304 research outputs found
A metric property of umbilic points
In the space of cubic forms of surfaces, regarded as a
-space and endowed with a natural invariant metric, the ratio of the volumes
of those representing umbilic points with negative to those with positive
indexes is evaluated in terms of the asymmetry of the metric, defined here. A
connection of this ratio with that reported by Berry and Hannay (1977) in the
domain of Statistical Physics, is discussed.Comment: 8 pages, 1 figur
Codimension two Umbilic points on Surfaces Immersed in R^3
In this paper is studied the behavior of lines of curvature near umbilic
points that appear generically on surfaces depending on two parameters.Comment: 19 pages, 10 figure
Geometric Mean Curvature Lines on Surfaces Immersed in R3
Here are studied pairs of transversal foliations with singularities, defined
on the Elliptic region (where the Gaussian curvature is positive)
of an oriented surface immersed in . The leaves of the foliations
are the lines of geometric mean curvature, along which the normal curvature is
given by , which is the geometric mean curvature of the
principal curvatures of the immersion. The singularities of the
foliations are the umbilic points and parabolic curves}, where and
, respectively. Here are determined the structurally stable
patterns of geometric mean curvature lines near the umbilic points, parabolic
curves and geometric mean curvature cycles, the periodic leaves of the
foliations. The genericity of these patterns is established. This provides the
three essential local ingredients to establish sufficient conditions, likely to
be also necessary, for Geometric Mean Curvature Structural Stability. This
study, outlined at the end of the paper, is a natural analog and complement for
the Arithmetic Mean Curvature and Asymptotic Structural Stability of immersed
surfaces studied previously by the authors.Comment: 21 pages, 5 figures. To appear in Annales de la Faculte de Sciences
de Toulous
Stable piecewise polynomial vector fields
Consider in R^2 the semi-planes N={y>0} and S={y<0}. In N and S are defined polynomial vector
fields X and Y, respectively, leading to a discontinuous piecewise polynomial
vector field Z=(X,Y). This work pursues the stability and the transition
analysis of solutions of Z between N and S, started by Filippov (1988) and
Kozlova (1984) and reformulated by Sotomayor-Teixeira (1995) in terms of the
regularization method. This method consists in analyzing a one parameter family
of continuous vector fields Z_{\epsilon}$, defined by averaging X and Y. This
family approaches Z when the parameter goes to zero. The results of
Sotomayor-Teixeira and Sotomayor-Machado (2002) providing conditions on (X,Y)
for the regularized vector fields to be structurally stable on planar compact
connected regions are extended to discontinuous piecewise polynomial vector
fields on R^2. Pertinent genericity results for vector fields satisfying the
above stability conditions are also extended to the present case. A procedure
for the study of discontinuous piecewise vector fields at infinity through a
compactification is proposed here
Lines of axial curvature at critical points on surfaces mapped into R4
In this paper are studied the simplest patterns of axial curvature lines
(along which the normal curvature vector is at a vertex of the ellipse of
curvature) near a critical point of a surface mapped into R4. These critical
points, where the rank of the mapping drops from 2 to 1, occur isolated in
generic one parameter families of mappings of surfaces into R4. As the
parameter crosses a critical bifurcation value, at which the mapping has a
critical point, it is described how the axial umbilic points, which are the
singularities of the axial curvature configurations at regular points, move
along smooth arcs to reach the critical point. The numbers of such arcs and
their axial umbilic types are fully described for a typical family of mappings
with a critical point.Comment: 19 pages, 12 figure
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