Contact Riemannian manifolds with Killing structure operators
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In contact geometry, the structure operator $h=\frac{1}{2}\mathcal{L}_\xi \phi$ and the characteristic Jacobi operator $\ell=R(\cdot,\xi)\xi$ have a fundamental role. In this paper, we find the necessary and sufficient conditions that the structure operator $h$ and the characteristic Jacobi operator $\ell$ are Killing for the Levi-Civita connection or the generalized Tanaka-Webster connection. Moreover, we determine the base manifold of the unit tangent sphere bundle whose structure operators are Killing for two connections, respectively.
Abstract
In contact geometry, the structure operator $h=\frac{1}{2}\mathcal{L}_\xi \phi$ and the characteristic Jacobi operator $\ell=R(\cdot,\xi)\xi$ have a fundamental role. In this paper, we find the necessary and sufficient conditions that the structure operator $h$ and the characteristic Jacobi operator $\ell$ are Killing for the Levi-Civita connection or the generalized Tanaka-Webster connection. Moreover, we determine the base manifold of the unit tangent sphere bundle whose structure operators are Killing for two connections, respectively.