4) 27(5), 611–660 (1994)
Gordon C.S., Kerr M.M.: New homogeneous Einstein metrics of negative Ricci curvature. Ann. Global Anal. Geom. 19(1), 75–101 (2001)
Gordon C.S., Wilson E.N.: Continuous families of isospectral Riemannian metrics which are not locally isometric. J. Differential Geom. 47(3), 504–529 (1997)
Heber J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133(2), 279–352 (1998)
Horn, R. A., Johnson, C. R.: Topics in matrix analysis. Cambridge University Press, Cambridge (1994) (Corrected reprint of the 1991 original)
Jablonski M.: Moduli of Einstein and non-Einstein nilradicals. Geom. Dedicata 152, 63–84 (2011)
Jensen G.R.: Homogeneous Einstein spaces of dimension four. J. Differential Geom. 3, 309–349 (1969)
Kaplan A.: Fundamental solutions for a class of hypoelliptic PDE generated by compositions of quadratic forms. Trans. Amer. Math. Soc. 258, 147–153 (1980)
Kerr, M. M.: A deformation of quaternionic hyperbolic space. Proc. Amer. Math. Soc. 134(2), 559–569 (electronic) (2006)
Lauret J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319(4), 715–733 (2001)
Lauret, J.: Einstein solvmanifolds and nilsolitons. In: New developments in Lie theory and geometry, volume 491 of Contemporary Mathematics, pp. 1–35. American Mathematical Society, Providence (2009)
Lawson H.B. Jr, Michelsohn M.-L.: Spin geometry, volume 38 of Princeton Mathematical Series. Princeton University Press, Princeton (1989)
Payne T.L.: The existence of soliton metrics for nilpotent Lie groups. Geom. Dedicata 145, 71–88 (2010)
Schueth D.: Continuous families of isospectral metrics on simply connected manifolds. Ann. of Math. (2) 149(1), 287–308 (1999)
Will C.: Rank-one Einstein solvmanifolds of dimension 7. Differential Geom. Appl. 19(3), 307–318 (2003)
Wilson E.N.: Isometry groups on homogeneous nilmanifolds. Geom. Dedicata 12(3), 337–346 (1982)
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Payne, T.L. Geometric invariants for nilpotent metric Lie algebras with applications to moduli spaces of nilsoliton metrics. Ann Glob Anal Geom 41, 139–160 (2012). https://doi.org/10.1007/s10455-011-9275-z
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DOI: https://doi.org/10.1007/s10455-011-9275-z
Keywords
- Nilmanifold
- Nilsoliton metric
- Soliton metric
- Algebra of type H
- Algebra of generalized Heisenberg type
- Uniform metric Lie algebra
- Geometric invariant