where Id denotes the identity tensor field of type ( 1, 1 ) on M and the indices are taken from { 1, 2, 3 } modulo 3. Then the bundle σ is called an almost quaternionic structure on M and { J 1, J 2, J 3 } is called a canonical local basis of σ.
Let M be an n-dimensional quaternionic CR-submanifold of a quaternionic space form M ¯ ( c ). Then: (i1) For each unit vector X ∈ D p ⊥, we have Ric ( X ) ≤ ( n − 1 ) c 4 + n 2 4 ∥ H ∥ 2. (2.).
A Riemannian metric g ¯ on M ¯ is said to be adapted to the almost quaternionic structure σ if it satisfies g ¯ ( J α X, J α Y ) = g ¯ ( X, Y ), ∀ α ∈ { 1, 2, 3 }.
Thus a proper slant submanifold M of a quaternionic Kähler manifold ( M ¯, σ, g ¯ ) is said to be a quaternionic slant submanifold if it satisfies the condition ( ∇ ¯ X P α ) Y = ω α + 2 ( X ) P α + 1 Y − ω α + 1 ( X ) P α + 2 Y. for all vector fields X, Y on M ¯, where the indices are taken from { 1, 2, 3 } modulo 3 and P α Y denotes the tangential component of J α Y.
Let ( M ¯, σ, g ¯ ) be a quaternionic Kähler manifold and let X be a non-null vector on M ¯.
If the bundle σ is parallel with respect to the Levi-Civita connection ∇ ¯ of g ¯, then ( M ¯, σ, g ¯ ) is said to be a quaternionic Kähler manifold.
Let ((overline{M},sigma,overline{g})) be a quaternionic Kähler manifold and let X be a non-null vector field on M̅.
A submanifold M of a quaternion Kähler manifold ( M ¯, σ, g ¯ ) is said to be a quaternionic CR-submanifold if there exist two orthogonal complementary distributions D and D ⊥ on M such that D is invariant under quaternionic structure and D ⊥ is totally real (see [40]).
