Isotropy Type Research Articles

ALGEBRAIC REALIZATION FOR CYCLIC GROUP ACTIONS WITH ONE ISOTROPY TYPE

Suppose G is a cyclic group and M a closed smooth G-manifold with exactly one isotropy type. We will show that there is a nonsingular real algebraic G-variety X such that X is equivariantly diffeomorphic to M and all G-vector bundles over X are strongly algebraic.

Pure Lovelock Kasner metrics

We study pure Lovelock vacuum and perfect fluid equations for Kasner-type metrics. These equations correspond to a single Nth order Lovelock term in the action in dimensions, and they capture the relevant gravitational dynamics when aproaching the big-bang singularity within the Lovelock family of theories. Pure Lovelock gravity also bears out the general feature that vacuum in the critical odd dimension, , is kinematic, i.e. we may define an analogue Lovelock–Riemann tensor that vanishes in vacuum for , yet the Riemann curvature is non-zero. We completely classify isotropic and vacuum Kasner metrics for this class of theories in several isotropy types. The different families can be characterized by means of certain higher order 4th rank tensors. We also analyze in detail the space of vacuum solutions for five- and six dimensional pure Gauss–Bonnet theory. It possesses an interesting and illuminating geometric structure and symmetries that carry over to the general case. We also comment on a closely related family of exponential solutions and on the possibility of solutions with complex Kasner exponents. We show that the latter imply the existence of closed timelike curves in the geometry.

Equivariant Bifurcation and Absolute Irreducibility in $$\mathbb {R}^8$$ R 8 : A Contribution to Ize Conjecture and Related Bifurcations

We refer to the hypotheses that for an absolutely irreducible representation of a compact Lie group there exists at least one subgroup with an odd dimensional fixed point space as the Ize conjecture (IC). If the IC is true, then it follows that loss of stability through an absolutely irreducible representation of a compact Lie group leads to bifurcation of steady states. Lauterbach and Matthews have shown that the (IC) is in general not true and have constructed three infinite families of finite subgroups of \(\mathop {\mathbf{SO}(4)}\) which act absolutely irreducibly on \(\mathbb {R}^4\) and have no odd dimensional fixed point space. They also have shown that in spite of this failure of the (IC) the nontrivial isotropy types are generically symmetry breaking at least for the groups in two of these three families. In this paper we show a similar bifurcation result for the third family defined by Lauterbach and Matthews. We go on and construct a family of groups acting absolutely irreducibly on \(\mathbb {R}^8\) which have only even dimensional fixed point spaces. Then we discuss the steady state bifurcations in this case. Key ingredients are an abstract group theoretic construction and a kind of inductive step reducing the issue of bifurcations to a problem in \(\mathbb {R}^4\). We end this paper with a discussion on how to extend the results by Lauterbach and Matthews to larger sets of groups which act on \(\mathbb {R}^{4}\) and \(\mathbb {R}^8\). In this context we point out, that the inductive step, which is important for our arguments, does not work in general and this gives rise to interesting new questions.

Formulation of the Riccati equation for the impedance operator in cylindrical coordinates for inhomogeneous anisotropic waveguides with the example of rectilinear anisotropy

The analysis of cylindrical waveguides with arbitrary anisotropy and inhomogeneities is more difficult than for isotropic or cylindrically anisotropic. The reason for this difficulty is because the independent consideration of different Fourier components of the wavefield is no longer possible. Taking into account the coupling between different Fourier components of the wavefield implies that the impedance tensor should be treated as the operator. Treating the impedance this way is the key difference between the isotropic and the cylindrical anisotropy case in which the independent consideration of the partial impedance matrices for each circumferential number n is possible. The natural approach to computing the impedance operator is to use the Riccati equation, which is presented by the authors in cylindrical coordinates. The matrix representations of the impedance operator and the Riccati equation over the basis of the time-harmonic cylindrical waves result in the infinite set of ordinary differential equations. The nondiagonal partial impedance matrices, which describe the coupling of different Fourier components, become nonzero. For the problems, which possess a symmetry, the formulation can be simplified by projecting the Fourier basis on that of the irreducible representations of the symmetry group. The numerical solution of the Riccati equation and its application to finding the dispersion curves of the eigenmodes of the waveguide have peculiarities due to the fact that the matrix representation is infinite. The feasibility of the proposed approach is demonstrated by computing the dispersion curves for the first monopole and dipole normal modes of a cylindrical waveguide in a homogeneous media possessing tilted transverse isotropy type of anisotropy.

Codimension-Two Bifurcations in Animal Aggregation Models with Symmetry

Pattern formation in self-organized biological aggregation is a phenomenon that has been studied intensively over the past 20 years. In general, the studies on pattern formation focus mainly on identifying the biological mechanisms that generate these patterns. However, identifying the mathematical mechanisms behind these patterns is equally important, since it can offer information on the biological parameters that could contribute to the persistence of some patterns and the disappearance of other patterns. Also, it can offer information on the mechanisms that trigger transitions between different patterns (associated with different group behaviors). In this article, we focus on a class of nonlocal hyperbolic models for self-organized aggregations and show that these models are ${{\bf O(2)}}$-equivariant. We then use group-theoretic methods, linear analysis, weakly nonlinear analysis, and numerical simulations to investigate the large variety of patterns that arise through ${{\bf O(2)}}$-symmetric codimension-two bifurcations (i.e., Hopf/Hopf, steady-state/Hopf, and steady-state/steady-state mode interactions). We classify the bifurcating solutions according to their isotropy types (subgroups), and we determine the criticality and stability of primary branches of solutions. We numerically show the existence of these solutions and determine scenarios of secondary bifurcations. Also, we discuss the secondary bifurcating solutions from the biological perspective of transitions between different group behaviors.

Equivariant Kählerian extensions of contact manifolds

For contact manifolds (M, η) a complexification Mc is constructed to which the contact form η extends such that the exterior derivative of the extended form is Kählerian. In the case of a proper action of an extendable Lie group this construction is realized in an equivariant way. In a simultaneous stratification of M and Mc according to the isotropy type, it is shown that the Kählerian reduction of the complexification can be seen as the complexification of the contact reduction. For this, the complexification is constructed as a stratified space.

Invariant frames for vector bundles and applications

This paper completes a proof of the Dirac reduction theorem by involutive tangent subbundles. As a consequence, Dirac reduction by a proper Lie group action having one isotropy type is carried out. The main technical tool in the proof is the notion of partial connections on suitable vector bundles.

The spectral function of a Riemannian orbifold

Hormander’s theorem on the asymptotics of the spectral function of an elliptic operator is extended to the setting of compact Riemannian orbifolds. In contrast to the manifold case, the asymptotics depend on the isotropy type of the point at which the spectral function is computed. It is shown that “on average” the eigenfunctions of the operator are larger at singular points than at manifold points, by a factor of the order of the isotropy type. A sketch of a more direct approach to the wave trace formula on orbifolds is also given, obtaining results already shown separately by M. Sandoval and Y. Kordyukov in the setting of Riemmannian foliations.

One cannot hear orbifold isotropy type

We show that the isotropy types of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. Indeed, we construct arbitrarily large families of mutually isospectral orbifolds with different isotropy types. Finally, we show that the corresponding singular strata of two isospectral orbifolds may not be homeomorphic.

Geometric depletion of vortex stretch in 3D viscous incompressible flow

New geometric constraints on vorticity are obtained which suppress possible development of finite-time singularity from the nonlinear vortex stretching mechanism. We find a new condition on the smoothness of the direction of vorticity in the vortical region which yields regularity. We also detect a regularity condition of isotropy type on vorticity in the intensive vorticity region via a new cancellation principle. This is in contrast with the one of isotropy type on the curl of vorticity obtained recently by A. Ruzmaikina and Z. Grujić [A. Ruzmaikina, Z. Grujić, On depletion of the vortex-stretching term in the 3D Navier–Stokes equations, Comm. Math. Phys. 247 (2004) 601–611]. We improve as well all of their results by eliminating their assumption that the initial vorticity ω 0 is required to be in L 1 .

Spectral Bounds on Orbifold Isotropy

We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the number of singular points in an orbifold in such an isospectral family is universally bounded above. These proofs employ spectral theory methods of Brooks, Perry and Petersen, as well as comparison geometry techniques developed by Grove and Petersen.

On the location of fixed points of G-deformations

Let G be a finite group and M be a compact smooth G-manifold. In this note, we show that there exist a G-deformation π of M with Fix π={ xϵM| π( x)= x}⊂⋃ M H where ( H) is maximal in Iso( M), the isotropy types of M. We compute the minimal number of fixed points in the G-homotopy class of the identity 1 M . Necessary and sufficient conditions are given for a nonempty closed invariant subset A⊂ M such that A= Fix f for some G-deformation f: M → M.

Weyl groups and the regularity properties of certain compact Lie group actions

The geometric weight system of a G G -manifold X X (acyclic or spherical) is the nonlinear analogue of the weight system of a linear representation. We study the possible realization of a given G G -weight pattern, via the interaction between roots, weights and the Weyl group, together with various fixed point results of P. A. Smith type. If the orbit structure is reasonably simple, then the G G -weight pattern must in fact coincide with that of a simple representation. This in turn implies that X X is (orthogonally) modeled on the linear G G -space, e.g., with the same orbit types. In particular, complete results in this direction are obtained for a certain family of G G -manifolds, G G a classical group. In this family the weight patterns are of 2 2 -parametric type, and it includes essentially all cases where the principal isotropy type is nontrivial. This family also covers many cases with trivial principal isotropy type.

On actions of regular type on complex Stiefel manifolds

The usual unitary representations of the special unitary, symplectic, or special orthogonal groups define a sequence of smooth actions on the complex Stiefel manifolds called the regular linear models. If one of the above groups acts smoothly on the complex Stiefel manifold of orthonormal 2 2 -frames in C n \mathbf C^n for odd n n , and if the identity component of the principal isotropy type is of regular type, then it is shown under mild dimension restrictions that the orbit structure and the cohomology structure of the fixed point varieties (over the mod 2 \mod 2 Steenrod algebra) resemble those of the regular linear models. The resemblance is complete in the cases of the special unitary and symplectic groups. There is an obstruction to complete resemblance in the case of the special orthogonal groups. An application of the above regularity theorems is given.

On actions of adjoint type on complex Stiefel manifolds

Let G ( m ) G(m) denote S U ( m ) {\rm {SU}}(m) or S p ( m ) {\rm {Sp}}(m) . It is shown that when m ≥ 5 G ( m ) m \geq 5\,G(m) cannot act smoothly on W n , 2 W_{n,2} , the complex Stiefel manifold of orthonormal 2 2 -frames in C n \mathbf C^n , for n n odd, with connected principal isotropy type equal to the class of maximal tori in G ( m ) G(m) . This demonstrates an important difference between W n , 2 W_{n,2} , n n odd, and S 2 n − 3 × S 2 n − 1 S^{2n-3}\times S^{2n-1} in the behavior of differentiable transformation groups. Exactly the same holds for S O ( m ) {\rm {SO}}(m) or Spin ( m ) (m) if it is further assumed that a maximal 2 2 -torus of S O ( m ) {\rm {SO}}(m) has fixed points. 2 ^{2}