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Contact metric manifolds satisfying a nullity condition. (English) Zbl 0837.53038

The authors study contact metric manifolds \(M^{2n+ 1}(\varphi, \xi, \eta, g)\) [for definitions see D. E. Blair, Contact manifolds in Riemannian geometry, Lect. Notes Math. 509, Springer (1976; Zbl 0319.53026)] for which the characteristic vector field \(\xi\) belongs to the so-called \((\kappa, \mu)\)-nullity distribution. This means that the curvature operator \(R(X, Y)\) of the manifold satisfies the condition \[ R(X, Y)\xi= \kappa (\eta(Y)X- \eta(X) Y)+ \mu(\eta(Y) hX- \eta(X) hY),\tag{\(*\)} \] where \(\kappa\), \(\mu\) are constants and \(2h\) is the Lie derivative of \(\varphi\) in the direction \(\xi\). It is proved that for \(\kappa< 1\), the curvature \(R\) is completely determined for such manifolds; in particular, they have constant scalar curvature. In the case of \(\dim M= 3\), contact metric manifolds satisfying \((*)\) are either Sasakian or locally isometric to one of the following Lie groups: \(\text{SO}(3)\), \(\text{SL}(2, R)\), \(E(2)\), \(E(1, 1)\) with a left invariant metric. The standard contact metric structure of the tangent sphere bundle \(T_1 M\) satisfies \((*)\) if and only if the base manifold is of constant sectional curvature.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

Citations:

Zbl 0319.53026
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References:

[1] C. Baikoussis, D. E. Blair and T. Koufogiorgos,A decomposition of the curvature tensor of a contact manifold satisfying R(X, Y){\(\zeta\)}={\(\kappa\)}({\(\eta\)}(Y)X-{\(\eta\)}(X)Y), Mathematics Technical Report, University of Ioannina, No 204, June 1992.
[2] D. E. Blair,Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics509, Springer-Verlag, Berlin, 1976. · Zbl 0319.53026
[3] D. E. Blair,Two remarks on contact metric structures, The Tôhoku Mathematical Journal29 (1977), 319–324. · Zbl 0376.53021
[4] D. E. Blair and J. N. Patnaik,Contact manifolds with characteristic vector field annihilated by the curvature, Bulletin of the Institute of Mathematics. Academia Sinica9 (1981), 533–545. · Zbl 0498.53029
[5] D. E. Blair,When is the tangent sphere bundle locally symmetric?, inGeometry and Topology, World Scientific, Singapore, 1989, pp. 15–30.
[6] D. E. Blair, T. Koufogiorgos and R. Sharma,A classification of 3-dimensional contact metric manifolds with Q{\(\phi\)}={\(\phi\)}Q, Kodai Mathematical Journal,13 (1990), 391–401. · Zbl 0716.53041
[7] D. E. Blair and H. Chen,A classification of 3-dimensional contact metric manifolds with Q{\(\phi\)}={\(\phi\)}Q, II, Bulletin of the Institute of Mathematics. Academia Sinica20 (1992), 379–383.
[8] O. Kowalski,Curvature of the induced Riemannian metric on the tangent bundle, Journal für die Reine und Angewandte Mathematik250 (1971), 124–129. · Zbl 0222.53044
[9] J. Milnor,Curvature of left invariant metrics on Lie groups, Advances in Mathematics21 (1976), 293–329. · Zbl 0341.53030
[10] Z. Olszak,On contact metric manifolds, The Tôhoku Mathematical Journal31 (1979), 247–253. · Zbl 0403.53018
[11] S. Tanno,The topology of contact Riemannian manifolds, Illinois Journal of Mathematics12 (1968), 700–717. · Zbl 0165.24703
[12] S. Tanno,Isometric immersions of Sasakian manifolds in spheres, Kodai Mathematical Seminar Reports21 (1969), 448–458. · Zbl 0196.25501
[13] S. Tanno,Ricci curvatures of contact Riemannian manifolds, The Tôhoku Mathematical Journal40 (1988), 441–448. · Zbl 0655.53035
[14] F. Trikerri and L. Vanhecke,Homogeneous structure on Riemannian manifolds, London Mathematical Society Lecture Note Series, 83, Cambridge Univ. Press, London, 1983.
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