Sturm-Liouville operator controlled by sectional curvature on Riemannian manifolds
نویسندگان
چکیده
The purpose of this paper is fourfold: (1) to introduce and study a second order PDE, determined accidentally by a Riemann wave, reflecting the connection between oriented parallelograms area and sectional curvature on Riemannian manifolds; (2) to introduce and study the asymptotic behavior of oriented parallelograms area controlled by the sectional curvature; (3) to study some partial differential inequalities describing the evolution of parallelogram area on pinched manifolds; (4) to find controlled minimum of total sectional curvature. This means to control some geometric quantities associated to a Riemannian metric as it evolves with respect to a parameter via a geometric PDE (partial differential equation) or PDI (partial differential inequality). This approach and our PDEs/PDIs on Riemannian manifolds inaugurate new understandings of certain interrelationships among fundamental geometrical concepts. M.S.C. 2010: 53C20, 53C44, 53C21.
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