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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

##### MSC:
 53C35 Differential geometry of symmetric spaces 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
##### Keywords:
Einstein manifold; pseudo symmetric manifold; Ricci tensor