JOURNAL Series A - Pure and Applied Mathematics

Some properties of one connection on spaces with an almost product structure 23Let us assume that ∇ i J k j= 0. Then the above identity implies∇ i J k j = g ij (ã k − b k ) + J ij (˜b k − a k ) + 1 2 δk i (b j − ã j ) + 1 2 J k i (a j − ˜b j ). (5)In (5) we contract with k = i and we obtain∇ i J i j = n 2 (b j − ã j ) + trJ2 (a j − ˜b j ). (6)The connection ∇ satisfies equation like (3), thus we have ∇ i Jk iequality and (6) we get the system:= 0. From the lastn(b j − ã j ) + trJ(a j − ˜b j ) = 0.n(˜b j − a j ) + trJ(ã j − b j ) = 0.The only solution of this system is ˜b j = a j , ã j = b j . After substituting the lastresult in (5), we obtain ∇ i Jj k = 0.3 The case a = 0Let M be in the class F AP , also ∇ and ∇ satisfy (4). If a kã k = 0), then (4) has the form:= 0 (consequentlyΓ k ij = Γ k ij + T kij,T kij = J ij b k − 1 2 δk i ˜b j − 1 2 δk j ˜b i . (7)For the curvature tensor fields R of ∇ and R of ∇ it is well known the identity:From (7) and (8) we obtainR h ijk = R h ijk + ∇ j T hik − ∇ k T hij + T sikT h sj − T sijT h sk. (8)R h ijk = R h ijk + J ik (∇ j b h + ˜b j b h − 1 2 δh j ˜b s b s )− J ij (∇ k b h + ˜b k b h − 1 2 δh k˜b s b s )− 1 2 δh i (∇ j˜bk − ∇ k˜bj ) − 1 4 δh k(2∇ j˜bi + ˜b i˜bj )(9)+ 1 4 δh j (2∇ k˜bi + ˜b i˜bk ) − b h (∇ k J ij − ∇ j J ik ).Let M be in the class F AP , also ∇ and ∇ satisfy (7). Then ∇ is an equiaffineconnection, if and only if, the vector field ˜b is gradient.