arXiv research

Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.

New biharmonic functions created on Lie groups.

Random convolutional networks can be fooled with adversarial examples.

An algorithm for efficient computation of equivariant neural network layers.

Characterizes group-equivariant neural networks for three groups.

A complete system of differential invariants for equivalence of curves in the $n$-dimensional pseudo-euclidean space with respect to the action of each of the groups $K^n \lhd O(n,p,K)$, $K^n \lhd SO(n,p,K)$, $O(n,p,K)$, and $SO(n,p,K)$, where $K = \mathbf{R}$, or $K = \mathbf{C}$, and $O(n,p,K)$ $\bigl($respectively, …

Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.

The paper creates a deformation retraction for homeomorphisms of the projective plane.

Study new metrics on tangent bundles from base manifolds.

Integrable geodesics found on special orthogonal group.

New representation theory for surface groups to SO0(2,3).

Study of skateboard flips as continuous curves in $SO(3)$ group.

New Einstein metrics found on orthogonal groups without natural reductivity.

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree $n$ of compact congruence $p$-dimensional hyperbolic manifolds "of simple type" a…

Paper derives local Plücker formulas for special orthogonal groups.

Special orthogonal representations from octonions have geometric properties linked to binary cubics.

The paper is based on relations between a ternary symmetric form defining the SO(3) geometry in dimension five and Cartan's works on isoparametric hypersurfaces in spheres. As observed by Bryant such a ternary form exists only in dimensions n_k=3k+2, where k=1,2,4,8. In these dimensions it reduces the orthogonal group …

We give a representation theoretical proof of Branson's classification of minimal elliptic sums of generalized gradients. The original proof uses tools of harmonic analysis, which as powerful as they are, seem to be specific for the structure groups SO(n) and Spin(n). The different approach we propose is based on the r…

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli …

We show that a finite group which admits a faithful, smooth, orientation-preserving action on a homology 4-sphere, and in particular on the 4-sphere, is isomorphic to a subgroup of the orthogonal group SO(5), by explicitly determining the various groups which can occur (up to an indetermination of index two in the case…

Chevalley theorems extended to isotropic functions on matrix spaces.

In this article, we study Einstein Kropina metrics on Lie groups and homogeneous spaces. We give a method to construct Einstein Kropina metrics on Lie groups. As an example of this method, a family of non-Riemannian Einstein Kropina metrics on the special orthogonal group $SO(n)$ is given. Then, we classify all left in…

In 1999 Chas and Sullivan showed that the homology of the free loop space of an oriented manifold admits the structure of a Batalin-Vilkovisky algebra. In this paper we give a direct description of this Batalin-Vilkovisky algebra in the case that the manifold is a compact Lie group G. Our answer is phrased in terms of …

Fast approximates orthogonal matrices for PCA.

Study on counting special-orthogonal Anosov orbits in geometric spaces.

A new method for optimizing neural networks with orthogonal constraints.

New algorithm speeds up group equivariant neural networks computations.

A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.

J.H.C. Whitehead defined a map $J_r:π_r(SO)\rightarrow π_r^s$ from the homotopy of the special orthogonal group to the stable homotopy of spheres. Within a toy model we show how the known computation for kernel$(J)$ leads to nonlinear $σ$-models with spherical source (space) and spherical target which admit false vacua…

The paper classifies topological holonomy groups in $SO(3)$.

We consider orientation-preserving actions of a finite group G on the 3-sphere S^3 (and also on Euclidean space R^3). By the geometrization of finite group actions on 3-manifolds, if such an action is smooth then it is conjugate to an orthogonal action, and in particular G is isomorphic to a subgroup of the orthogonal …

We investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric.

Researchers study the conformal geometry of bivariate Gaussian manifolds.

We classify the effective and transitive actions of a Lie group $G$ on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that $G$ is a closed, connected Lie subgroup of $SO_0(n-r,r+1)$, the connected component of the indefinite special orthogonal group. Assumin…

The paper classifies actions of a specific group on certain manifolds.

New optimization algorithms on orthogonal group for machine learning.

Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real orthogonal group with signature $(p,q)$. We give a complete description of the spaces of continuous and generalized translation- and $\mathrm{SO}^+(p,q)$-invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invari…

The paper proves stability for Möbius transformations in high dimensions.

A left-invariant sub-Riemannian metric $d$ on the shortened Lorentz group $SO_0(2,1)$ under the condition that $d$ is right-invariant relative to the orthogonal Lie subgroup $1\otimes SO(2)$ is studied. The distance between arbitrary two elements, the cut locus (as the union of the subgroup $1\otimes SO(2)$ with the an…

Investigates harmonic self-maps' stability on cohomogeneity one manifolds.

We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B…

We provide new conditions for the Strong Atiyah conjecture to lift to finite group extensions. In particular, we show cocompact special groups satisfy these conditions, so the Strong Atiyah conjecture holds for virtually cocompact special groups.

Paper addresses group synchronization with incomplete measurements and proves linear convergence of GPM.

We propose studies of special Riemannian geometries with structure groups $H_1=SO(3)\subset SO(5)$, $H_2=SU(3)\subset SO(8)$, $H_3=Sp(3)\subset SO(14)$ and $H_4=F_4\subset SO(26)$ in respective dimensions 5, 8, 14 and 26. These geometries, have torsionless models with symmetry groups $G_1=SU(3)$, $G_2=SU(3)\times SU(3)…

In higher dimensions, Schottky spaces have unique topological properties.

Paper refines Alesker-Bernig-Schuster theorem, proving Hodge-Riemann relations for Euclidean balls.

Let $R$ be an infinite commutative ring with identity and $n\geq 2$ be an integer. We prove that for each integer $i=0,1,\cdots ,n-2,$ the $L^{2}$-Betti number $b_{i}^{(2)}(G)=0,$ $\$when $G=\mathrm{GL}_{n}(R)$ the general linear group, $\mathrm{SL}_{n}(R)$ the special linear group, $$ the group generated by…

Classifies symmetries of knots using group actions and orthogonal representation theory.