28
Barnali Laha and Arindam Bhattacharyya
Corollary 4.1 The distribution D1 ⊕ D2 on a hemislant submanifold of an (ǫ)-para Sasakian manifold M̃ is integrable. Corollary 4.2 The distribution D1 and D2 on a hemislant submanifold of an (ǫ)-para Sasakian manifold M̃ is integrable. Proposition 4.1 Let M be a hemislant submanifold of an (ǫ)-para Sasakian manifold M̃ , then for any Z, W ∈ D2 , the anti-invariant distribution D2 ⊕ < ξ > is integrable iff ⊥ AφZ W − AφW Z + ∇⊥ Z φW − ∇W φZ − ǫη(W )Z + ǫη(Z)W = 0.
Proof For any Z, W ∈ D2
L
ξ we know
˜ Z φW = (∇ ˜ Z φW ) + φ∇ ˜ Z W = (∇ ˜ Z φW ) + φ∇Z W + φh(Z, W ). ∇
(4.6)
Using (2.16) and (2.17) we have ˜ ˜ ˜ −AφW Z + ∇⊥ Z φW = (∇Z φW ) + φ∇Z W = (∇Z φW ) + φ∇Z W + φh(Z, W ).
(4.6)
Interchanging Z and W , we obtain ˜ ˜ ˜ −AφZ W + ∇⊥ W φZ = (∇W φZ) + φ∇W Z = (∇W φZ) + φ∇W Z + φh(W, Z).
(4.8)
Then from (4.7) and (4.8) we calculate ⊥ ˜ ˜ AφZ W − AφW Z + ∇⊥ Z φW − ∇W φZ = (∇Z φW ) − (∇W φZ) + φ[Z, W ].
(4.9)
From (2.15) we obtain ⊥ 2 2 AφZ W − AφW Z + ∇⊥ Z φW − ∇W φZ = φ[Z, W ] − ǫη(W )φ Z + ǫη(Z)φ W.
(4.10)
Taking inner product with φX, for any X ∈ D1 we obtain ⊥ g(AφZ W − AφW Z + ∇⊥ Z φW − ∇W φZ, φX)
= g(φ[Z, W ], φX) − ǫη(W )g(φ2 Z, φX) + ǫη(Z)g(φ2 W, φX).
(4.11)
Thus from (2.5) the above equation takes the form g(φ[Z, W ], φX)
⊥ = g(AφZ W − AφW Z + ∇⊥ Z φW − ∇W φZ − ǫη(W )Z
(4.12)
+ǫη(Z)W, φX). The distribution D2 zero.
L
< ξ > is integrable iff the right hand side of the above equation is