Unit tangent sphere boundles and two-point homogeneous spaces (Q1307329)

Consider a Riemannian manifold \((M,G)\) and its unit tangent bundle \(T_1 M\) endowed with its standard contact metric structure \((\omega,g,\phi,\xi)\). It will be shown that \((M,G)\) is locally isomorphic to a two-point homogeneous space iff \((T_1M,\omega,g)\) satisfies \(\nabla_\xi h=2ah\phi +2b\phi S\), where \(h=\frac 12 \mathcal L_\xi\phi\), \(a\) and \(b\) are functions on \(T_1M\) which are constant on the fibers, and \(S\) acts as the identity on horizontal vectors and as minus the identity on vertical vectors. If \(\dim M\) is odd, then \(\nabla_\xi h=2a h\phi\) characterizes the geometric properties: (1) \((T_1M,\omega,g)\) is locally \(\varphi\)-symmetric; (2) the associated CR-structure of \((T_1M,\omega,g)\) is integrable. In the last section, unit tangent sphere bundles are considered which are homogeneous contact metric manifolds. It is shown that under various conditions on the manifold \((M,G)\), \((M,G)\) is necessarily isometric to a two-point homogeneous space.