On harmonic and asymptotically harmonic homogeneous spaces

Heber, Jens

doi:10.1007/s00039-006-0569-4

Cited by 41 publications

(41 citation statements)

References 30 publications

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“…(a) Let T r be defined as in (13). Then F T r (λ) = 2 n π n/2 Γ 1+ 1 2 n sinh n 1 2 r cosh q−1 1 2 r ϕ (p,q+2) λ (r).…”

Section: Applications: Two-radius Theorems In Damek-ricci Spacesmentioning

confidence: 99%

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Spherical spectral synthesis and two-radius theorems on Damek–Ricci spaces

Peyerimhoff¹,

Samiou

2010

Ark. Mat.

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“…(a) Let T r be defined as in (13). Then F T r (λ) = 2 n π n/2 Γ 1+ 1 2 n sinh n 1 2 r cosh q−1 1 2 r ϕ (p,q+2) λ (r).…”

Section: Applications: Two-radius Theorems In Damek-ricci Spacesmentioning

confidence: 99%

“…Recently, Heber [13] proved that the non-flat simply-connected homogeneous harmonic spaces are precisely the symmetric spaces of rank one and the non-symmetric Damek-Ricci spaces.…”

Section: Introductionmentioning

confidence: 99%

Spherical spectral synthesis and two-radius theorems on Damek–Ricci spaces

Peyerimhoff¹,

Samiou

2010

Ark. Mat.

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“…This class contains the symmetric spaces of noncompact type and more generally, the irreducible nonflat homogeneous spaces of nonpositive curvature which are Einstein (by a result of [10]), where Damek-Ricci spaces are a distinguished subclass. Furthermore, D'Atri spaces of Iwasawa type of rank one and homogeneous (simply connected) harmonic spaces are Damek-Ricci spaces as is shown in [8] and [11], respectively. In particular, we focus our attention on those S that are Carnot spaces, defined as solvable extensions of codimension one of a two-step nilpotent Lie group with left invariant metrics.…”

mentioning

confidence: 95%

Geometry of D’Atri spaces of type k

Druetta

2010

Ann Glob Anal Geom

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A Riemannian n-dimensional manifold M is a D'Atri space of type k (or k-D'Atri space), 1 ≤ k ≤ n − 1, if the geodesic symmetries preserve the k-th elementary symmetric functions of the eigenvalues of the shape operators of all small geodesic spheres in M. Symmetric spaces are k-D'Atri spaces for all possible k ≥ 1 and the property 1-D'Atri is the D'Atri condition in the usual sense. In this article we study some aspects of the geometry of k-D'Atri spaces, in particular those related to properties of Jacobi operators along geodesics. We show that k-D'Atri spaces for all k = 1, . . ., l satisfy that tr(R k v ), v a unit vector in T M, is invariant under the geodesic flow for all k = 1, . . ., l. Further, if M is k-D'Atri for all k = 1, . . ., n − 1, then the eigenvalues of Jacobi operators are constant functions along geodesics. In the case of spaces of Iwasawa type, we show that k-D'Atri spaces for all k = 1, . . ., n − 1 are exactly the symmetric spaces of noncompact type. Moreover, in the class of Damek-Ricci spaces, the symmetric spaces of rank one are characterized as those that are 3-D'Atri.A Riemannian manifold M n is called a D'Atri space of type k or a k-D'Atri space, 1 ≤ k ≤ n − 1, if the geodesic symmetries preserve the k-th elementary symmetric functions of the eigenvalues of the shape operators of all small geodesic spheres. This definition was introduced in [12] and it is a natural generalization of the concept of D'Atri spaces.Recall, that the k-th elementary symmetric function σ k , k = 1, . . ., n, of the eigenvalues of a symmetric endomorphism A on a n -dimensional real vectorial space are determined by its characteristic polynomial as follows, M. J. Druetta (B)

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“…It is an open problem if these examples exhaust all asymptotically harmonic Hadamard manifolds. See [9] for the homogeneous case.…”

Section: Introductionmentioning

confidence: 99%