On weakly pseudo quasi-conformally symmetric manifolds.

*(English)*Zbl 1167.53023A non-flat Riemannian manifold (\(M^n,g)\), \(n>2\), is called a weakly symmetric manifold if its curvature tensor \(R\) satisfies the condition

\[ \begin{split} (\nabla _XR)(Y,Z,U,V)=A(X)R(Y,Z,U,V)+B(Y)R(X,Z,U,V)+\\ +H(Z)R(Y,X,U,V)+D(U)R(Y,Z,X,V)+E(V)R(Y,Z,U,V), \end{split} \] where \(A, B, H, D, E\) are 1-forms non-zero simultaneously. Such a manifold is denoted by \((WS)_n.\)

The aim of this paper is to study non-flat Riemannian manifolds (\(M^n,g)\), \(n>3\), whose quasi-conformal curvature tensor \(W\) is not identically zero and satisfies a similar condition. Such a manifold is called a weakly pseudo quasi-conformally symmetric manifold, denoted \((WPQCS)_n.\)

The quasi-conformal curvature tensor curvature tensor [A. A. Shaikh and S. K. Jana, South East Asian J. Math. Math. Sci. 4, No. 1, 15–20 (2005; Zbl 1116.53016)] is defined by

\[ \begin{split} W(X,Y,Z,U)=(p+d)R(X,Y,Z,U)+\big(q-\tfrac{d}{ n-1}\big) [S(Y,Z)g(X,U)- S(X,Z)g(Y,U)]+\\ +q[S(X,U)g(Y,Z)-S(Y,U)g(X,Z)]- \tfrac{r}{n(n-1)} [p+2(n-1)q][g(Y,Z)g(X,U)-g(X,Z)g(Y,U)],\end{split} \] where \(R\) is the curvature tensor of type (0,4), \(S\) is the Ricci tensor, \(r\) is the scalar curvature and \(p, q, d\) are arbitrary constants non-zero simultaneously. This notion includes the conformal, concircular and projective curvature tensor as special cases.

In the present paper the nature of the scalar curvature of \((WPQCS)_n\) is studied. Every \((WS)_n\) is a \((WPQCS)_n\) but not conversely. A sufficient condition for a \((WPQCS)_n\) to be \((WS)_n\) is obtained. Moreover a sufficient condition for an Einstein \((WPQCS)_n\) to be \((WS)_n\) is determined. Finally the paper deals with several non-trivial examples of \((WPQCS)_n\).

\[ \begin{split} (\nabla _XR)(Y,Z,U,V)=A(X)R(Y,Z,U,V)+B(Y)R(X,Z,U,V)+\\ +H(Z)R(Y,X,U,V)+D(U)R(Y,Z,X,V)+E(V)R(Y,Z,U,V), \end{split} \] where \(A, B, H, D, E\) are 1-forms non-zero simultaneously. Such a manifold is denoted by \((WS)_n.\)

The aim of this paper is to study non-flat Riemannian manifolds (\(M^n,g)\), \(n>3\), whose quasi-conformal curvature tensor \(W\) is not identically zero and satisfies a similar condition. Such a manifold is called a weakly pseudo quasi-conformally symmetric manifold, denoted \((WPQCS)_n.\)

The quasi-conformal curvature tensor curvature tensor [A. A. Shaikh and S. K. Jana, South East Asian J. Math. Math. Sci. 4, No. 1, 15–20 (2005; Zbl 1116.53016)] is defined by

\[ \begin{split} W(X,Y,Z,U)=(p+d)R(X,Y,Z,U)+\big(q-\tfrac{d}{ n-1}\big) [S(Y,Z)g(X,U)- S(X,Z)g(Y,U)]+\\ +q[S(X,U)g(Y,Z)-S(Y,U)g(X,Z)]- \tfrac{r}{n(n-1)} [p+2(n-1)q][g(Y,Z)g(X,U)-g(X,Z)g(Y,U)],\end{split} \] where \(R\) is the curvature tensor of type (0,4), \(S\) is the Ricci tensor, \(r\) is the scalar curvature and \(p, q, d\) are arbitrary constants non-zero simultaneously. This notion includes the conformal, concircular and projective curvature tensor as special cases.

In the present paper the nature of the scalar curvature of \((WPQCS)_n\) is studied. Every \((WS)_n\) is a \((WPQCS)_n\) but not conversely. A sufficient condition for a \((WPQCS)_n\) to be \((WS)_n\) is obtained. Moreover a sufficient condition for an Einstein \((WPQCS)_n\) to be \((WS)_n\) is determined. Finally the paper deals with several non-trivial examples of \((WPQCS)_n\).

Reviewer: Iulia Hirică (Bucureşti)

##### MSC:

53B20 | Local Riemannian geometry |

53B05 | Linear and affine connections |

53B35 | Local differential geometry of Hermitian and Kählerian structures |