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Stability of contact metric manifolds and unit vector fields of minimum energy

Published online by Cambridge University Press:  17 April 2009

D. Perrone  and
L. Vergori
D. Perrone
Affiliation:
Dipartimento di Matematica “Ennio De Giorgi”, Università di Lecce, 73100, Lecce, Italy, e-mail: [email protected], [email protected]
L. Vergori
Affiliation:
Dipartimento di Matematica “Ennio De Giorgi”, Università di Lecce, 73100, Lecce, Italy, e-mail: [email protected], [email protected]
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In this paper we obtain criteria of stability for ηEinstein k-contact manifolds, for Sasakian manifolds of constant ϕ-sectional curvature and for 3-dimensional Sasakian manifolds. Moreover, we show that a stable compact Einstein contact metric manifold M is Sasakian if and only if the Reeb vector field ξ minimises the energy functional. In particular, the Reeb vector field of a Sasakian manifold M of constant ϕ-holomorphic sectional curvature +1 minimises the energy functional if and only if M is not simply connected.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 2 , October 2007 , pp. 269 - 283
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Besse, A., Einstein manifolds (Springer-Verlag, Berlin, 1987).CrossRefGoogle Scholar
[2]Blair, D.E., Riemannian geometry of contact and symplectic manifold, Progress in Math. 203(Birkhäuser, Boston, Basel, Berlin, 2002).CrossRefGoogle Scholar
[3]Blair, D.E., Koufogiorgos, T. and Papantoniou, B.J., ‘Contact metric manifolds satisfying a nullity condition’, Israel J. Math. 91 (1995), 189–214.CrossRefGoogle Scholar
[4]Boeckx, E. and Gherge, C., ‘Harmonic maps and cosymplectic manifolds’, J. Austral. Math. Soc. 76 (2004), 75–92.CrossRefGoogle Scholar
[5]Boyer, C.P. and Galicki, K., ‘Einstein manifolds and contact geometry’, Proc. Amer. Math. Soc. 129 (2001), 2419–2430.CrossRefGoogle Scholar
[6]Boyer, C.P., Galicki, K. and Matzeu, P., ‘On eta-Einstein Sasakian geometry’, Comm. Math. Phys. 262 (2006), 177–208.CrossRefGoogle Scholar
[7]Borrelli, V., ‘Stability of the Reeb vector field of a Sasakian manifold’, Soochow J. Math. 30 (2004), 283–292.Google Scholar
[8]Brito, F.B., ‘Total bending of flows with mean curvature correction’, Differential Geom. Appl. 12 (2000), 157–163.CrossRefGoogle Scholar
[9]Chern, S.S. and Hamilton, R.S., ‘On Riemannian metrics adapted to three-dimensional contact manifolds’, in Lect. Notes in Math. 1111 (Springer-Verlag, Berlin, Heidelberg, New York, 1985), pp. 279–305.Google Scholar
[10]Gherghe, C., Ianus, S. and Pastore, A.M., ‘CR-manifolds, harmonic maps and stability’, J. Geom. 71 (2001), 42–53.CrossRefGoogle Scholar
[11]Gil-Medrano, O. and Llinares-Fuster, E., ‘Second variation of volume and energy of vector fields. Stability of Hopf vector fields’, Math. Ann. 320 (2001), 531–545.CrossRefGoogle Scholar
[12]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, (Kowalski, O., Musso, E., Perrone, D., Editors), Progress in Math. 234 (Birkhäuser, Boston, Basel, Berlin, 2005), pp. 165–186.CrossRefGoogle Scholar
[13]Gluck, H. and Ziller, W., ‘On the volume of a unit vector field on the three sphere’, Comment. Math. Helv. 61 (1986), 177–192.CrossRefGoogle Scholar
[14]Ianus, S. and Pastore, A.M., ‘Harmonic maps on contact metric manifolds’, Ann. Math. Blaise Pascal 2 (1995), 43–53.CrossRefGoogle Scholar
[15]Nagano, T., ‘Stability of harmonic maps between symmetric spaces’, in Harmonic Maps, Lecture Notes in Math. 949 (Springer-Verlag, Berlin and New York, 1982).CrossRefGoogle Scholar
[16]Perrone, D., ‘Harmonic characteristic vector fields on contact metric three-manifolds’, Bull. Austral. Math. Soc. 67 (2003), 305–315.CrossRefGoogle Scholar
[17]Perrone, D., ‘Contact metric manifolds whose characteristic vector field is a harmonic vector field’, Differential Geom. Appl. 20 (2004), 367–378.CrossRefGoogle Scholar
[18]Perrone, D., ‘Stability of the Reeb vector field of a H-contact manifold’, (preprint 2006).Google Scholar
[19]Poor, W.A., Differential geometric structures (McGraw Hill, New York, 1981).Google Scholar
[20]Smith, R.T., ‘The second variation formula for harmonic mappings’, Proc. Amer. Math. Soc. 47 (1975), 229–236.CrossRefGoogle Scholar
[21]Takahashi, J., ‘On the gap between the first eigenvalue of the Laplacian on functions and 1-forms’, J. Math. Soc. Japan 53 (2001), 307–319.CrossRefGoogle Scholar
[22]Tanno, S., ‘Sasakian manifolds with constant φ-holomorphic sectional curvature’, Tohoku Math. J. 16 (1993), 171–180.Google Scholar
[23]Urakawa, H., ‘The first eigenvalue of the Laplacian for a positively curved homogeneous Riemannian maniolds’, Compositio Math. 59 (1986), 57–71.Google Scholar
[24]Urakawa, H., ‘Stability of harmonic maps and eigenvalues of the Laplacian’, Trans. Amer. Math. Soc. 301 (1987), 557–589.CrossRefGoogle Scholar
[25]Urakawa, H., Calculus of variations and harmonic maps, Transl. Math. Monogr. 132 (Amer. Math. Soc., Providence, R.I., 1993).Google Scholar
[26]Wiegmink, G., ‘Total bending of vector fields on Riemannian manifolds’, Math. Ann. 303 (1995), 325–344.CrossRefGoogle Scholar
[27]Wiegmink, G., ‘Total bending of vector fields on the sphere S3’, Differential Geom. Appl. 6 (1996), 219–236.CrossRefGoogle Scholar
[28]Wolf, J.A., ‘A contact structure for odd dimensional spherical spaces forms’, Proc. Amer. Math. Soc. 19 (1968), 196.Google Scholar
[29]Wood, C.M., ‘On the energy of a unit vector field’, Geom. Dedicata 64 (1997), 319–330.CrossRefGoogle Scholar
[30]Yano, K. and Nagano, T., ‘On geodesic vector fields in a compact orientable Riemannian space’, Comm. Math. Helv. 35 (1961), 55–64.CrossRefGoogle Scholar