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On pseudo symmetric manifolds. (English) Zbl 0626.53037
An n-dimensional pseudo symmetric manifold (denoted by \((PS)_ n)\) is defined as a nonflat Riemannian manifold \((M^ n,g)\), \(n\geq 2\), whose curvature tensor R satisfies the condition: \[ (\nabla_ XR)(Y,Z)W=2A(X)R(Y,Z)W+A(Y)R(X,Z)W+A(Z)R(Y,X)W+A(W)R(Y,Z)X+g(R(Y,Z)W,X)\rho, \] for all X,Y,Z,W\(\in {\mathcal X}(M)\), where A is a nonzero 1-form, \(g(X,\rho)=A(X)\), for all \(X\in {\mathcal X}(M)\). When A is zero, M becomes a symmetric manifold. For \(n=2\), it is shown that A is closed. For \(n>2\), a \((PS)_ n\) cannot be of constant curvature, since an Einstein \((PS)_ n\) \((n>2)\) is of zero scalar curvature. Next, a \((PS)_ n\) with Ricci tensor of Codazzi type is studied. Under a certain condition, a simply connected conformally flat \((PS)_ n\) (n\(\geq 3)\) can be isometrically immersed in an Euclidean space \(R^{n+1}\).
Reviewer: C.L.Bejan

53C35 Differential geometry of symmetric spaces
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)