Some properties of transversally symmetric foliation (Q2706963)

Let \((M,g_M,F)\) be a \((p+q)\)-dimensional Riemannian manifold with a foliation \(F\) of codimension \(q\) and a bundle-like metric \(g_M\) with respect to a foliation \(F\). Then there exists an exact sequence of vector bundles \(0\to L\to TM\to Q\to 0\), where \(L\) is the tangent bundle and \(Q\) the normal bundle of \(F\) with respect to \(g_M\), \(TM\equiv L\oplus Z^\perp\), \(Q\equiv L^\perp\). A geodesic \(\gamma\) on \(M\) is called a transversal geodesic if \(\gamma\) is orthogonal to the leaves at all points of \(\gamma\). \(F\) is transversally symmetric if its transversal geometry is locally modeled on a Riemannian symmetric space. Theorem 3.5. Let \((M,g_M,F)\) be a Riemannian symmetric space with a bundle-like metric \(g_M\). Then \(F\) is transversally symmetric if and only if NEWLINE\[NEWLINEg_M(A_XY,T_{A_XY}X) +2g_M\bigl((D_XA)_X Y,A_XY\bigr) =0NEWLINE\]NEWLINE for all \(X,Y\in \Gamma L^\perp\).NEWLINENEWLINENEWLINETheorem 3.7. Let \(F\) be a Kähler foliation on \((M,g_M)\), and \(g_M\) be a bundle-like metric. Then the following conditions are equivalent:NEWLINENEWLINENEWLINE(1) \(F\) is transversally symmetric;NEWLINENEWLINENEWLINE(2) \(\nabla_X R^\nabla_{X,JX,X,JX} =0\) for all \(X\in\Gamma L^\perp\);NEWLINENEWLINENEWLINE(3) \(D_XR_{X, JX,X,JX}+ 2R_{X,A_XJX, X,JX}=-3Xg_M(A_XJX, A_XJX)\) for all \(X\in\Gamma L^\perp\). Other results are about umbilic Kähler foliations.