Geodesic symmetries in spaces with special curvature tensors

Abstract. We prove that a four-dimensional Hermitian Einstein space is weakly *-Einsteinian and use this result to show that all geodesic symmetries are volume-preserving (up to sign) if and only if it is local symmetric. §1. IntroductionRiemannian manifolds such that all (local) geodesic symmetries are volume-preserving (up to sign) or equivalently, are divergence-preserving, have been introduced in [5] and are called DΆtri spaces [27]. The first examples which are not locally symmetric were discovered in [4], [6]. These are the naturally reductive homogeneous spaces. Since then, many other classes of examples has been found and studied. The main classes are the following: Riemannian g.o. spaces (i.e., spaces such that every geodesic is an orbit of a one-parameter group of isometries), commutative spaces (i.e., homogeneous spaces whose algebra of all differential operators which are invariant under all isometries is commutative), generalized Heisenberg groups, harmonic spaces (in particular, the Damek-Ricci examples), weakly symmetric spaces, 5C-spaces (i.e., spaces such that the principal curvatures of small geodesic spheres have antipodal symmetry), probabilistic commutative spaces, TC-and Co-spaces (see [1], [2] for more details). Of course, any manifold which is locally isometric to one of these examples is also a DΆtri space. We refer to

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“…The first examples which are not locally symmetric were discovered in [4], [6]. These are the naturally reductive homogeneous spaces.…”

Section: §1 Introductionmentioning

confidence: 99%

Volume-preserving geodesic symmetries on four-dimensional Hermitian Einstein spaces

Cho

Sekigawa

Vanhecke

1997

Nagoya Mathematical Journal

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“…D'Atri and Nickerson (see [6]) proved that every naturally reductive Riemannian manifold has this property. See [10] for a survey of the whole topic.…”

Section: Introductionmentioning

confidence: 99%

Classification of 4-dimensional homogeneous D’Atri spaces

Arias-Marco

Kowalski

2008

Czech Math J

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“…However, a product formula for two MZV fractions is known only in this case and in the case of Eq. (4). In this paper we will provide a product formula for any two MZV fractions making use of the general double shuffle framework introduced in our previous work [9] which is obtained with motivation from the shuffle relation and quasi-shuffle (stuffle) relation of MZVs.…”

Section: Introductionmentioning

confidence: 99%

“…In the spacial case when s i = 1, 1 i k, such fractions appeared in connection with differential geometry [3,4] and polylogarithms [6] where their products were shown to satisfy the shuffle relation. For example, In general, such fractions occur naturally from multiple zeta values which, since their introduction in the early 1990s, have attracted much attention from a wide range of areas in mathematics and mathematical physics [1,2,5,7,10,11,12,13,14,17].…”

Section: Introductionmentioning

confidence: 99%

The shuffle relation of fractions from multiple zeta values

Guo

Xie

2011

Ramanujan J

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Abstract. Partial fraction methods play an important role in the study of multiple zeta values. One class of such fractions is related to the integral representations of MZVs. We show that this class of fractions has a natural structure of shuffle algebra. This finding conceptualizes the connections among the various methods of stuffle, shuffle and partial fractions in the study of MZVs. This approach also gives an explicit product formula of the fractions.

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Geodesic symmetries in spaces with special curvature tensors

Cited by 50 publications

References 5 publications

Volume-preserving geodesic symmetries on four-dimensional Hermitian Einstein spaces

Volume-preserving geodesic symmetries on four-dimensional Hermitian Einstein spaces

Classification of 4-dimensional homogeneous D’Atri spaces

The shuffle relation of fractions from multiple zeta values

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