Bi-invariant Riemannian Metric Research Articles

On the δ-Pinching Function of the Sectional Curvature of a Compact Connected Lie Group G with a Bi-Invariant Riemannian Metric and a Vectorial Torsion Connection

One of the important problems of Riemannian geometry is the problem of establishing connections between curvature and the topology of a Riemannian manifold, and, in particular, the influence of the sign of sectional curvature on the topological structure of a Riemannian manifold. Of particular importance in these studies is the question of the influence of d-pinching of Riemannian metrics of positive sectional curvature on the geometric and topological structure of the Riemannian manifold. This question is most studied for the homogeneous Riemannian case. In this direction, the classification of homogeneous Riemannian manifolds of positive sectional curvature, obtained by M. Berger, N. Wallach, L. Bergeri, as well as a number of results on d- pinching of homogeneous Riemannian metrics of positive sectional curvature, is well known.
 In this paper, we investigate Riemannian manifolds with metric connection being a connection with vectorial torsion. The Levi-Civita connection falls into this class of connections. Although the curvature tensor of these connections does not possess the symmetries of the Levi-Civita connection curvature tensor, it seems possible to determine sectional curvature. This paper studies the d-pinch function of the sectional curvature of a compact connected Lie group G with a biinvariant Riemannian metric and a connection with vectorial torsion. It is proved that it takes the values d(||V ||)∈(0,1].

The cubic de Casteljau construction and Riemannian cubics

The classical de Casteljau construction of cubic polynomials in Euclidean space Em has been generalised to Riemannian settings since the 1980's. The generalisations replace line segments by geodesic arcs, yielding elegant methods for Hermite interpolation in Riemannian manifolds M, such as spheres and rotation groups. Unlike the classical algorithm however, it is not so easy to analyse the resulting curves. Even when M is the unit m-sphere Sm in Em+1, the mathematical properties of generalised cubic de Casteljau curves are not well understood.There is another class of curves called Riemannian cubics which can also be used for Hermite interpolation in Riemannian manifolds. Unlike generalised cubic de Casteljau curves, Riemannian cubics are defined as critical curves for a variational problem, and their mathematical properties are much better understood. Riemannian cubics are also more difficult to construct, whereas generalised cubic de Casteljau curves have a simple geometrical construction.It is well-known that, when M is curved, generalised cubic de Casteljau curves and Riemannian cubics are different. On the other hand their general appearance is somewhat similar. In the classical situation where M=Em, both cubic de Casteljau curves and Riemannian cubics reduce to curves that are cubic polynomial in each coordinate.The present paper analyses the differences between generalised cubic de Casteljau curves and Riemannian cubics. We also modify the generalised cubic de Casteljau algorithm to yield curves that are much closer to Riemannian cubics. In this way the elegant geometry of the de Casteljau algorithm is adjusted to better approximate a class of curves that are better understood from a mathematical point of view. Examples are given of the modified cubic de Casteljau algorithm for curves in the 2-dimensional unit sphere S2⊂E3, and in the special orthogonal group SO(3) with bi-invariant Riemannian metric.

SPECTRAL UNIQUENESS OF BI-INVARIANT METRICS ON SYMPLECTIC GROUPS

In this short note, we prove that a bi-invariant Riemannian metric on $\mathrm{Sp}(n)$ is uniquely determined by the spectrum of its Laplace-Beltrami operator within the class of left-invariant metrics on $\mathrm{Sp}(n)$. In other words, on any of these compact simple Lie groups, every left-invariant metric which is not right-invariant cannot be isospectral to a bi-invariant metric. The proof is elementary and uses a very strong spectral obstruction proved by Gordon, Schueth, and Sutton.

Left Lie reduction for curves in homogeneous spaces

Let H be a closed subgroup of a connected finite-dimensional Lie group G, where the canonical projection π : G → G/H is a Riemannian submersion with respect to a bi-invariant Riemannian metric on G. Given a C∞ curve x : [a, b] → G/H, let $\tilde {x}:[a,b]\rightarrow G$ be the horizontal lifting of x with $\tilde {x}(a)=e$ , where e denotes the identity of G. When (G, H) is a Riemannian symmetric pair, we prove that the left Lie reduction $V(t):=\tilde x(t)^{-1}\dot {\tilde x}(t)$ of $\dot {\tilde x}(t)$ for t ∈ [a, b] can be identified with the parallel pullbackP(t) of the velocity vector $\dot {x}(t)$ from x(t) to x(a) along x. Then left Lie reductions are used to investigate Riemannian cubics, Riemannian cubics in tension and elastica in homogeneous spaces G/H. Simplifications of reduced equations are found when (G, H) is a Riemannian symmetric pair. These equations are compared with equations known for curves in Lie groups, focusing on the special case of Riemannian cubics in the 3-dimensional unit sphere S3.

The spectrum of the Laplace operator on connected compact simple Lie groups of rank four

In the present article, we explicitly compute the spectrum of the Laplace operator on smooth real-valued and complex-valued functions on connected compact simple Lie groups of rank four with a bi-invariant Riemannian metrics that correspond to the root systems B 4, C 4, and D 4. We also find a connection between the obtained formulas, number theory, and integral quadratic forms in two, three, and four variables.

INTEGRABILITY OF GEODESIC FLOWS FOR METRICS ON SUBORBITS OF THE ADJOINT ORBITS OF COMPACT GROUPS

Let G/K be an orbit of the adjoint representation of a compact connected Lie group G, σ be an involutive automorphism of G and \( \tilde{G} \) be the Lie group of fixed points of σ. We find a sufficient condition for the complete integrability of the geodesic ow of the Riemannian metric on \( \tilde{G}/\left(\tilde{G}\cap K\right) \) which is induced by the bi-invariant Riemannian metric on \( \tilde{G} \). The integrals constructed here are real analytic functions, polynomial in momenta. It is checked that this sufficient condition holds when G is the unitary group U(n) and σ is its automorphism determined by the complex conjugation.

Riemannian L p Averaging on Lie Group of Nonzero Quaternions

This paper discusses quaternion $L^p$ geometric weighting averaging working on the multiplicative Lie group of nonzero quaternions $\mathbb{H}^{*}$, endowed with its natural bi-invariant Riemannian metric. Algorithms for computing the Riemannian $L^p$ center of mass of a set of points, with $1 \leq p \leq \infty$ (i.e., median, mean, $L^p$ barycenter and minimax center), are particularized to the case of $\mathbb{H}^{*}$. Two different approaches are considered. The first formulation is based on computing the logarithm of quaternions which maps them to the Euclidean tangent space at the identity $\mathbf{1}$, associated to the Lie algebra of $\mathbb{H}^{*}$. In the tangent space, Euclidean algorithms for $L^p$ center of mass can be naturally applied. The second formulation is a family of methods based on gradient descent algorithms aiming at minimizing the sum of quaternion geodesic distances raised to power $p$. These algorithms converges to the quaternion Fr\'{e}chet-Karcher barycenter ($p=2$), the quaternion Fermat-Weber point ($p=1$) and the quaternion Riemannian 1-center ($p=+\infty$). Besides giving explicit forms of these algorithms, their application for quaternion image processing is shown by introducing the notion of quaternion bilateral filtering.

CLIFFORD-WOLF TRANSLATIONS OF LEFT INVARIANT RANDERS METRICS ON COMPACT LIE GROUPS

A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the sam distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous if for any two point $x_1, x_2\in M$ there is a Clifford-Wolf translation $\rho$ such that $\rho(x_1)=x_2$. In this paper, we study Clifford-Wolf translations of left invariant Randers metrics on compact Lie groups. The mian result is that a left invariant Randers metric on a connected compact simple Lie group is Clifford-Wolf homogeneous if and only if the indicatrix of the metric is a round sphere with respect to a bi-invariant Riemannian metric. This presents a large number of examples of non-reversible Finsler metrics which are Clifford-Wolf homogeneous.

Statistical analysis of relative pose information of subcortical nuclei: Application on ADNI data

Many brain morphometry studies have been performed in order to characterize the brain atrophy pattern of Alzheimer's disease (AD). The earliest studies focused on the volume of particular brain structures, such as hippocampus and entorhinal cortex. Even though volumetry is a powerful, robust and intuitive technique that has yielded a wealth of findings, more complex shape descriptors have been used to perform statistical shape analysis of particular brain structures. However, in shape analysis studies of brain structures the information of the relative pose between neighbor structures is typically disregarded. This work presents a framework to analyse pose information including the following approaches: similarity transformations with either pseudo-Riemannian or left-invariant Riemannian metric, and centered transformations with a bi-invariant Riemannian metric. As an illustration, an analysis of covariance (ANCOVA) and a discrimination analysis were performed on Alzheimer's Disease Neuroimaging Initiative (ADNI) data.

The laplace operator on normal homogeneous Riemannian manifolds

The article presents an information about the Laplace operator defined on the real-valued mappings of compact Riemannian manifolds, and its spectrum; some properties of the latter are studied. The relationship between the spectra of two Riemannian manifolds connected by a Riemannian submersion with totally geodesic fibers is established. We specify a method of calculating the spectrum of the Laplacian for simply connected simple compact Lie groups with biinvariant Riemannian metrics, by representations of their Lie algebras. As an illustration, the spectrum of the Laplacian on the group SU(2) is found.

YANG-MILLS CONNECTIONS ON A COMPACT CONNECTED SEMISIMPLE LIE GROUP

Abstract. Let G be a compact connected semisimple Lie group, g theLie algebra of G, g the canonical metric (the biinvariant Riemannianmetric which is induced from the Killing form of g), and r be the Levi-Civita connection for the metric g. Then, we get the fact that the Levi-Civita connection r in the tangent bundle TG over (G;g) is a Yang-Millsconnection. 1. IntroductionThe problem of nding metrics and connections which are critical pointsof some functional plays an important role in global analysis and Riemann-ian geometry. A Yang-Mills connection is a critical point of the Yang-MillsfunctionalYM(D) =12Z M kR D k 2 v g (1:1)on the space C E of all connections in a smooth vector bundle Eover a closed(compact and connected) Riemannian manifold (M;g), where R D is the cur-vature of D2C E . Equivalently, Dis a Yang-Mills connection if it satis es theYang-Mills equation (cf. [1, 5, 6]) D R D = 0; (1:2)(the Euler-Lagrange equations of the variational principle associated with (1.1)).If Dis a connection in a vector bundle Ewith bundle metric hover a Riemann-ian manifold (M;g), then the connection D given byh(D

Elastica in SO(3)

AbstractIn a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group SO(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = SO(3), we solve the Euler-Lagrange equation by quadratures.

The spectral geometry of the canonical Riemannian submersion of a compact Lie group

Let G be a compact connected Lie group which is equipped with a bi-invariant Riemannian metric. Let m ( x , y ) = x y be the multiplication operator. We show the associated fibration m : G × G → G is a Riemannian submersion with totally geodesic fibers and we study the spectral geometry of this submersion. We show that the pull-backs of eigenforms on the base have finite Fourier series on the total space and we give examples where arbitrarily many Fourier coefficients can be non-zero. We give necessary and sufficient conditions for the pull-back of a form on the base to be harmonic on the total space.

Duality and Riemannian cubics

Riemannian cubics are curves used for interpolation in Riemannian manifolds. Applications in trajectory planning for rigid bodiy motion emphasise the group SO(3) of rotations of Euclidean 3-space. It is known that a Riemannian cubic in a Lie group G with bi-invariant Riemannian metric defines a Lie quadraticV in the Lie algebra, and satisfies a linking equation. Results of the present paper include explicit solutions of the linking equation by quadrature in terms of the Lie quadratic, when G is SO(3) or SO(1,2). In some cases we are able to give examples where the Lie quadratic is also given in closed form. A basic tool for constructing solutions is a new duality theorem. Duality is also used to study asymptotics of differential equations of the form $\dot{x}(t)=(\beta_{0}+t\beta_{1})x(t)$ , where β0,β1 are skew-symmetric 3×3 matrices, and x :ℝ→ SO(3). This is done by showing that the dual of β0+tβ1 is a null Lie quadratic. Then results on asymptotics of x follow from known properties of null Lie quadratics.

Asymptotic Expansions of Hermite Functions on Compact Lie Groups

Let G be a compact, connected Lie group endowed with a bi-invariant Riemannian metric. Let ρt be the heat kernel on G; that is, ρt is the fundamental solution to the heat equation on the group determined by the Laplace–Beltrami operator. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new family of functions on G. These functions, obtained from ρt and its derivatives, are the compact group analogs of the classical Hermite polynomials on \(\mathbb{R}^n \). Previous work of this author has established that these Hermite functions approach the classical Hermite polynomials on \(\mathfrak{g}\;\; = \;\;Lie(G)\) in the limit of small t, where the Hermite functions are viewed as functions on \(\mathfrak{g}\) via composition with the exponential map. The present work extends these results by showing that these Hermite functions can be expanded in an asymptotic series in powers of \(\sqrt t \). For symmetrized derivatives, it is shown that the terms with fractional powers of t vanish. Additionally, the asymptotic series for Hermite functions associated to powers of the Laplacian are computed explicitly. Remarkably, these asymptotic series terminate, yielding a polynomial in t.

Optimal pinching constants of odd dimensional homogeneous spaces

In this article we compute the pinching constants of all invariant Riemannian metrics on the Berger space B13=SU(5)/(Sp(2)×ℤ2S1) and of all invariant U(2)-biinvariant Riemannian metrics on the Aloff–Wallach space W7 1,1=SU(3)/S1 1,1. We prove that the optimal pinching constants are precisely in both cases. So far B13 and W7 1,1 were only known to admit Riemannian metrics with pinching constants.¶We also investigate the optimal pinching constants for the invariant metrics on the other Aloff–Wallach spaces W7 k,l =SU(3)/S1 k,l . Our computations cover the cone of invariant T2-biinvariant Riemannian metrics. This cone contains all invariant Riemannian metrics unless k/l=1. It turns out that the optimal pinching constants are given by a strictly increasing function in k/l∈[0,1]. Thus all the optimal pinching constants are ≤.¶In order to determine the extremal values of the sectional curvature of an invariant Riemannian metric on W7 k,l we employ a systematic technique, which can be applied to other spaces as well. The computation of the pinching constants for B13 is reduced to the curvature computation for two proper totally geodesic submanifolds. One of them is diffeomorphic to ℂℙ3/ℤ2 and inherits an Sp(2)-invariant Riemannian metric, and the other is W7 1,1 embedded as recently found by Taimanov. This approach explains in particular the coincidence of the optimal pinching constants for W7 1,1 and the Berger space B13.