At last, we note here that the global exponential stability of traveling curved fronts in the sense of Theorem 3isa difficult problem, since the level set of the traveling curved fronts [PHI](z, y) of (1) have two asymptotic directions
as [absolute value of (z)] [right arrow] +[infinity], and both directions make an angle with the negative y-axis, which is different from the case of planar traveling fronts (see ).
In the differential geometry of surfaces, an asymptotic curve is formally defined as a curve on a regular surface such that the normal curvature is zero in the asymptotic direction
. Asymptotic directions
can only occur when the Gaussian curvature on surface is negative or zero along the asymptotic curve [1, 4, 5].
From the definitions of the conjugate vectors, the asymptotic directions and the equation II*(X; Y) = II(X; Y), we say the conjugate vectors and asymptotic directions in M are also the conjugate vectors and asymptotic directions in M*.
(iii) Asymptotic directions in M are also the asymptotic directions in M*.