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On some pseudo-symmetric Riemann spaces. (English) Zbl 1190.53012

A Riemannian manifold \((M,g)\) with curvature \(R\) is said to be pseudo-symmetric if at every point of \(M\) the tensor \(R\cdot R\) and the Tachibana tensor are linearly dependent. This concept, which generalizes that of a semisymmetric space, arose during the study of totally umbilical submanifolds in semi-symmetric spaces and in the study of subgeodesic mappings.
In this paper the author studies conharmonic semi-symmetric spaces which are geodesically related.
A Riemannian manifold \((M,g)\), \(\dim M= n\geq 3\), is called conharmonic semi-symmetric if \(R\cdot C= 0\), \(C\) denoting the conharmonic tensor, which is defined by means of the Ricci operator \(A\) according to the formula: \[ C(X,Y, Z)= R(X,Y, Z)-{1\over n-2} (g (Y,Z)AX- g(AX, Z)Y). \] In particular, one considers two Riemannian metrics \(g\), \(\overline g\) on \(M\) which are non-trivially geodesically related. The author proves that, if \((M,\overline g)\) is conharmonic semi-symmetric, then either \(g\), \(\overline g\) have constant sectional curvature or \(g\), \(\overline g\) are special geodesically related. Moreover, both metrics \(g\), \(\overline g\) have constant sectional curvature if \((M,g)\) has irreducible curvature.
Finally, the author finds suitable conditions of pseudo-symmetric type for sub-geodesically related spaces.

MSC:

53B20 Local Riemannian geometry
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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