A characterization of revolution quadrics by a system of partial differential equations

Abstract

Abstract. It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such a solution, its reciprocal is the radial function of only one of the following rotationally symmetric hypersurfaces in R n+1 : paraboloid, ellipsoid, one sheet of a two-sheeted hyperboloid, and a hyperplane. Main resultLet M be a C ∞ complete Riemannian manifold of dimension n ≥ 2 with metric h. Denote by ∇ 2 the Hessian matrix of the second covariant derivatives in the metric h. The well known theorem of Obata [9] states that if the system [15]. The proofs usually require substantial efforts. By contrast, the characterization given in the following theorem uses a nonlinear second order system of PDE's, ultimately connected with first order spherical harmonics and it is obtained by rather simple means akin to the original Obata's theorem. In fact, the proof relies on Obata's theorem.Recall that a hypersurface F in R n+1 is defined by its radial function if F is a graph of a positive function ρ over some domain ø ⊂ S n , where S n is a unit sphere in R n+1 with the center at the origin of a Cartesian coordinate system. The position

Far-field reflector problem and intersection of paraboloids

“…, λ N such that the surface ∂(∩ 1 i N P (y i , λ i )) reflects exactly the amount α i in each direction y i . Other types of inverse problems in geometric optics can be formulated as Minkowski-type problems involving the union of confocal solid paraboloids, and the union or intersection of confocal ellipsoids [17,14].…”