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 [20]).
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*.