Structure of a leaf of some codimension one Riemannian foliation
Let \(M\) be a smooth connected open orientable finite-dimensional manifold endowed with a smooth codimension one Riemannian foliation \(F\). The purpose of this paper is to study the geometric structure of a leaf \(L\) of \(F\) when \(F\) is a totally geodesic foliation. Let \(f\) be a map given by \(f(x) = (x,n(x),m \rho)\), \(x \in L\), where \(x \to n(x)\) is a normal field of unit vectors, \(m \in Z^+\) and \(\rho\) is a positive constant depending on \(F\). The author shows that: 1) There exists a vector \(V\) on \(L\) such that \(f(x) = \exp e_L(x)V(x)\), where the range \(e_L(x)\) is the greatest lower bound of the lengths of minimizing geodesics joining two variable points in the \(L\)-submanifold; 2) The leaf \(L\) is of fibred type over a complete Riemannian manifold \(N\) with boundary.
scientific article; zbMATH DE number 4063972