Compact Riemannian Symmetric Spaces Research Articles

Laplace Eigenfunctions on Riemannian Symmetric Spaces and Borel–Weil Theorem

AbstractWe identify a geometric relation between the Laplace–Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel–Weil theory using ideas from symplectic geometry and geometric quantization. This is done by associating with each compact Riemannian symmetric space, via Marsden–Weinstein reduction, a generalized flag manifold that covers the space parametrizing all of its maximal totally geodesic tori. In the process, we notice a direct relation between the Satake diagram of the symmetric space and the painted Dynkin diagram of its associated flag manifold. We consider in detail the examples of the classical simply connected spaces of rank one and the space $SU(3)/SO(3)$. In the 2nd part of the paper, with the aid of harmonic polynomials, we induce Laplace–Beltrami eigenfunctions on the symmetric space from holomorphic sections of the associated line bundle on the generalized flag manifold. In the examples we consider, we show that our construction provides all of the eigenfunctions.

Explicit Proper p-Harmonic Functions on the Riemannian Symmetric Spaces mathbf{SU} (n)/mathbf{SO} (n), mathbf{Sp} (n)/mathbf{U} (n), mathbf{SO} (2n)/mathbf{U} (n), mathbf{SU} (2n)/mathbf{Sp} (n)

In this work we construct explicit complex-valued p-harmonic functions on the compact Riemannian symmetric spaces mathbf{SU} (n)/mathbf{SO} (n), mathbf{Sp} (n)/mathbf{U} (n), mathbf{SO} (2n)/mathbf{U} (n), mathbf{SU} (2n)/mathbf{Sp} (n). We also describe how the same can be manufactured on their non-compact symmetric dual spaces.

On Weakly Reflective Submanifolds in Compact Isotropy Irreducible Riemannian Homogeneous Spaces

We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.

Periodic solutions of Hilbert’s fourth problem

It is shown that a possibly irreversible [Formula: see text] Finsler metric on the torus, or on any other compact Euclidean space form, whose geodesics are straight lines is the sum of a flat metric and a closed [Formula: see text]-form. This is used to prove that if [Formula: see text] is a compact Riemannian symmetric space of rank greater than one and [Formula: see text] is a reversible [Formula: see text] Finsler metric on [Formula: see text] whose unparametrized geodesics coincide with those of [Formula: see text], then [Formula: see text] is a Finsler symmetric space.

Positive scalar curvature on simply connected spin pseudomanifolds

Let [Formula: see text] be an [Formula: see text]-dimensional Thom–Mather stratified space of depth [Formula: see text]. We denote by [Formula: see text] the singular locus and by [Formula: see text] the associated link. In this paper, we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge [Formula: see text]-class [Formula: see text]. In order to establish a sufficient condition, we need to assume additional structure: we assume that the link of [Formula: see text] is a homogeneous space of positive scalar curvature, [Formula: see text], where the semisimple compact Lie group [Formula: see text] acts transitively on [Formula: see text] by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these assumptions, when [Formula: see text] and [Formula: see text] are spin, we reinterpret our obstruction in terms of two [Formula: see text]-classes associated to the resolution of [Formula: see text], [Formula: see text], and to the singular locus [Formula: see text]. Finally, when [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are simply connected and [Formula: see text] is big enough, and when some other conditions on [Formula: see text] (satisfied in a large number of cases) hold, we establish the main result of this paper, showing that the vanishing of these two [Formula: see text]-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature.

Homogeneous Finsler spaces with only one orbit of prime closed geodesics

When a closed Finsler manifold admits continuous isometric actions, estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics. To extend the results of Duan, Long, Rademacher, Wang and others on the existence of two prime closed geodesics to the equivariant situation, we propose the question if a closed Finsler manifold has only one orbit of prime closed geodesics if and only if it is a compact rank-one Riemannian symmetric space. In this paper, we study this problem in homogeneous Finsler geometry, and get a positive answer when the dimension is even or the metric is reversible. We guess the rank inequality and the algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.

Relations Between Laplace Spectra and Geometric Quantization of Reimannian Symmetric Spaces

We consider a modified Kostant-Souriau geometric quantization scheme due to Czyz and Hess for Hamiltonian systems on the cotangent bundles of compact rank-one Riemannian symmetric spaces (CROSS). It is used, together with a symplectic reduction process, to relate its energy spectrum to the spectrum of the Laplace-Beltrami operator. Moreover, the corresponding eigenspaces have real dimension equal to the complex dimension of the space of the holomorphic sections of the quantum bundle which is obtained after the quantization. The relation between the two constructions was first noticed by Mladenov and Tsanov for the case of the spheres. In addition to the CROSS case, we announce preliminary results related to the case of compact Riemannian symmetric spaces of higher rank.

Non-flat totally geodesic surfaces in symmetric spaces of classical type

We give a classification of non-flat totally geodesic surfaces in compact Riemannian symmetric spaces of classical type.

Paley–Wiener theorem for line bundles over compact symmetric spaces and new estimates for the Heckman–Opdam hypergeometric functions

AbstractPaley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space . It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K. An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix.

Quantization of compact Riemannian symmetric spaces

The phase space of a compact, irreducible, simply connected, Riemannian symmetric space admits a natural family of Kähler polarizations parametrized by the upper half plane S . Using this family, geometric quantization, including the half-form correction, produces the field H c o r r → S of quantum Hilbert spaces. We show that projective flatness of H c o r r implies, that the symmetric space must be isometric to a compact Lie group equipped with a biinvariant metric. In the latter case the flatness of H c o r r was previously established.

Biharmonic hypersurfaces in Riemannian symmetric spaces I

We classify biharmonic geodesic spheres in the Cayley projective plane. Our results completes the classification of all biharmonic homogeneous hypersurfaces in simply connected compact Riemannian symmetric spaces of rank 1. In addition we show that complex Grassmannian manifolds, and exceptional Lie groups $F_4$ and $G_2$ admit proper biharmonic real hypersurfaces.

Some remarks on polar actions

We classify infinitesimally polar actions on compact Riemannian symmetric spaces of rank one. We also prove that every polar action on one of those spaces has the same orbits as an asystatic action.

Riesz Transforms on Compact Riemannian Symmetric Spaces of Rank One

In this paper we prove mixed norm estimates for Riesz transforms related to Laplace–Beltrami operators on compact Riemannian symmetric spaces of rank one. These operators are closely related to the Riesz transforms for Jacobi polynomials expansions. The key point is to obtain sharp estimates for the kernel of the Jacobi–Riesz transforms with uniform control on the parameters, together with an adaptation of Rubio de Francia’s extrapolation theorem. The latter results are of independent interest.

Fractional integrals on compact Riemannian symmetric spaces of rank one

In this paper we study mixed norm boundedness for fractional integrals related to Laplace–Beltrami operators on compact Riemannian symmetric spaces of rank one. The key point is the analysis of weighted inequalities for fractional integral operators associated to trigonometric Jacobi polynomials expansions. In particular, we find a novel sharp estimate for the Jacobi fractional integral kernel with explicit dependence on the type parameters.

Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces

Let \(M=G/H\) be a compact connected isotropy irreducible Riemannian homogeneous manifold, where \(G\) is a compact Lie group (may be, disconnected) acting on \(M\) by isometries. This class includes all compact irreducible Riemannian symmetric spaces and, for example, the tori \(\mathbb{R }^n/\mathbb{Z }^n\) with the natural action on itself extended by the finite group generated by all permutations of the coordinates and inversions in circle factors. We say that \(u\) is a polynomial on \(M\) if it belongs to some \(G\)-invariant finite dimensional subspace \(\mathcal{E }\) of \(L^2(M)\). We compute or estimate from above the averages over the unit sphere \(\mathcal{S }\) in \(\mathcal{E }\) for some metric quantities such as Hausdorff measures of level set and norms in \(L^p(M)\), \(1\le p\le \infty \), where \(M\) is equipped with the invariant probability measure. For example, the averages over \(\mathcal{S }\) of \(\Vert u\Vert _{L^p(M)}\), \(p\ge 2\), are less than \(\sqrt{\frac{p+1}{e}}\) independently of \(M\) and \(\mathcal{E }\).

Ramanujanʼs Master Theorem for Riemannian symmetric spaces

Ramanujanʼs Master Theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of compact and noncompact reductive Riemannian symmetric spaces inside a common complexification, we prove an analogue of Ramanujanʼs Master Theorem for the spherical Fourier transform of a spherical Fourier series. This extends the results proven by Bertram for Riemannian symmetric spaces of rank-one.

Infinite dimensional multiplicity free spaces III: matrix coefficients and regular functions

In earlier papers we studied direct limits \({(G,\,K) = \varinjlim\, (G_n,K_n)}\) of two types of Gelfand pairs. The first type was that in which the G n /K n are compact Riemannian symmetric spaces. The second type was that in which \({G_n = N_n\rtimes K_n}\) with N n nilpotent, in other words pairs (G n , K n ) for which G n /K n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections \({\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}\) for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit \({L^2(G/K) := \varinjlim L^2(G_n/K_n)}\) is multiplicity-free. This left open questions concerning the nature of the elements of L 2(G/K). Here we define spaces \({\mathcal{A}(G_n/K_n)}\) of regular functions on G n /K n and injections \({\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}\) for m ≧ n related to restriction by \({\nu_{m,n}(f)|_{G_n/K_n} = f}\). Thus the direct limit \({\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}\) sits as a particular G-submodule of the much larger inverse limit \({\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}\). Further, we define a pre Hilbert space structure on \({\mathcal{A}(G/K)}\) derived from that of L 2(G/K). This allows an interpretation of L 2(G/K) as the Hilbert space completion of the concretely defined function space \({\mathcal{A}(G/K)}\), and also defines a G-invariant inner product on \({\mathcal{A}(G/K)}\) for which the left regular representation of G is multiplicity-free.

ADDING A UNITON VIA THE DPW METHOD

In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces from certain holomorphic 1-forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we prove that the Gauss bundle of a harmonic map of finite type into a Grassmannian is also of finite type.

On harmonic tori in compact rank one symmetric spaces

In this paper we prove that, in contrast with the S n and C P n cases, there are harmonic 2-tori into the quaternionic projective space H P n which are neither of finite type nor of finite uniton number; we also prove that any harmonic 2-torus in a compact Riemannian symmetric space which can be obtained via the twistor construction is of finite type if and only it is constant; in particular, we conclude that any harmonic 2-torus in C P n or S n which is simultaneously of finite type and of finite uniton number must be constant.

Lorentzian Lie 3-algebras and their Bagger-Lambert moduli space

We classify Lie 3-algebras possessing an invariant lorentzian inner product. The indecomposable objects are in one-to-one correspondence with compact real forms of metric semisimple Lie algebras. We analyse the moduli space of classical vacua of the Bagger-Lambert theory corresponding to these Lie 3-algebras. We establish a one-to-one correspondence between one branch of the moduli space and compact riemannian symmetric spaces. We analyse the asymptotic behaviour of the moduli space and identify a large class of models with moduli branches exhibiting the desired N^{3/2} behaviour.