On certain locally homogeneous Clifford manifolds

Barberis, M. L.; Miatello, Isabel G. Dotti; Miatello, Roberto J.

doi:10.1007/bf00773661

Cited by 46 publications

(53 citation statements)

References 8 publications

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“…, J m } is isomorphic to the real Clifford algebra associated to R m with the Euclidean inner product. Observe that a Cl 2 -structure is precisely a hypercomplex structure on g. Cl m -structures on certain solvable Lie algebras were constructed in [3]. If G is a Lie group with Lie algebra g then a Cl m -structure on g can be left translated to all of G.…”

Section: Preliminariesmentioning

confidence: 99%

Complex structures on affine motion groups

Barberis

Dotti

2004

The Quarterly Journal of Mathematics

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Abstract. We study existence of complex structures on semidirect products g ⊕ ρ v where g is a real Lie algebra and ρ is a representation of g on v. Our first examples, the Euclidean algebra e(3) and the Poincaré algebra e(2, 1), carry complex structures obtained by deformation of a regular complex structure on sl(2, C). We also exhibit a complex structure on the Galilean algebra G(3, 1). We construct next a complex structure on g ⊕ ρ v starting with one on g under certain compatibility assumptions on ρ.As an application of our results we obtain that there exists k ∈ {0, 1} such that (S 1 ) k × E(n) admits a left invariant complex structure, where S 1 is the circle and E(n) denotes the Euclidean group. We also prove that the Poincaré group P 4k+3 has a natural left invariant complex structure.In case dim g = dim v, then there is an adapted complex structure on g ⊕ ρ v precisely when ρ determines a flat, torsion-free connection on g. If ρ is self-dual, g ⊕ ρ v carries a natural symplectic structure as well. If, moreover, ρ comes from a metric connection then g ⊕ ρ v possesses a pseudo-Kähler structure.We prove that the tangent bundle T G of a Lie group G carrying a flat torsion free connection ∇ and a parallel complex structure possesses a hypercomplex structure. More generally, by an iterative procedure, we can obtain Lie groups carrying a family of left invariant complex structures which generate any prescribed real Clifford algebra.

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Section: Preliminariesmentioning

confidence: 99%

Complex structures on affine motion groups

Barberis

Dotti

2004

The Quarterly Journal of Mathematics

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“…If an almost complex structure J satisfies the condition [Jx, Jy] = [x, y] for all x, y ∈ g, then it is automatically integrable, and it is called abelian [7]. This name follows from the fact that the eigenspaces corresponding to the eigenvalues ±i of the natural extension J : g C → g C are abelian Lie subalgebras of the complex Lie algebra g C .…”

Section: Preliminariesmentioning

confidence: 99%

Complex product structures on 6-dimensional nilpotent Lie algebras

Andrada

2008

Forum Mathematicum

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Abstract. We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex product structure on a 6-dimensional nilpotent Lie algebra must be nilpotent in the sense of Cordero-Fernández-Gray-Ugarte. A study is made of the torsion-free connection associated to the complex product structure and we consider also the associated hypercomplex structures on the 12-dimensional nilpotent Lie algebras obtained by complexification.

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“…1 We now consider a family of Z 2 -manifolds, of cardinality quadratic in n, which are pairwise not isospectral on functions, but which are f -isospectral, according to Theorem 2.1.…”

Section: Examples and Counterexamplesmentioning

confidence: 99%

“…C n−1 and we take the respective column vector c n ≡ c 1 + · · · + c n−1 mod and having coordinates in 0, 1 2 . In (4.1), in the case when all * 's in the first n − 1 columns equal zero-and thus the * 's in the n-th column are 1 2 's, except for the first one, which is zero-the corresponding group was introduced in [11] and is known to be torsion-free, i.e., a Bieberbach group. We will denote this group by K n (see Fig.…”

Section: Large Families Of Manifolds Isospectral On Formsmentioning

confidence: 99%

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$${\mathbb{Z}}_2^k$$ -Manifolds are isospectral on forms

2007

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We obtain a simple formula for the multiplicity of eigenvalues of the HodgeLaplace operator, f , acting on sections of the full exterior bundle * (TM) = n p=0 p (TM)over an arbitrary compact flat Riemannian n-manifold M with holonomy group Z k 2 , 1 ≤ k ≤ n − 1. This formula implies that any two such manifolds having isospectral lattices of translations are isospectral with respect to f . As a consequence, we construct a large family of pairwise f -isospectral and nonhomeomorphic n-manifolds of cardinality greater than 2 (n−1)(n−2) 2 .

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On certain locally homogeneous Clifford manifolds

Cited by 46 publications

References 8 publications

Complex structures on affine motion groups

Complex structures on affine motion groups

Complex product structures on 6-dimensional nilpotent Lie algebras

$${\mathbb{Z}}_2^k$$ -Manifolds are isospectral on forms

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