These E(c, [mu])-spaces are 3-manifolds admitting a global unit Killing vector field whose

integral curves are the fibers of a certain Riemannian submersion over the simply connected constant Gaussian curvature surface M(c) and, therefore, determine Killing submersions [pi] : E(c, [mu]) [right arrow] M(c) (for more details, see [25]).

Observe that the

integral curves of the system do not depend on [eta], so without loss of generality we may assume that [eta] = 1.

Several authors studied the

integral curves by using matrix of a linear vector field in Euclidean and 3-dimensional Lorentz-Minkowski spaces.

In Section 3, we make the numerical simulation for bounded

integral curves. In Section 4, we derive the exact implicit or parameter expressions of the kink-like wave and compacton-like wave solutions.

Note that the periodic waves of (9) correspond to the periodic

integral curves of (40), and the periodic

integral curves correspond to the closed orbits of system (41).

The

integral curves for -1/(1 - t)F'[(x).sup.-1]F(x) are the paths implicitly determined by

Sometimes, M is a union of family of subsets (a plane as union of straight lines, an integral manifold of a Pfaff equation as union of

integral curves, and so on.

Key words: roughnes of the cut surface, differential equation,

integral curvesPaper topics include studies of a singularly perturbed Riccati equation, asymptotic differential algebra, non- oscillating

integral curves and o-minimal structures, topological construction of transseries and an introduction of generalized Borel summability, higher-order norms for the Moivre-Laplace theorem, matching the similarities of canard values, and generalizing surreal numbers.

Our complete line of floors offers a wide variety of options, including performance characteristics, thickness,

integral curves, color and texture.

This paper considers the problem of minimizing 1/2 [integral of] [k.sub.2] between limits 0 and T (s) ds over all

integral curves g(t), satisfying the prescribed boundary con ditions g(0) = [g.sub.0], g(T) = [g.sub.l], of the following differential system:

Integral curves of the vector fields [e.sub.1] and [e.sub.2], passing through the point ([t.sub.0], [x.sub.0]) [member of] [R.sup.2], are determined by solutions of the ordinary differential equations