199 research outputs found
Zindler-type hypersurfaces in R^4
In this paper the definition of Zindler-type hypersurfaces is introduced in as a generalization of planar Zindler curves. After recalling some properties of planar Zindler curves, it is shown that Zindler hypersurfaces satisfy similar properties. Techniques from quaternions and symplectic geometry are used. Moreover, each Zindler hypersurface is fibrated by space Zindler curves that correspond, in the convex case, to some space curves of constant width lying on the associated hypersurface of constant width and with the same symplectic area
Concerning the semistability of tensor products in Arakelov geometry
We study the semistability of the tensor product of hermitian vector bundles
by using the -tensor product and the geometric (semi)stability of
vector subspaces in the tensor product of two vector spaces
Weighted composition operators preserving various Lipschitz constants
Let , ,
and be the
vector spaces of Lipschitz, bounded Lipschitz,
locally Lipschitz and pointwise Lipschitz
(real-valued) functions defined on a metric space , respectively.
We show that if a weighted composition operator
defines a bijection between such vector spaces preserving Lipschitz constants,
local Lipschitz constants or pointwise Lipschitz constants, then is a constant function for some scalar and is
an -dilation.
Let be open connected and be open, or both are convex bodies,
in normed linear spaces , respectively. Let be
a bijective weighed composition operator between the vector spaces
and , and
, and
, or and
, preserving the Lipschitz, locally Lipschitz, or
pointwise Lipschitz constants, respectively. We show that there is a linear
isometry , an and a vector such that
, and is a constant function assuming value . More concrete results are obtained for the special cases when
, or when are -dimensional flat manifolds.Comment: to appear in "Annals of Mathematical Sciences and Applications
Nijenhuis tensors in pseudoholomorphic curves neighborhoods
Normal forms of almost complex structures in a neighborhood of
pseudoholomorphic curve are considered. We define normal bundles of such curves
and study the properties of linear bundle almost complex structures. We
describe 1-jet of the almost complex structure along a curve in terms of its
Nijenhuis tensor. For pseudoholomorphic tori we investigate the problem of
pseudoholomorphic foliation of the neighborhood. We obtain some results on
nonexistence of the tori deformation.Comment: 27 page
Quasisymmetric Embeddability of Weak Tangents
In this paper, we study the quasisymmetric embeddability of weak tangents of
metric spaces. We first show that quasisymmetric embeddability is hereditary,
i.e., if can be quasisymmetrically embedded into , then every weak
tangent of can be quasisymmetrically embedded into some weak tangent of
, given that is proper and doubling. However, the converse is not true
in general; we will illustrate this with several counterexamples. In special
situations, we are able to show that the embeddability of weak tangents implies
global or local embeddability of the ambient space. Finally, we apply our
results to expanding dynamics and establish several results on Gromov
hyperbolic groups and visual spheres of expanding Thurston maps.Comment: 35 pages, 6 figure
Uniqueness of Tangent Cones to Positive-(p,p) Integral Cycles
Let (M, \om) be a symplectic manifold, endowed with a compatible almost
complex structure J and the associated metric g . For any p \in {1, 2, ... (dim
M)/2} the form \Om := \frac{\om^p}{p!} is a calibration. More generally,
dropping the closedness assumption on \om, we get an almost hermitian
manifold (M, \om, J, g) and then \Om is a so-called semi-calibration. We
prove that integral cycles of dimension 2p (semi-)calibrated by \Om possess
at every point a unique tangent cone. The argument relies on an algebraic blow
up perturbed in order to face the analysis issues of this problem in the almost
complex setting.Comment: 22 page
An Invitation to Generalized Minkowski Geometry
The present thesis contributes to the theory of generalized Minkowski spaces as a continuation of Minkowski geometry, i.e., the geometry of finite-dimensional normed spaces over the field of real numbers.
In a generalized Minkowski space, distance and length measurement is provided by a gauge, whose definition mimics the definition of a norm but lacks the symmetry requirement.
This seemingly minor change in the definition is deliberately chosen.
On the one hand, many techniques from Minkowski spaces can be adapted to generalized Minkowski spaces because several phenomena in Minkowski geometry simply do not depend on the symmetry of distance measurement.
On the other hand, the possible asymmetry of the distance measurement set up by gauges is nonetheless meaningful and interesting for applications, e.g., in location science.
In this spirit, the presentation of this thesis is led mainly by minimization problems from convex optimization and location science which are appealing to convex geometers, too.
In addition, we study metrically defined objects, which may receive a new interpretation when we measure distances asymmetrically.
To this end, we use a combination of methods from convex analysis and convex geometry to relate the properties of these objects to the shape of the unit ball of the generalized Minkowski space under consideration
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