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Remarks on some almost Hermitian structure on the tangent bundle. II. (English) Zbl 1306.53019

Let \(M\) and \(TM\) be an almost Hermitian manifold and its tangent bundle respectively. M. Tahara et al. [Note Mat. 18, No. 1, 131–141 (1998; Zbl 0964.53021)] constructed three almost complex structures \(J_1,J_2,J_3\) on \(TM\), which define in special cases an almost hyper-complex structure. They also found a Riemannian metric \(G_1\) such that \((J_1,G_1)\) is an almost Hermitian structure on \(TM\).
Here, the authors construct Riemannian metrics \(G_2\) and \(G_3\) on \(TM\) such that \((J_2,G_2)\) and \((J_3,G_3)\) are almost Hermitian structures on \(TM\). For these structures \((J_i,G_i)\), \(i=1,2,3\), they find conditions such that \((TM,J_i,G_i)\) belongs to the classes constructed by A. Gray and L. M. Hervella [Ann. Mat. Pura Appl. (4) 123, 35–58 (1980; Zbl 0444.53032)].

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C56 Other complex differential geometry
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Full Text: Euclid

References:

[1] P. Dombrowski, On the geometry of the tangent bundle , J. Reine Angew. Math. 210 (1962), 73-88. · Zbl 0105.16002 · doi:10.1515/crll.1962.210.73
[2] A. Gray and L.M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants , Ann. Mat. Pura Appl. 123 (1980), 35-58. · Zbl 0444.53032 · doi:10.1007/BF01796539
[3] T. Koike, T. Oguro and N. Watanabe, Remarks on some almost Hermitian structure on the tangent bundle , Nihonkai Math. J. 20 (2009), 25-32. · Zbl 1179.53017
[4] M. Tahara, L. Vanhecke and Y. Watanabe, New structures on tangent bundles , Note di Mathematica 18 (1998), 131-141. · Zbl 0964.53021
[5] M. Tahara and Y. Watanabe, Natural almost Hermitian, Hermitian and Kähler metrics on the tangent bundles , Math. J. Toyama Univ. 20 (1997), 149-160. · Zbl 1076.53516
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