12 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
Time-like Weingarten surfaces with real principal curvatures in the three-dimensional Minkowski space and their natural partial differential equations
We study time-like surfaces in the three-dimensional Minkowski space with
diagonalizable second fundamental form. On any time-like W-surface we introduce
locally natural principal parameters and prove that such a surface is
determined uniquely (up to motion) by a special invariant function, which
satisfies a natural non-linear partial differential equation. This result can
be interpreted as a solution of the Lund-Regge reduction problem for time-like
W-surfaces with real principal curvatures in Minkowski space. We apply this
theory to linear fractional time-like W-surfaces and obtain the natural partial
differential equations describing them.Comment: 17 pages. arXiv admin note: substantial text overlap with
arXiv:1105.364
Kaehler Manifolds of Quasi-Constant Holomorphic Sectional Curvatures
The Kaehler manifolds of quasi-constant holomorphic sectional curvatures are
introduced as Kaehler manifolds with complex distribution of codimension two,
whose holomorphic sectional curvature only depends on the corresponding point
and the geometric angle, associated with the section. A curvature identity
characterizing such manifolds is found. The biconformal group of
transformations whose elements transform Kaehler metrics into Kaehler ones is
introduced and biconformal tensor invariants are obtained. This makes it
possible to classify the manifolds under consideration locally. The class of
locally biconformal flat Kaehler metrics is shown to be exactly the class of
Kaehler metrics whose potential function is only a function of the distance
from the origin in complex Euclidean space. Finally we show that any rotational
even dimensional hypersurface carries locally a natural Kaehler structure,
which is of quasi-constant holomorphic sectional curvatures.Comment: 36 page