Harmonicity of vector fields on four-dimensional generalized symmetric spaces

• Autorzy: Giovanni Calvaruso
• Języki publikacji: EN

Abstrakty

EN

Let (M = G/H;g)denote a four-dimensional pseudo-Riemannian generalized symmetric space and g = m + h the corresponding decomposition of the Lie algebra g of G. We completely determine the harmonicity properties of vector fields belonging to m. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also classified, and the energy of these vector fields is explicitly calculated.

Słowa kluczowe

EN

Harmonic vector fields; Harmonic maps; Tangent sphere bundle; Generalized symmetric spaces; Pseudo-Riemannian homogeneous spaces

Kategorie tematyczne

• 53C43: Differential geometric aspects of harmonic maps
• 53C20: Global Riemannian geometry, including pinching
• 58E20: Harmonic maps , etc.
• 53C50: Lorentz manifolds, manifolds with indefinite metrics

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 2
• Strony: 411-425

Daty

wydano

2012-04-01

online

2012-01-18

Twórcy

autor

Giovanni Calvaruso

• Università del Salento

Bibliografia

• [1] Abbassi M.T.K., Calvaruso G., Perrone D., Harmonicity of unit vector fields with respect to Riemannian g-natural metrics, Differential Geom. Appl., 2009, 27(1), 157–169 http://dx.doi.org/10.1016/j.difgeo.2008.06.016
• [2] Abbassi M.T.K., Calvaruso G., Perrone D., Some examples of harmonic maps for g-natural metrics, Ann. Math. Blaise Pascal, 2009, 16(2), 305–320 http://dx.doi.org/10.5802/ambp.269
• [3] Abbassi M.T.K., Calvaruso G., Perrone D., Harmonic maps defined by the geodesic flow, Houston J. Math, 2010, 36(1), 69–90
• [4] Abbassi M.T.K., Calvaruso G., Perrone D., Harmonic sections of tangent bundles equipped with Riemannian g-natural metrics, Q. J. Math., 2011, 62(2), 259–288 http://dx.doi.org/10.1093/qmath/hap040
• [5] Apostolov V., Armstrong J., Drăghici T., Local rigidity of certain classes of almost Kähler 4-manifolds, Ann. Global Anal. Geom., 2002, 21(2), 151–176 http://dx.doi.org/10.1023/A:1014779405043
• [6] Benyounes M., Loubeau E., Wood C.M., Harmonic sections of Riemannian vector bundles, and metrics of Cheeger-Gromoll type, Differential Geom. Appl., 2007, 25(3), 322–334 http://dx.doi.org/10.1016/j.difgeo.2006.11.010
• [7] Calvaruso G., Harmonicity properties of invariant vector fields on three-dimensional Lorentzian Lie groups, J. Geom. Phys., 2011, 61(2), 498–515 http://dx.doi.org/10.1016/j.geomphys.2010.11.001
• [8] Calvaruso G., De Leo B., Curvature properties of four-dimensional generalized symmetric spaces, J. Geom., 2008, 90, 30–46 http://dx.doi.org/10.1007/s00022-008-2046-8
• [9] Calvaruso G., Marinosci R.A., Homogeneous geodesics in five-dimensional generalized symmetric spaces, Balkan J. Geom. Appl., 2002, 8(1), 1–19
• [10] Calviño-Louzao E., García-Río E., Vázquez-Abal M.E., Vázquez-Lorenzo R., Local rigidity and nonrigidity of symplectic pairs, Ann. Global Anal. Geom. (in press), DOI: 10.1007/s10455-011-9279-8
• [11] Černý J., Kowalski O., Classification of generalized symmetric pseudo-Riemannian spaces of dimension n ≤ 4, Tensor (N.S.), 1982, 38, 256–267
• [12] Chaichi M., García-Río E., Matsushita Y., Curvature properties of four-dimensional Walker metrics, Classical Quantum Gravity, 2005, 22(3), 559–577 http://dx.doi.org/10.1088/0264-9381/22/3/008
• [13] Chaichi M., García-Río E., Vázquez-Abal M.E., Three-dimensional Lorentz manifolds admitting a parallel null vector field, J. Phys. A, 2005, 38(4), 841–850 http://dx.doi.org/10.1088/0305-4470/38/4/005
• [14] De Leo B., Marinosci R.A., Homogeneous geodesics of four-dimensional generalized symmetric pseudo-Riemannian spaces, Publ. Math. Debrecen, 2008, 73(3–4), 341–360
• [15] De Leo B., Van der Veken J., Totally geodesic hypersurfaces of four-dimensional generalized symmetric spaces, preprint available at https://perswww.kuleuven.be/~u0043959/Artikels
• [16] Gil-Medrano O., Relationship between volume and energy of vector fields, Differential Geom. Appl., 2001, 15(2), 137–152 http://dx.doi.org/10.1016/S0926-2245(01)00053-5
• [17] Gil-Medrano O., Unit vector fields that are critical points of the volume and of the energy: characterization and examples, In: Complex, Contact and Symmetric Manifolds, Progr. Math., 234, Birkhäuser, Boston, 2005, 165–186 http://dx.doi.org/10.1007/0-8176-4424-5_12
• [18] Gil-Medrano O., Gonzáles-Dávila J.C., Vanhecke L., Harmonic and minimal invariant unit vector fields on homogeneous Riemannian manifolds, Houston J. Math., 2001, 27(2), 377–409
• [19] Gil-Medrano O., Hurtado A., Spacelike energy of timelike unit vector fields on a Lorentzian manifold, J. Geom. Phys., 2004, 51(1), 82–100 http://dx.doi.org/10.1016/j.geomphys.2003.09.008
• [20] Gil-Medrano O., Hurtado A., Volume, energy and generalized energy of unit vector fields on Berger spheres: stability of Hopf vector fields, Proc. Roy. Soc. Edinburgh Sect. A, 2005, 135(4), 789–813 http://dx.doi.org/10.1017/S0308210500004121
• [21] González C., Chinea D., Homogeneous structures on generalized symmetric spaces, In: Proceedings of the 12th Portuguese-Spanish Conference on Mathematics, Vol. II, Braga, 1987, Univ. Minho, Braga, 1987, 572–578 (in Spanish)
• [22] Han D.-S., Yim J.-W., Unit vector fields on spheres, which are harmonic maps, Math. Z., 1998, 227(1), 83–92 http://dx.doi.org/10.1007/PL00004369
• [23] Ishihara T., Harmonic sections of tangent bundles, J. Math. Tokushima Univ., 1979, 13, 23–27
• [24] Komrakov B. Jr., Einstein-Maxwell equation on four-dimensional homogeneous spaces, Lobachevskii J. Math., 2001, 8, 33–165
• [25] Kotschick D., Terzic S., On formality of generalized symmetric spaces, Math. Proc. Cambridge Philos. Soc., 2003, 134(3), 491–505 http://dx.doi.org/10.1017/S0305004102006540
• [26] Kowalski O., Generalized Symmetric Spaces, Lectures Notes in Math., 805, Springer, Berlin-Heidelberg-New York, 1980
• [27] Nouhaud O., Applications harmoniques d’une variété riemannienne dans son fibré tangent. Généralisation, C. R. Acad. Sci. Paris Sér. A-B, 1977, 284(14), A815–A818
• [28] O’Neill B., Semi-Riemannian Geometry, Pure Appl. Math., 103, Academic Press, New York, 1983
• [29] Terzich S., Real cohomology of generalized symmetric spaces, Fundam. Prikl. Mat., 2001, 7(1), 131–157 (in Russian)
• [30] Terzich S., Pontryagin classes of generalized symmetric spaces, Mat. Zametki, 2001, 69(3–4), 613–621 (in Russian)
• [31] Wiegmink G., Total bending of vector fields on Riemannian manifolds, Math. Ann., 1995, 303(2), 325–344 http://dx.doi.org/10.1007/BF01460993
• [32] Wood C.M., On the energy of a unit vector field, Geom. Dedicata, 1997, 64(3), 319–330 http://dx.doi.org/10.1023/A:1017976425512

Identyfikatory

DOI

10.2478/s11533-011-0109-9