Slant Submanifolds of Quaternion Kaehler and HyperKaehler Manifolds

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 .