Computing isometry groups of Hermitian maps BROOKSBANK, PETER A.; WILSON, JAMES B.
Transactions of the American Mathematical Society,
04/2012, Letnik:
364, Številka:
4
Journal Article
We give two analogs of Korn’s inequality on Heisenberg groups. First, the norm of the horizontal differential is estimated in terms of the symmetric part of the differential. Second, Korn’s ...inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for the operator.
A pair of bodies rolling on each other is an interesting example of nonholonomic systems in control theory. There is a geometric condition equivalent to the rolling constraint which enables us to ...generalize the rolling motions for any two-dimensional Riemannian manifolds. This system has a five-dimensional phase space. In order to study the controllability of the rolling surfaces, we lift the system to a six-dimensional space and show that the lifted system is controllable unless the two surfaces have isometric universal covering spaces. In the non-controllable case there are some three-dimensional orbits each of which corresponds to an isometry of the universal covering spaces. KCI Citation Count: 0
On contact sub-pseudo-Riemannian isometries Grochowski, Marek; Kryński, Wojciech
ESAIM. Control, optimisation and calculus of variations,
10/2017, Letnik:
23, Številka:
4
Journal Article
Recenzirano
Odprti dostop
We study isometries in contact sub-pseudo-Riemannian geometry. In particular we give an upper bound on the dimension of the isometry group of a general sub-pseudo-Riemannian manifold and prove that ...the maximal dimension is attained for the left invariant structures on the Heisenberg group.
A classification is performed of the spaces with plane-wave metrics in accordance with the presence of three-dimensional subgroups of the group of space-time motion with three-dimensional spatial ...orbits. In order to obtain spatially-homogeneous models, exact solutions to Einstein’s vacuum equations are found. To obtain these solutions, the Petrov and Bianchi classifications are performed. The results obtained on the classification of the plane-wave metrics using the assumed or three-dimensional subgroups of the group of motion with three-dimensional spatial orbits can be used for any metric gravitational theories.
We study the Lorentzian manifolds
M
1
,
M
2
,
M
3
, and
M
4
obtained by small changes of the standard Euclidean metric on ℝ
4
with the punctured origin
O
. The spaces
M
1
and
M
4
are closed isotropic ...space-time models. The manifolds
M
3
and
M
4
(respectively,
M
1
and
M
2
) are geodesically (non)complete;
M
1
are
M
4
are globally hyperbolic, while
M
2
and
M
3
are not chronological. We found the Lie algebras of isometry and homothety groups for all manifolds; the curvature, Ricci, Einstein, Weyl, and energy-momentum tensors. It is proved that
M
1
and
M
4
are conformally flat, while
M
2
and
M
3
are not conformally flat and their Weyl tensor has the first Petrov type.
A two-valued fitness landscape is introduced for the classical Eigen’s quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single- or sharply ...peaked landscape. A general, non-permutation invariant quasispecies model is studied, and therefore the dimension of the problem is
2
N
×
2
N
, where
N
is the sequence length. It is shown that if the fitness function is equal to
w
+
s
on a
G
-orbit
A
and is equal to
w
elsewhere, then the mean population fitness can be found as the largest root of an algebraic equation of degree at most
N
+
1
. Here
G
is an arbitrary isometry group acting on the metric space of sequences of zeroes and ones of the length
N
with the Hamming distance. An explicit form of this exact algebraic equation is given in terms of the spherical growth function of the
G
-orbit
A
. Motivated by the analysis of the two-valued fitness landscapes, an abstract generalization of Eigen’s model is introduced such that the sequences are identified with the points of a finite metric space
X
together with a group of isometries acting transitively on
X
. In particular, a simplicial analog of the original quasispecies model is discussed, which can be considered as a mathematical model of the switching of the antigenic variants for some bacteria.
Maximal couplings are (probabilistic) couplings of Markov processes such that the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. ...Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before making the next transition. Markovian couplings are typically easier to construct and analyze than general couplings, and play an important role in many branches of probability and analysis. Hsu and Sturm, in a preprint circulating in 2007, but later published in 2013, proved that the
reflection-coupling
of Brownian motion is the
unique Markovian maximal coupling
(
MMC
) of Brownian motions starting from two different points. Later, Kuwada (Electron J Probab 14(25), 633–662,
2009
) proved that the existence of a MMC for Brownian motions on a Riemannian manifold enforces existence of a reflection structure on the manifold. In this work, we investigate suitably regular elliptic diffusions on manifolds, and show how consideration of the diffusion geometry (including dimension of the isometry group and flows of isometries) is fundamental in classification of the space and the generator of the diffusion for which an MMC exists, especially when the MMC also holds under local perturbations of the starting points for the coupled diffusions. We also describe such diffusions in terms of
Killing vectorfields
(generators of isometry groups) and
dilation vectorfields
(generators of scaling symmetry groups). This permits a complete characterization of those possible manifolds and their diffusions for which there exists a MMC under local perturbations of the starting points of the coupled diffusions. For example, in the time-homogeneous case it is shown that the only possible manifolds that may arise are Euclidean space, hyperbolic space and the hypersphere. Moreover the permissible drifts can then derive only from rotation isometries of these spaces (and dilations, in the Euclidean case). In this sense, a geometric rigidity phenomenon holds good.
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