Pseudo-symmetric contact \(3\)-manifolds. II. When is the tangent sphere bundle over a surface pseudo-symmetric? (English) Zbl 1150.53006

A Riemannian manifold \((M,g)\) is said to be pseudo-symmetric if there exists a function \(L\) such that \(R(X,Y)\cdot R = L \left\{ (X\wedge Y)\cdot R\right\}\) for all vector fields \(X,Y\) on \(M\), where \(R\) is the curvature tensor and \(X\wedge Y\) denotes the endomorphism field defined by \(X\wedge Y=g(Y,\, )X- g(X,\, )Y\). A pseudo-symmetric space with \(L=0\) is said to be semi-symmetric. G. Calvaruso and D. Perrone [Yokohama Math. J. 49, No. 2, 149–161 (2002; Zbl 1047.53017)] proved that: if \(\dim M=2\), then the unit tangent sphere bundle \(T^1M\), equipped with its standard contact metric, is semi-symmetric if and only if \(M\) is flat or has Gaussian curvature \(1\). In the present paper, the authors prove the following result: if \(\dim M=2\), then the tangent sphere bundle \(T^rM\) of radius \(r\), equipped with the Sasaki metric, is pseudo-symmetric if and only if \(M\) has constant Gaussian curvature.
Reviewer: D. Perrone (Lecce)


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)


Zbl 1047.53017