Curvature of a hyperdistribution and contact metric manifolds
Let \(\xi\) be a unit vector field on a Riemannian manifold M and \(\Delta\) the codimension one distribution orthogonal to \(\xi\) (which is non- integrable in general). The author studies properties of the second fundamental form, the sectional torsion and the curvature of \(\Delta\) in the sense stated by himself in [Mat. Zametki 31, 139-146 (1982; Zbl 0492.53028), 35, 111-124 (1984; Zbl 0553.53017)]. In particular, the Gauss and Codazzi equations for \(\Delta\) are obtained. Next, applying the above theory to contact metric manifolds, he proves among others the following: Let M be a Sasakian manifold, dim \(M\geq 5\), and (\(\phi\),\(\xi\),\(\eta\),g) its Sasakian structure. If the sectional curvatures \(K_{x\phi x}\), \(x\in \Delta_ p\), of its contact distribution \(\Delta\) depend only on points \(p\in M\), then 1/\(\Delta\) has constant \(\phi\)-sectional curvature and its curvature tensor is of the form \[ K(X,Y)=\lambda /4\{X\wedge Y+\phi X\wedge \phi Y+2<X,\phi Y>\phi \}, \] where X,Y\(\in \Gamma (\Delta)\) and \(\lambda =const.\), 2/\(\Delta\) has 1/4-bounded curvature for \(\lambda\) \(\neq 0\) and is of zero curvature if \(\lambda =0\). This is a non-holonomic analog of a similar result known for Kähler manifolds.
scientific article; zbMATH DE number 3960472