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July 2011
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47 Reads
Inverse spectral theory addresses the question of which geometrical data of a Riemannian manifold (M, M, g) with some extra geometrical structure can be recovered or not recovered from the spectra of naturally associated differential operators.
December 2006
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11 Reads
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1 Citation
Monatshefte für Mathematik
We study some scalar curvature invariants on geodesic spheres and use them to characterize several kinds of Riemannian manifolds such as homogenous manifolds and in particular, the two-point homogeneous spaces and the Damek-Ricci spaces.
November 2006
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16 Reads
November 2006
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17 Reads
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2 Citations
November 2006
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21 Reads
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19 Citations
November 2006
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8 Reads
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2 Citations
January 2006
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47 Reads
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3 Citations
Advances in Geometry
An n-dimensional Riemannian manifold is called k-harmonic for some integer k, 1 ≤ k ≤ n - 1, if the k-th elementary symmetric functions of the principal curvatures of small geodesic spheres are radial functions. We prove that k-harmonic manifolds are necessarily 2-stein and show that locally symmetric manifolds which are k-harmonic for one k, are k-harmonic for all k. We then establish some results relating the harmonic and k-harmonic conditions for the class of non-compact harmonic non-symmetric spaces constructed by Damek and Ricci. We also discuss other notions of k-harmonicity and the problem of their equivalence.
October 2005
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15 Reads
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8 Citations
Bulletin of the Belgian Mathematical Society - Simon Stevin
We show that on any locally conformal Kähler (l.c.K.) manifold (M,J,g) with parallel Lee form the unit anti-Lee vector field is harmonic and minimal and determines a harmonic map into the unit tangent bundle. Moreover, the canonical distribution locally generated by the Lee and anti-Lee vector fields is also harmonic and minimal when seen as a map from (M,g) with values in the Grassmannian endowed with the Sasaki metric. As a particular case of l.c.K. manifolds, we look at locally conformal hyperkähler manifolds and show that, if the Lee form is parallel (that is always the case if the manifold is compact), the naturally associated three- and four-dimensional distributions are harmonic and minimal when regarded as maps with values in appropriate Grassmannians. As for l.c.K. manifolds without parallel Lee form, we consider the Tricerri metric on an Inoue surface and prove that the unit Lee and anti-Lee vector fields are harmonic and minimal and the canonical distribution is critical for the energy functional, but when seen as a map with values in it is minimal, but not harmonic.
July 2005
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10 Reads
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3 Citations
December 2004
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19 Reads
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23 Citations
Israel Journal of Mathematics
We consider an oriented distribution as a section of the corresponding Grassmann bundle and, by computing the tension of this map for conveniently chosen metrics, we obtain the conditions which the distribution must satisfy in order to be critical for the functionals related to the volume or the energy of the map. We show that the three-dimensional distribution ofS 4m+3 tangent to the quaternionic Hopf fibration defines a harmonic map and a minimal immersion and we extend these results to more general situations coming from 3-Sasakian and quaternionic geometry.
... In [Sza85] it was shown that all such locally irreducible and complete examples must be three dimensional and he constructed many local examples. To our knowledge, the only known complete examples are submanifolds in Euclidean space, see [Tak72], [DF01], and [BKV96]. ...
November 1996
... The vector field N p = v v is a unit normal of T 1 N. In contrast with the horizontal lift of a vector field, the vertical lift is not in general tangent to T 1 N [3]; for this reason, it was defined the tangential lift of Υ 1 ∈ T p N to p ∈ T 1 N as following [3] ...
June 1997
Houston Journal of Mathematics
... Remark. This is a special case of a more general result concerning isometric re¯ections on special semi-direct product Lie groups; (see the forthcoming paper [6] by the same authors). ...
December 1999
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
... These transformations are extensions of geodesic symmetries and local reflections with respect to submanifolds. They have been introduced in [5], [13] (see also [3]). In [5] [8], we studied conformal and divergence-preserving geodesic transformations and used them to characterize real, complex and quaternionic space forms and harmonic spaces as well as special classes of submanifolds. ...
Reference:
Holomorphic geodesic transformations
August 1997
... Harmonic vector fields X : M —> TM on a Riemannian manifold, when TM carries the Sasaki metric, are studied in [10]. We also recall that any endomorphism field F on a (semi-)Riemannian manifold (M,g) gives rise to a map F : (TM,g C ) —> (TM,g C ), so that it was possible in [6,7,8] to introduce and study harmonic endomorphism fields, harmonic connections and harmonic tensor fields. ...
Reference:
Harmonic Maps From The Tangent Bundle
January 1999
... Among others, for semi-parallel submanifolds in real space forms see References [184][185][186][187][188][189][190][191][192][193]; for semi-parallel submanifolds of indefinite space forms see References [194,195]; for semi-parallel submanifolds in Kaehler manifolds see [196][197][198][199]; for semi-parallel submanifolds in reducible spaces see Reference [117]; for manifold with semi-parallel geodesic spheres or semi-parallel tubes see Reference [200]; for semi-parallel submanifolds in contact metric manifolds see References [201,202]; and for semi-parallel submanifolds in other Riemannian manifolds see References [203][204][205]. For some further results on semi-parallel submanifolds see Reference [2]. ...
March 1996
Indian Journal of Pure and Applied Mathematics
... The The following interesting subclass of transversally symmetric (Riemannian) foliations was introduced and studied in [111][112][113][114]. These foliated spaces were studied in [111][112][113][114] and in [110] with the use of the extrinsic and the intrinsic geometry of geodesic spheres and tubular hypersurfaces around flow lines and around geodesics orthogonal to flow lines. ...
January 1996
Publicationes Mathematicae Debrecen
... If one assumes that the complex structure involved is in fact integrable, Gray [12] showed one of the components does not appear so there are 9 irreducible unitary modules in the decomposition in the context of Hermitian geometry if m ≥ 8, 8 if m = 6, and 6 if m = 4. Kähler geometry remains a field of active investigation in many different contexts [16, 17, 21, 26]; the Riemannian Kähler curvature tensors have 3 factors in their decomposition (m ≥ 4) as unitary modules. Note that Sasakian geometry is intimately linked with Kähler geometry – see, for example, the discussion in [6, 8] – so odd dimensional phenomena can also appear in this setting. De Smedt [7] showed there are 37 modules in the decomposition of R under the action of the symplectic group in the hyper-Hermitian setting for m ≥ 16 (the number drops to 36 if m = 12 and to 32 if m = 8). ...
January 1996
Balkan Journal of Geometry and Its Applications
... Denoting by (J, g) the Kähler structure of M (n+p)/2 (c), we find by definition (cf. [4,6,11,12,14]) that the maximal J-invariant subspace Hence we have, for any tangent vector field X and for a local orthonormal basis {ξ α ; α = 1, . . . , p} (ξ 1 := ξ) of normal vectors to M , the following decomposition in tangential and normal components: ...
May 1994
Rendiconti del Circolo Matematico di Palermo
... Let V be a unit Killing vector field on an arbitrary Riemannian manifold (M, g). Then A V is skewsymmetric and we have [19]). Hence, we obtain ...
January 1995
Kyungpook mathematical journal
