Slant Submanifolds of Quaternion Kaehler and HyperKaehler Manifolds

Mohammad Hasan Shahid (), Falleh R. Al-Solamy () and Mohammad Jamali ()
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Mohammad Hasan Shahid: Jamia Millia Islamia, Department of Mathematics, Faculty of Natural Sciences
Falleh R. Al-Solamy: King Abdul Aziz University, Department of Mathematics
Mohammad Jamali: Al-Falah University, Department of Mathematics

A chapter in Complex Geometry of Slant Submanifolds, 2022, pp 251-280 from Springer

Abstract: Abstract The aim of this chapter is to discuss and survey briefly some results on slant submanifolds of quaternion Kaehler and hyperKaehler manifolds. For this, let $$\overline{M}$$ M ¯ be a 4m-dimensional Riemannian manifold with metric tensor g. Then $$\overline{M}$$ M ¯ is said to be a quaternion Kaehler manifold if there exists a three-dimensional vector bundle E consisting of tensors of type (1,1) with local basis of almost Hermitian structures $$J_{1}, J_{2}, J_{3}$$ J 1 , J 2 , J 3 such that (a) $$J_{1}^{2}=-I,J_{2}^{2}=-I,J_{3}^{2}=-I$$ J 1 2 = - I , J 2 2 = - I , J 3 2 = - I (b) $$J_{1}J_{2}=-J_{2}J_{1}=J_{3},J_{2}J_{3}=-J_{3}J_{2}=J_{1},J_{3}J_{1}=-J_{1}J_{3}=J_{2}$$ J 1 J 2 = - J 2 J 1 = J 3 , J 2 J 3 = - J 3 J 2 = J 1 , J 3 J 1 = - J 1 J 3 = J 2 where I is the identity tensor of type (1,1) on $$\overline{M}$$ M ¯ . (c) $$\overline{\nabla }_{X}J_{a}=\sum _{b=1}^{3}Q_{ab}(X)J_{b},a=1,2,3$$ ∇ ¯ X J a = ∑ b = 1 3 Q ab ( X ) J b , a = 1 , 2 , 3 for all vector fields X tangent to $$\overline{M}$$ M ¯ , where $$\overline{\nabla }$$ ∇ ¯ denotes the Riemannian connection in $$\overline{M}$$ M ¯ and $$Q_{ab}$$ Q ab are 1-forms defined locally on $$\overline{M}$$ M ¯ such that $$Q_{ab}+Q_{ba}=0.$$ Q ab + Q ba = 0 .

Keywords: 53C15; 53C25; 53C40 (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-16-0021-0_9

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DOI: 10.1007/978-981-16-0021-0_9

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