910 research outputs found
Natural PDE's of Linear Fractional Weingarten surfaces in Euclidean Space
We prove that the natural principal parameters on a given Weingarten surface
are also natural principal parameters for the parallel surfaces of the given
one. As a consequence of this result we obtain that the natural PDE of any
Weingarten surface is the natural PDE of its parallel surfaces. We show that
the linear fractional Weingarten surfaces are exactly the surfaces satisfying a
linear relation between their three curvatures. Our main result is
classification of the natural PDE's of Weingarten surfaces with linear relation
between their curvatures.Comment: 16 page
An Invariant Theory of Spacelike Surfaces in the Four-dimensional Minkowski Space
We consider spacelike surfaces in the four-dimensional Minkowski space and
introduce geometrically an invariant linear map of Weingarten-type in the
tangent plane at any point of the surface under consideration. This allows us
to introduce principal lines and an invariant moving frame field. Writing
derivative formulas of Frenet-type for this frame field, we obtain eight
invariant functions. We prove a fundamental theorem of Bonnet-type, stating
that these eight invariants under some natural conditions determine the surface
up to a motion. We show that the basic geometric classes of spacelike surfaces
in the four-dimensional Minkowski space, determined by conditions on their
invariants, can be interpreted in terms of the properties of the two geometric
figures: the tangent indicatrix, and the normal curvature ellipse. We apply our
theory to a class of spacelike general rotational surfaces.Comment: 23 pages; to appear in Mediterr. J. Math., Vol. 9 (2012
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