Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
-
Esteban Calviño-Louzao
,
María Ferreiro-Subrido
,
Eduardo Garcia-Rio
and
Ramon Vázquez-Lorenzo
Abstract
We show that a four-dimensional homogeneous manifold with half-harmonic Weyl curvature tensor is symmetric or homothetic either to the only nonsymmetric anti-self-dual homogeneous manifold or to the 3-symmetric space.
Funding source: Agencia Estatal de Investigación
Award Identifier / Grant number: PID2019-105138GB-C21
Funding source: Xunta de Galicia
Award Identifier / Grant number: ED431C 2019/10
Award Identifier / Grant number: ED431F 2020/04
Funding statement: Supported by the projects PID2019-105138GB-C21 (AEI/FEDER, Spain) and ED431C 2019/10, ED431F 2020/04 (Xunta de Galicia, Spain).
-
Communicated by: Jan Frahm
References
[1] V. Apostolov, J. Armstrong and T. Drăghici, Local rigidity of certain classes of almost Kähler 4-manifolds, Ann. Global Anal. Geom. 21 (2002), no. 2, 151–176. 10.1023/A:1014779405043Search in Google Scholar
[2] V. Apostolov, D. M. J. Calderbank and P. Gauduchon, The geometry of weakly self-dual Kähler surfaces, Compos. Math. 135 (2003), no. 3, 279–322. 10.1023/A:1022251819334Search in Google Scholar
[3] L. Bérard-Bergery, Les espaces homogènes riemanniens de dimension 4, Riemannian Geometry in Dimension 4 (Paris 1978/1979), Textes Math. 3, CEDIC, Paris (1981), 40–60. Search in Google Scholar
[4] A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer, Berlin, 1987. 10.1007/978-3-540-74311-8Search in Google Scholar
[5] E. Calviño Louzao, X. García-Martínez, E. García-Río, I. Gutiérrez-Rodríguez and R. Vázquez-Lorenzo, Conformally Einstein and Bach-flat four-dimensional homogeneous manifolds, J. Math. Pures Appl. (9) 130 (2019), 347–374. 10.1016/j.matpur.2019.01.005Search in Google Scholar
[6] E. Calviño Louzao, E. García-Río, I. Gutiérrez-Rodríguez and R. Vázquez-Lorenzo, Four-dimensional homogeneous Kähler Ricci solitons, Mexican Mathematicians in the World—Trends and Recent Contributions, Contemp. Math. 775, American Mathematical Society, Providence (2021), 31–39. 10.1090/conm/775/15587Search in Google Scholar
[7] X. Cao and H. Tran, The Weyl tensor of gradient Ricci solitons, Geom. Topol. 20 (2016), no. 1, 389–436. 10.2140/gt.2016.20.389Search in Google Scholar
[8] Q. Chen and M. Zhu, On rigidity of gradient Kähler–Ricci solitons with harmonic Bochner tensor, Proc. Amer. Math. Soc. 140 (2012), no. 11, 4017–4025. 10.1090/S0002-9939-2012-11648-XSearch in Google Scholar
[9] D. A. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 4th ed., Undergrad. Texts Math., Springer, Cham, 2015. 10.1007/978-3-319-16721-3Search in Google Scholar
[10] J. E. D’Atri, Codazzi tensors and harmonic curvature for left invariant metrics, Geom. Dedicata 19 (1985), no. 3, 229–236. 10.1007/BF00149363Search in Google Scholar
[11] W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-3-1—A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, 2022. Search in Google Scholar
[12] A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compos. Math. 49 (1983), no. 3, 405–433. Search in Google Scholar
[13] V. De Smedt and S. Salamon, Anti-self-dual metrics on Lie groups, Differential Geometry and Integrable Systems (Tokyo 2000), Contemp. Math. 308, American Mathematical Society, Providence (2002), 63–75. 10.1090/conm/308/05312Search in Google Scholar
[14] E. Griffin, Gradient ambient obstruction solitons on homogeneous manifolds, Ann. Global Anal. Geom. 60 (2021), no. 3, 469–499. 10.1007/s10455-021-09784-3Search in Google Scholar
[15] W. Jelonek, Compact Kähler surfaces with harmonic anti-self-dual Weyl tensor, Differential Geom. Appl. 16 (2002), no. 3, 267–276. 10.1016/S0926-2245(02)00076-1Search in Google Scholar
[16] G. R. Jensen, Homogeneous Einstein spaces of dimension four, J. Differential Geom. 3 (1969), 309–349. 10.4310/jdg/1214429056Search in Google Scholar
[17] J. Kim, On Kähler manifolds with harmonic Bochner curvature tensor, Ann. Global Anal. Geom. 35 (2009), no. 4, 339–343. 10.1007/s10455-008-9134-8Search in Google Scholar
[18] R. S. Kulkarni, Curvature and metric, Ann. of Math. (2) 91 (1970), 311–331. 10.2307/1970580Search in Google Scholar
[19] J. Lauret, Ricci soliton homogeneous nilmanifolds, Math. Ann. 319 (2001), no. 4, 715–733. 10.1007/PL00004456Search in Google Scholar
[20] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), no. 3, 293–329. 10.1016/S0001-8708(76)80002-3Search in Google Scholar
[21] B. L. Neto, Generalized quasi-Einstein manifolds with harmonic anti-self dual Weyl tensor, Arch. Math. (Basel) 106 (2016), no. 5, 489–499. 10.1007/s00013-016-0896-0Search in Google Scholar
[22] P. Petersen and W. Wylie, Rigidity of homogeneous gradient soliton metrics and related equations, Differential Geom. Appl. 84 (2022), Paper No. 101929. 10.1016/j.difgeo.2022.101929Search in Google Scholar
[23] F. Podestà and A. Spiro, Four-dimensional Einstein-like manifolds and curvature homogeneity, Geom. Dedicata 54 (1995), no. 3, 225–243. 10.1007/BF01265339Search in Google Scholar
[24] J.-Y. Wu, P. Wu and W. Wylie, Gradient shrinking Ricci solitons of half harmonic Weyl curvature, Calc. Var. Partial Differential Equations 57 (2018), no. 5, Paper No. 141. 10.1007/s00526-018-1415-xSearch in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
- The Singer transfer for infinite real projective space
- The Stiefel–Whitney classes of moment-angle manifolds are trivial
-
On almost p-rational characters of
- Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
- Construction of a class of spectral measures
- On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
- Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations
- Conway invariant Jacobi forms on the Leech lattice
- Geometric characterizations of Gromov hyperbolic Hölder domains
- Improved quantitative unique continuation for complex-valued drift equations in the plane
- On uniqueness of branching to fixed point Lie subalgebras
- Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
Articles in the same Issue
- Frontmatter
- Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
- The Singer transfer for infinite real projective space
- The Stiefel–Whitney classes of moment-angle manifolds are trivial
-
On almost p-rational characters of
- Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
- Construction of a class of spectral measures
- On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
- Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations
- Conway invariant Jacobi forms on the Leech lattice
- Geometric characterizations of Gromov hyperbolic Hölder domains
- Improved quantitative unique continuation for complex-valued drift equations in the plane
- On uniqueness of branching to fixed point Lie subalgebras
- Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
