Consider the operator ?r 2 ?q(), where ?r 2 is the (positive) Laplace-Beltrami operator on a closed manifold of the topological type of the two-sphere S 2 and q is a symmetric non-negative quadratic form in the principal curvatures. Generalizing a well-known theorem of J. Hersch for the Laplace-Beltrami operator alone, it is shown in this note that the second eigenvalue 1 is uniquely maximized,...

In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in cases of both symmetric and non-symmetric connection. Although curvature-flatness idea classic for connection, our main theorems about flatness solutions completely new, leaving a while point view geometry entering PDEs. second int...