Cyclic homogeneous Riemannian manifolds

Gadea, Pedro M.; González–Dávila, J. C.; Oubiña, José A.

doi:10.1007/s10231-015-0534-7

Cited by 8 publications

(6 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the latter paper, the classification of simply connected traceless cyclic homogeneous Riemannian manifolds of dimension 4 was given. The present authors extended it to the cyclic case in [6]. The first examples which are not cyclic metric Lie groups [5] do appear in dimension four.…”

Section: Introductionmentioning

confidence: 88%

See 1 more Smart Citation

Homogeneous spin Riemannian manifolds with the simplest Dirac operator

Gadea

González–Dávila

Oubiña

2018

Advances in Geometry

Self Cite

View full text Add to dashboard Cite

We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds (M, g) which are traceless cyclic with respect to some quotient expression M = G/K and reductive decomposition g = k⊕m. Using transversally symmetric fibrations of noncompact type, we give a list of them.Keywords: Dirac operator, homogeneous spin Riemannian manifolds, traceless cyclic homogeneous Riemannian manifolds, Riemannian symmetric spaces.MSC 2010: 53C30, 53C35, 34L40.Acknowledgments: The first and second authors have been supported by DGI (Spain) Project MTM2013-46961-P.

show abstract

Section: Introductionmentioning

confidence: 88%

“…To this end, we recall that a homogeneous Riemannian manifold (M, g) is said [6] to be cyclic if there exists a quotient expression M = G/K and a reductive decomposition g = k ⊕ m satisfying (1.2)…”

Section: Introductionmentioning

confidence: 99%

Homogeneous spin Riemannian manifolds with the simplest Dirac operator

Gadea

González–Dávila

Oubiña

2018

Advances in Geometry

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is known that cyclic homogeneous manifolds are never compact [TV88], which fits into the general picture (see the examples) below that all interesting examples are non compact. For dimension n ≤ 4, they are classified in [GGO14]. Examples on Lie groups can be found in [GGO15].…”

Section: Curvaturementioning

confidence: 99%

Manifolds with vectorial torsion

Agricola

Kraus

2016

Differential Geometry and its Applications

View full text Add to dashboard Cite

The present note deals with the properties of metric connections ∇ with vectorial torsion V on semi-Riemannian manifolds (M n , g). We show that the ∇-curvature is symmetric if and only if V ♭ is closed, and that V ⊥ then defines an (n − 1)-dimensional integrable distribution on M n . If the vector field V is exact, we show that the V -curvature coincides up to global rescaling with the Riemannian curvature of a conformally equivalent metric. We prove that it is possible to construct connections with vectorial torsion on warped products of arbitrary dimension matching a given Riemannian or Lorentzian curvature-for example, a V -Ricci-flat connection with vectorial torsion in dimension 4, explaining some constructions occurring in general relativity. Finally, we investigate the Dirac operator D of a connection with vectorial torsion. We prove that for exact vector fields, the V -Dirac spectrum coincides with the spectrum of the Riemannian Dirac operator. We investigate in detail the existence of V -parallel spinor fields; several examples are constructed. It is known that the existence of a V -parallel spinor field implies dV ♭ = 0 for n = 3 or n ≥ 5; for n = 4, this is only true on compact manifolds. We prove an identity relating the V -Ricci curvature to the curvature in the spinor bundle. This result allows us to prove that if there exists a nontrivial V -parallel spinor, then Ric V = 0 for n = 4 and Ric V (X) = X dV ♭ for n = 4. We conclude that the manifold is conformally equivalent either to a manifold with Riemannian parallel spinor or to a manifold whose universal cover is the product of R and an Einstein space of positive scalar curvature. We also prove that if dV ♭ = 0, there are no non-trivial ∇-Killing spinor fields.

show abstract

“…4. We prove (Lemma 4.1, Theorem 4.2) that every semisimple cyclic metric Lie group (and, more generally, every nonabelian cyclic metric Lie group) is not compact.…”

Section: Introductionmentioning

confidence: 98%

“…The homogeneous Riemannian spaces G/H that generalize in a natural way the cyclic metric Lie groups are the cyclic homogeneous Riemannian manifolds, that we have studied in [4]. Note that Pfäffle and Stephan [12, p. 267] (see also Friedrich [3]) proved that the Dirac operator on a Riemannian manifold (M, g) does not contain any information on the Cartan-type component of the torsion of a generic Riemannian connection.…”

Section: Introductionmentioning

confidence: 98%

Cyclic metric Lie groups

Gadea¹,

González–Dávila

Oubiña

2014

Monatsh Math

Self Cite

View full text Add to dashboard Cite

Cyclic metric Lie groups are Lie groups equipped with a left-invariant metric which is in some way far from being biinvariant, in a sense made explicit in terms of Tricerri and Vanhecke's homogeneous structures. The semisimple and solvable cases are studied. We extend to the general case, Kowalski-Tricerri's and Bieszk's classifications of connected and simply-connected unimodular cyclic metric Lie groups for dimensions less than or equal to five.

show abstract

Cyclic homogeneous Riemannian manifolds

Cited by 8 publications

References 21 publications

Homogeneous spin Riemannian manifolds with the simplest Dirac operator

Homogeneous spin Riemannian manifolds with the simplest Dirac operator

Manifolds with vectorial torsion

Cyclic metric Lie groups

Contact Info

Product

Resources

About