A proximal regularization of the steepest descent method in Riemannian manifold.
We apply Cartan’s method of equivalence to find a Bäcklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.
We shall prove the following Theorem. Let Fs and Fu be two continuous transverse foliations with uniformly smooth leaves, of some manifold. If f is uniformly smooth along the leaves of Fs and Fu, then f is smooth.
We prove a formula relating the index of a solution and the rotation number of a certain complex vector along bifurcation diagrams.