Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 44 (2003)
Abstracts
G. Preissler:
Isothermic Surfaces and Hopf Cylinders,
44 (2003) 001--008
- Based on the work of Pinkall, characterizations of spherical curves are given
whose corresponding Hopf cylinders are isothermic surfaces in the three-dimensional
sphere. Comparing these characterizations with results of Langer and Singer about
elastic spherical curves we determine all isothermic Willmore Hopf tori.
Keywords: isothermic surface, Hopf cylinder, Clifford torus, Willmore surface.
- In another paper ["Convolution of Riemannian manifolds and its applications", to
appear], we introduced the notion of convolution of Riemannian manifolds. We also
provided some examples and applications of convolution manifolds.
In this paper we use the tensor product to construct more examples of convolution
manifolds and investigates fundamental properties of convolution manifolds. In
particular, we study the relationship between convolution manifolds and the gradient
of their scale functions. Moreover, we obtain a necessary and sufficient condition
for a factor of a convolution Riemannian manifold to be totally geodesic. We also
completely classify flat convolution Riemannian surfaces.
Keywords: convolution manifold, convolution Riemannian manifold, convolution metric,
conic submanifold, totally geodesic submanifolds, flat convolution Riemannian surface,
tensor product immersion.
MSC 2000: 53B20, 53C50; 53C42, 53C17.
- The classification of finite flag-transitive linear spaces is almost complete. For the
thick case, this result was announced by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck
and Saxl, and in the thin case (where the lines have 2 points), it amounts to the
classification of 2-transitive groups, which is generally considered to follow from the
classification of finite simple groups. These two classifications actually leave an open
case, which is the so-called 1-dimensional case. In this paper, we work with two
additional assumptions. These two conditions, namely (2T)1 and RWP
RI, are taken from another field of study in Incidence Geometry and allow us to
obtain a complete classification, which we present at the end of this paper. In particular,
for the 1-dimensional case, we show that the only (2T)1 flag-transitive linear
spaces are AG(2, 2) and AG(2, 4), with AGL(1,4) and
AGL(1,16) as
respective automorphism groups.
- The set-theoretical circle-squaring problem goes back to Tarski: Can a circle be
partitioned into sets that can be reassembled to form a square? We give a short survey
on results to this question and add new claims concerning "scissor congruence" of circle
and square with respect to particular affine transformations.
Keywords: squaring the circle, equidecomposable, congruent by dissection, homothety,
similarity, equiaffine map, affine map.
MSC 2000: 52B45.
- We study isometric actions on compact symmetric spaces for which the principal orbits
are tubular hypersurfaces around totally geodesic singular orbits. We show that in these
cases the symmetric space can be thought of as a compact tube the radius of which is
determined by the curvature tensor. Since the constant principal curvatures of the
tubular orbits can explicitly be expressed, we obtain a simple method to determine
volumes of symmetric spaces by using volumes of lower dimensional ones. Finally, we
discuss the classical irreducible symmetric spaces of types I and II, each of which
admits such special hyperpolar actions.
MSC 2000: 53C35, 53C40, 57S15.
- Greatest common divisors and least common multiples of quotients of elements of
integral domains have been investigated by Lüneburg and further by Jäger.
In this paper we extend these results to invertible fractional ideals. We also lift
our earlier study of the greatest common divisor and least common multiple of finitely
generated faithful multiplication ideals to finitely generated projective ideals.
Keywords: greatest common divisor, least common multiple, invertible ideal, projective
ideal, multiplication ideal, flat ideal, Prüfer domain, semihereditary ring, Bezout
domain, p. p. ring.
MSC 2000: 13A15, 13F05.
- The dimension (or "minimal base size") of a finite permutation group action is
defined to be the smallest power of the action that contains a regular orbit. Although
the concept has appeared before in various contexts, the intention of the current paper
is to survey it from a slightly different viewpoint, with particular emphasis on its
behaviour with respect to G-set constructions. Elementary inequalities relate the
dimension to the degree and closure properties of the action. The dimension is also
expressed exactly in terms of the Möbius function of the subgroup lattice of the
permutation group. For geometric permutation actions, the dimension is related to the
geometric dimension of the space being acted on. The behaviour of the dimension is
studied with respect to disjoint unions, Cartesian products, and wreath products of
actions. Use of the wreath product construction exhibits permutation group actions with
arbitrary positive integral dimension and degree-to-dimension ratio.
- We study projective non-degenerate closed subschemes X of Pn having
degenerate general hyperplane section, continuing our earlier work. We find inequalities
involving three relevant integers, namely: the dimensions of the spans of Xred
and of the general hyperplane section of X, and a measure of the "fatness" of X, which is
introduced in this paper. We prove our results first for curves and then for higher
dimensional schemes by induction, via hyperplane sections. All our proofs and results
are characteristic free. We add also many clarifying examples.
MSC 2000: 14H50, 14N05; 14M99.
- A hypermap H is a cellular embedding of a 3-valent graph G into a closed surface
which cells are 3-coloured (adjacent cells have different colours). The vertices of G
are called flags of H and let us denote by F the set of flags. An automorphism of the
underlying graph which extends to a colour preserving self-homeomorphism of the surface
is called an automorphism of the hypermap. If the surface is orientable the automorphisms
of H split into two classes, orientation preserving and orientation reversing automorphisms.
It is not difficult to observe that |Aut(H)| <= |F| while for the group of orientation
preserving automorphisms we have |Aut+(H)| <= |F|/2. A hypermap satisfying
|Aut+(H)| = |F|/2 = |Aut(H)| will be called chiral. Hence chiral hypermaps have
maximum number of orientation preserving symmetries but they are not "mirror symmetric".
>br>
The main goal of this paper is to classify all chiral hypermaps on surfaces of genus at most
four. It follows that they consist of the infinite families of chiral toroidal hypermaps of
types (2, 3, 6), (2, 4, 4), (3, 3, 3), and their duals, and two exceptional chiral hypermaps
(up to duality) of types (3, 3, 7) and (4, 4, 5). These exceptional chiral hypermaps are
members of regular hypermaps with metacyclic oriented monodromy groups.
- We investigate the measure of unbounded polyhedra in three-dimensional
hyperbolic space. This measure is decomposition-invariant. We obtain an
inner characterisation of this measure by using the notions of endlike
decomposition-equal and of endlike composition-equal.
- The paper concerns the star mappings understood as topological embedding of
Rn into itself preserving the class of bodies which are star shaped
at point 0. The main result is full characterization of star mappings. At the
end we give a solution of some related problem.
Keywords: star set, star body, star mapping.
MSC 2000: 52A30, 54C99, 54C05.
- The authors study smooth lines on projective planes over the algebra C of complex
numbers, the algebra C1 of double numbers, and the algebra C0
of dual numbers. In the space RP5, to these smooth lines there correspond
families of straight lines forming point three-dimensional submanifolds X3
with degenerate Gauss maps of rank r <= 2 . The authors study focal properties of
these submanifolds and prove that they represent examples of different types of
submanifolds X3 with degenerate Gauss maps. Namely, the submanifold
X3, corresponding in RP5 to a smooth line
g of the projective plane CP2, does not have real singular points,
the submanifold X3, corresponding in RP5 to a smooth line
g the projective plane C1P2, bears
two plane singular lines, and finally the submanifold X3, corresponding in
RP5 to a smooth line g of the projective plane
C0P2, bears one singular line. It is also proved that in all
three cases, the rank of X3 is equal to the rank of the curvature of the
line g .
Keywords: smooth line, projective plane over an algebra, submanifold with degenerate
Gauss map, hypersurface of Sacksteder, hypersurface of Bourgain.
MSC 2000: 53A20; 14M99.
- We prove that a unit-regular ring R is isomorphic to a Busque ring S if
its matrix rings Mn(R) and Mn(S) are isomorphic. This
gives a partial answer to a matrix isomorphism question for unit-regular rings
proposed in the text of K. Goodearl.
Keywords: unit-regular rings; directly finite Aleph0-continuous regular
rings; Replacement Lemma.
MSC 2000: 16A30.
- Let M be a closed connected oriented topological 4-manifold with fundamental
group p1. Let L
be the integral group ring of p1 . Suppose
that f from M to P is a degree one map inducing an isomorphism on
p1 . We give a homological condition on the intersection forms
lMZ and
pML under
which M is homotopy
equivalent to a connected sum P # M' for some simply-connected closed (non-trivial)
topological 4-manifold M'. This gives a partial solution to a conjecture of J. A. Hillman
[Free products and 4-dimensional connected sums, Bull. London Math. Soc. 27 (1995) 387--391]
on the classification of closed 4-manifolds with vanishing second homotopy group. Then
some splitting results for closed 4-manifolds with special homotopy complete the paper.
Keywords: four-manifolds, homotopy type, connected sum, obstruction theory, intersection
forms, homology with local coefficients, degree one map.
MSC 2000: 57N65, 57R67, 57Q10.
- We study cylindrical helices and Bertrand curves as curves on ruled surfaces. Some
results in this paper clarify that the cylindrical helix is related to Gaussian curvature
and the Bertrand curve is related to mean curvature of ruled surfaces. All arguments in
this paper are elementary and classical. There are some articles which investigate curves
on ruled surfaces of H. Brauner [Neuere Untersuchungen über windschiefe Flächen, Jahresber.
DMV (1967) 61--85] and R. Koch [Regelflächen in euklidischen Räumen, in: O.Giering und J.
Hoschek (eds.), Geometrie und ihre Anwendungen, München 1994, 71--106]. However, the main
results are not obtained in these articles and do shade new light on an old subject.
Keywords: cylindrical helix, Bertrand curve, ruled surfaces.
MSC 2000: 58C27, 53A25, 53A05.
- [Abstract-pdf]
Let $\Phi$ be an irreducible, spherical, possibly nonreduced root system
of rank $\ell \geq 2$ and $G$ a group generated by subgroups $A_r, r \in \Phi$
satisfying:
\begin{enumerate}
\item[(i)] $X_r = \langle A_r, A_{-r} \rangle$ is a rank one group (in the sense
of [Ti1]) for $r \in \Phi$.
\item[(ii)] If $r, s \in \Phi$ with $r \not= -s$ or $-2s$ (resp. $s \not= -r$
or $-2r$), then $[A_r, A_s] \leq \langle A_{\lambda r + \mu s} \mid \lambda,
\mu \in {\Bbb N}$ with $\lambda r + \mu s \in \Phi \rangle$.
\end{enumerate}
Clearly, by the Steinberg-Presentation, all Chevalley-groups satisfy these
conditions. Conversely, by [Mü], [Ti2,3 and 4] and the above paper the structure
of a group satisfying (i) and (ii) is determined, apart from the special cases
when $\Phi$ is of type $G_2$ or $^2F_4$. The above paper treats the final case,
when $\Phi$ is of type $BC_\ell$, $\ell \geq 2$, which corresponds to unitary
groups which are not of maximal Witt-index.
\medskip
\begin{description}
\item[[M\"u]] M\"uller, C., On the Steinberg-presentation for Lie-type-groups of type
$C_2$, J. of Algebra, {\bf 252} (2002), 150--160
\item[[Ti1]] Timmesfeld, F. G., Abstract Root Subgroups and simple groups of Lie-type,
Monographs in Mathematics 95, Birkhäuser Verlag, 2001
\item[[Ti2]] Timmesfeld, F. G., On the Steinberg-Presentation of Lie-type groups,
to appear in Forum Math. (2002)
\item[[Ti3]] Timmesfeld, F. G., A remark on Presentations of certain Chevalley
groups, to appear in Archiv der Mathematik. (2002)
\item[[Ti4]] Timmesfeld, F. G., Groups with a central factor of Lie-type,
to appear in J. of Algebra.
\end{description}
- This article classifies the even lattices L such that each coset in L/2L
contains a vector of square length less or equal to 6. In particular all
even lattices with covering radius less than square root of 2 have this
property.
- A polytope is perfect if its shape cannot be changed without changing the
action of its symmetry group on its face-lattice. There was a conjecture by
which perfect 4-polytopes formed a rather limited class of Wythoffian polytopes.
It was disproved in a preceding paper of the author by showing that this class
is much more wide. In the present paper we go even further by giving a construction
that provides non-Wythoffian perfect 4-polytopes. The construction is based on
including the copies of a suitable 3-polytope into the facets of a facet-transitive
4-polytope in a symmetry-preserving way.
Keywords: nodal polytope, perfect polytope, regular polytope, semi-nodal polytope,
Wythoff's construction.
- The natural mapping of the right quaternion vector space H2 onto the
quaternion projective line (identified with the four-sphere) can be defined for complex
quaternions H otimesR C as well. We discuss its exceptional set, the fiber
subspaces, and how the linear automorphism groups of two-dimensional quaternion vector
spaces and modules induce groups of projective automorphisms of the image quadrics.
- We study groupoids with the following property: every element is idempotent and
for all elements a, b, c we have (ab)c = ac . We also obtain a few results if the
operations are continuous or differentiable. Some open problems are formulated.
- It is well known that for 2-codimensional aCM subschemes of Pr with
a fixed Hilbert function H there are all the possible graded Betti numbers between
suitable bounds depending on H. For aCM subschemes of codimension c >= 3 with
Hilbert function H it is just known that there are upper bounds for the graded
Betti numbers depending on H and these can be reached; but what are the graded
Betti numbers which can be realized is not yet completely understood. The aim of
the paper is to construct c-codimensional subschemes of Pr which could
recover as many graded Betti numbers as possible generalizing both the 2-codimensional
case and the maximal case.
MSC 2000): 13D40, 13H10.