II-orthogonal two-curve condition forces quadrics
Prove that if M is an open patch of a C^2 hypersurface in R^n (n ≥ 3) and Γ1, Γ2 are space curves such that for every p in M the tangent hyperplane T_pM meets Γ1 and Γ2 at unique points u1 and u2, and the vectors u1−p and u2−p in T_pM are orthogonal with respect to the second fundamental form II_p, then M is contained in a quadric.
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However, it seems to us that this obstacle may occur only on a nowhere dense set, unless the hypersurface is locally contained in a quadric. More precisely, we believe that the following conjecture holds: Let M⊂ℝn, n≥3, be an open patch of a hypersurface of class C2, and let Γ1,Γ2 be space curves. Suppose that for every point p∈M, there exist unique intersection points u1=Γ1∩T_pM,u2=Γ2∩T_pM of the tangent hyperplane T_pM with Γ1,Γ2, respectively, and that the vectors u1−p,u2−p∈T_pM are orthogonal with respect to the second fundamental form II_p of M at p. Then M is contained in a quadric.