325 research outputs found
Laguerre Geometry of Hypersurfaces in
Laguerre geometry of surfaces in is given in the book of Blaschke [1],
and have been studied by E.Musso and L.Nicolodi [5], [6], [7], B. Palmer [8]
and other authors. In this paper we study Laguerre differential geometry of
hypersurfaces in . For any umbilical free hypersurface with
non-zero principal curvatures we define a Laguerre invariant metric on
and a Laguerre invariant self-adjoint operator , and
show that is a complete Laguerre invariant system for
hypersurfaces in with . We calculate the Euler-Lagrange equation
for the Laguerre volume functional of Laguerre metric by using Laguerre
invariants. Using the Euclidean space , the Lorentzian space and
the degenerate space we define three Laguerre space forms ,
and and define the Laguerre embedding
and , analogue to the Moebius geometry where we have Moebius
space forms , \H^n and (spaces of constant curvature) and
conformal embedding \H^n\to S^n and (cf. [4], [10]). Using
these Laguerre embedding we can unify the Laguerre geometry of hypersurfaces in
, and . As an example we show that minimal surfaces in
or are Laguerre minimal in .Comment: 24 page
Deformation of Hypersurfaces Preserving the Moebius Metric and a Reduction Theorem
A hypersurface without umbilics in the n+1 dimensional Euclidean space is
known to be determined by the Moebius metric and the Moebius second fundamental
form up to a Moebius transformation when n>2. In this paper we consider Moebius
rigidity for hypersurfaces and deformations of a hypersurface preserving the
Moebius metric in the high dimensional case n>3. When the highest multiplicity
of principal curvatures is less than n-2, the hypersurface is Moebius rigid.
Deformable hypersurfaces and the possible deformations are also classified
completely. In addition, we establish a Reduction Theorem characterizing the
classical construction of cylinders, cones, and rotational hypersurfaces, which
helps to find all the non-trivial deformable examples in our classification
with wider application in the future.Comment: 51 pages. A mistake in the proof to Theorem 9.2 has been fixed.
Accepted by Adv. in Mat
M\"obius and Laguerre geometry of Dupin Hypersurfaces
In this paper we show that a Dupin hypersurface with constant M\"{o}bius
curvatures is M\"{o}bius equivalent to either an isoparametric hypersurface in
the sphere or a cone over an isoparametric hypersurface in a sphere. We also
show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre
equivalent to a flat Laguerre isoparametric hypersurface. These results solve
the major issues related to the conjectures of Cecil et al on the
classification of Dupin hypersurfaces.Comment: 45 pages. arXiv admin note: text overlap with arXiv:math/0512090 by
other author
M\"{o}bius Homogeneous Hypersurfaces in
Let denote the M\"{o}bius transformation group
of the -dimensional sphere . A hypersurface is called a M\"{o}bius homogeneous hypersurface if there
exists a subgroup of such that the orbit
. In this paper, the M\"{o}bius homogeneous
hypersurfaces are classified completely up to a M\"{o}bius transformation of
.Comment: 35 pages, comments welcom
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