On the stability of \(T\)-space forms. (English) Zbl 1539.53081
A Riemannian manifold is called stable if the identity map \(1_{M}\) is stable as a harmonic self-map, i.e., its second variation is non-negative. An \(f\)-structure on a Riemannian manifold is a reduction of its holonomy to \(\mathrm{U}(n)\times\mathrm{O}(s)\). According to [D. E. Blair, J. Differ. Geom. 4, 155–167 (1970; Zbl 0202.20903)], there are three important classes of \(f\)-structures, called \(K\)-, \(S\)- and \(T\)-manifolds.
The main purpose of this paper is to investigate the stability property of \(T\)-space forms, i.e., \(T\)-manifolds of constant \(\varphi\)-sectional curvature. The main results of the paper are as follows.
Theorem. Any compact \(T\)-space form \(M\) of constant \(\varphi\)-sectional curvature \(c\geq0\) is stable.
Theorem. Let \(M\) be a \((2n+s)\)-dimensional \(T\)-space form of constant \(\varphi\)-sectional curvature \(c\leq0\). If the first eigenvalue \(\lambda_{1}\) of the Laplace-Beltrami operator satisfies \[ \lambda_{1}<-c(n+1), \] then \(1_{M}\) is unstable.
The main purpose of this paper is to investigate the stability property of \(T\)-space forms, i.e., \(T\)-manifolds of constant \(\varphi\)-sectional curvature. The main results of the paper are as follows.
Theorem. Any compact \(T\)-space form \(M\) of constant \(\varphi\)-sectional curvature \(c\geq0\) is stable.
Theorem. Let \(M\) be a \((2n+s)\)-dimensional \(T\)-space form of constant \(\varphi\)-sectional curvature \(c\leq0\). If the first eigenvalue \(\lambda_{1}\) of the Laplace-Beltrami operator satisfies \[ \lambda_{1}<-c(n+1), \] then \(1_{M}\) is unstable.
Reviewer: Shaolin Chen (Hengyang)
Citations:
Zbl 0202.20903References:
| [1] | Akyol, M. A.; Şahin, B., Conformal semi-invariant submersions, Commun. Contemp. Math., 19, 2, Article 1650011 pp., 2017 · Zbl 1361.53023 |
| [2] | Aktan, N.; Sariikaya, M. Z.; Özüsaǧlam, E., B. Y. Chen’s inequality for semi-slant submanifolds in T-space forms, Balk. J. Geom. Appl., 13, 1, 1-10, 2008 · Zbl 1158.53010 |
| [3] | Al-Jedani, A.; Al-Ghefari, R., CR-product of T-manifold, Far East J. Math. Sci., 38, 1, 39-48, 2010 · Zbl 1186.53023 |
| [4] | Ali, A.; Othman, W. A.M., Geometric aspects of CR-warped product submanifolds of T-manifolds, Asian-Eur. J. Math., 10, 4, Article 1750067 pp., 2017 · Zbl 1378.53065 |
| [5] | Ali, R.; Ali, A.; Khan, K.; Alkhaldi, A. H., Applications of Hopf’s lemma in contact CR-warped products of T-space forms, C. R. Acad. Bulgare Sci., 72, 12, 1616-1625, 2019 · Zbl 1463.53071 |
| [6] | Aquib, M., Bounds for generalized normalized δ-Casorati curvatures for bi-slant submanifolds in T-space forms, Filomat, 32, 1, 329-340, 2018 · Zbl 1488.53029 |
| [7] | Baird, P.; Wood, J., Harmonic Morphisms between Riemannian manifolds, 2003, Oxford University Press: Oxford University Press Oxford · Zbl 1055.53049 |
| [8] | Blair, D. E., Geometry of manifolds with structural group \(U(n) \times O(s)\), J. Differ. Geom., 4, 155-167, 1970 · Zbl 0202.20903 |
| [9] | Boeckx, E.; Gherghe, C., Harmonic maps and cosymplectic manifolds, J. Aust. Math. Soc., 76, 1, 75-92, 2004 · Zbl 1063.53069 |
| [10] | Burns, D.; Burstall, F.; De Bartolomeis, P.; Rawnsley, J., Stability of harmonic maps of Kähler manifolds, J. Differ. Geom., 30, 2, 579-594, 1989 · Zbl 0678.53062 |
| [11] | Călin, C., CR-submanifolds of a T-manifold, Math. J. Toyama Univ., 25, 53-63, 2002 · Zbl 1035.53076 |
| [12] | Chen, B.-Y., Differential geometry of identity maps: a survey, Mathematics, 2020, 8, 1264, 2020 |
| [13] | De, U. C.; Matsuyama, Y.; Sengupta, A. K., Generalized CR-submanifolds of a T-manifold, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math., 11, 3, 175-187, 2004 · Zbl 1140.53304 |
| [14] | Gherghe, C., Harmonic maps and stability on locally conformal Kähler manifolds, J. Geom. Phys., 70, 48-53, 2013 · Zbl 1280.53065 |
| [15] | Gherghe, C.; Ianus, S.; Pastore, A. M., CR-manifolds, harmonic maps and stability, J. Geom., 71, 1-2, 42-53, 2001 · Zbl 1038.58014 |
| [16] | Goldberg, S.; Yano, K., Globally framed f-manifolds, Ill. J. Math., 15, 456-474, 1971 · Zbl 0215.23002 |
| [17] | Ianuş, S.; Mazzocco, R.; Vîlcu, G.-E., Harmonic maps between quaternionic Kähler manifolds, J. Nonlinear Math. Phys., 15, 1, 1-8, 2008 · Zbl 1165.53353 |
| [18] | Kaushal, R.; Gupta, G.; Sachdeva, R.; Kumar, R., Conformal slant Riemannian maps with totally umbilical fibers, Mediterr. J. Math., 20, 44, 2023 · Zbl 1518.53031 |
| [19] | Khan, V. A.; Khan, M. A., Semi-slant submanifolds of T-manifolds, Demonstr. Math., 39, 4, 907-918, 2006 · Zbl 1118.53014 |
| [20] | Kobayashi, M.; Tsuchiya, S., Invariant submanifolds of an f-manifold with complemented frames, Kōdai Math. Semin. Rep., 24, 430-450, 1972 · Zbl 0246.53038 |
| [21] | Kobayashi, M., Semi-invariant submanifolds in an f-manifold with complemented frames, Tensor (N.S.), 49, 2, 154-177, 1990 · Zbl 0738.53024 |
| [22] | Kobayashi, M., Totally umbilical submanifolds and extrinsic spheres in T-manifolds, Tensor (N.S.), 60, 2, 189-194, 1998 · Zbl 1040.53073 |
| [23] | Mazet, E., La formule de la variation secondere de l’energie au voisinage d’un application harmonique, J. Differ. Geom., 8, 279-296, 1973 · Zbl 0301.53012 |
| [24] | Misner, C. W., Harmonic maps as models for physical theories, Phys. Rev. D, 18, 4510, 1978 |
| [25] | Perrone, D.; Vergori, L., Stability of contact metric manifolds and unit vector fields of minimum energy, Bull. Aust. Math. Soc., 76, 2, 269-283, 2007 · Zbl 1135.53042 |
| [26] | Rehman, N. A., Stability on generalized Sasakian space forms, Math. Rep., 17(67), 1, 57-64, 2015 · Zbl 1374.53112 |
| [27] | Rehman, N. A., Stability on S-space form, Indian J. Pure Appl. Math., 50, 4, 1087-1096, 2019 · Zbl 1444.53041 |
| [28] | Şahin, B., Riemannian submersions, Riemannian maps in Hermitian geometry, and their applications, 2017, Elsevier/Academic Press: Elsevier/Academic Press London · Zbl 1378.53003 |
| [29] | Schimming, R.; Hirschmann, T., Harmonic maps from spacetimes and their coupling to gravitation, Astron. Nachr., 309, 5, 311-321, 1998 · Zbl 0665.58050 |
| [30] | Smith, R. T., The second variation formula for harmonic mappings, Proc. Am. Math. Soc., 47, 229-236, 1975 · Zbl 0303.58008 |
| [31] | Tanveer, S., Singularities in the classical Rayleigh-Taylor flow: formation and subsequent motion, Proc. R. Soc. A, Math. Phys. Eng. Sci., 441, 501-525, 1993 · Zbl 0789.76031 |
| [32] | Vîlcu, G.-E., Horizontally conformal submersions from CR-submanifolds of locally conformal Kähler manifolds, Mediterr. J. Math., 17, 26, 2020 · Zbl 1433.53051 |
| [33] | Vîlcu, G.-E.; Gherghe, C., Harmonic maps on locally conformal almost cosymplectic manifolds, Commun. Contemp. Math., 2024, in press · Zbl 07900926 |
| [34] | Yano, K., On a structure defined by a tensor field f of type \((1, 1)\) satisfying \(f^3 + f = 0\), Tensor (N.S.), 14, 99-109, 1963 · Zbl 0122.40705 |
| [35] | Wani, T. A.; Lone, M. A., Horizontally conformal submersions from CR-submanifolds of locally conformal quaternionic Kaehler manifolds, Mediterr. J. Math., 19, 114, 2022 · Zbl 1492.53078 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
