Tangent sphere bundles satisfying \(\nabla_ \xi \tau=0\)
Let \((N,G)\) be a Riemannian manifold and denote by \(T_ 1 N\) the unit tangent sphere bundle of it. Then \(T_ 1 N\) is endowed with the standard contact metric structure which we denote by \((\omega, g,\xi)\). Further, let \(\tau\) denote the Lie derivative of \(g\) with respect to the characteristic field \(\xi\). The main theorem of the paper states that \(\nabla_ \xi \tau=0\) on \((T_ 1 N,g)\) with respect to the Levi Civita connection \(\nabla\) if and only if \((N,g)\) has constant sectional curvature 0 or 1. The author also derives several equivalent conditions. They yield another proof of D. E. Blair's result: \((T_ 1 N,g)\) is locally symmetric if and only if \((N,G)\) is flat or two-dimensional and of constant sectional curvature 1.
scientific article; zbMATH DE number 558741
scientific article