Geometric Realizations of Para-Hermitian Curvature Models

Brozos‐Vázquez, Miguel; Gilkey, Peter Β.; Nikčević, Stana; Vázquez-Lorenzo, Ramón

doi:10.1007/s00025-009-0403-z

Cited by 16 publications

(31 citation statements)

References 57 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Geometric realizability. Theorem 1.1 and results described in [4] also yield the following result which formed part of the motivation of our paper; we will present several examples illustrating this result in Section 4: Theorem 1.3. Every algebraic model of the curvature tensor of a structure from Definition 1.1 is geometrically realizable by a compact manifold.…”

Section: Introductionmentioning

confidence: 64%

“…y) so that Γ 1 (P 1 ) = 0 (resp. Γ 2 (P 2 ) = 0); see, for example, the discussion in [4]. Having chosen x and y, we now identify P 1 = P 2 = 0 and U 1 = U 2 = B 3r .…”

Section: The Proof Of Theorem 11mentioning

confidence: 99%

“…Let ∇ be the Levi-Civita connection of g. One has the following useful result -see, for example, the discussion in [4,21].…”

Section: 5mentioning

confidence: 99%

“…We refer to [4,16] for background information concerning Weyl geometry. A Weyl structure on M is a pair (M, ∇) where g is a pseudo-Riemannian metric on M and where ∇ is a torsion free connection on M so that ∇g = −2ω ⊗ g for some smooth 1-form ω.…”

Section: Weyl Structuresmentioning

confidence: 99%

“…One then wants to show these identities hold more generally and this involves a transplanting problem. Similarly, when establishing geometric realization results, one often constructs examples which are only defined on a small neighborhood of the origin -the discussion in [4] provides a nice summary of these problems and we refer to other results in [5,6,15,16]. And one wants to then deduce these geometric realization results also hold in the compact setting.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Transplanting geometrical structures

Euh

Gilkey

Park

et al. 2013

Differential Geometry and its Applications

View full text Add to dashboard Cite

We say that a germ G of a geometric structure can be transplanted into a manifold M if there is a suitable geometric structure on M which agrees with G on a neighborhood of some point P of M . We show for a wide variety of geometric structures that this transplantation is always possible provided that M does in fact admit some such structure of this type. We use this result to show that a curvature identity which holds in the category of compact manifolds admitting such a structure holds for germs as well and we present examples illustrating this result. We also use this result to show geometrical realization problems which can be solved for germs of structures can in fact be solved in the compact setting as well. MSC 2010: 53B20 and 53B35.

show abstract

Section: Introductionmentioning

confidence: 64%

“…y) so that Γ 1 (P 1 ) = 0 (resp. Γ 2 (P 2 ) = 0); see, for example, the discussion in [4]. Having chosen x and y, we now identify P 1 = P 2 = 0 and U 1 = U 2 = B 3r .…”

Section: The Proof Of Theorem 11mentioning

confidence: 99%

“…Let ∇ be the Levi-Civita connection of g. One has the following useful result -see, for example, the discussion in [4,21].…”

Section: 5mentioning

confidence: 99%

Section: Weyl Structuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Transplanting geometrical structures

Euh

Gilkey

Park

et al. 2013

Differential Geometry and its Applications

View full text Add to dashboard Cite

show abstract

(para)-Kähler Weyl Structures

Gilkey

Nikčević

2012

Recent Trends in Lorentzian Geometry

View full text Add to dashboard Cite

Abstract. We work in both the complex and in the para-complex categories and examine (para)-Kähler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kähler Weyl algebraic curvature tensor is in fact Riemannian in dimension m ≥ 6; this yields as a geometric consequence that any (para)-Kähler Weyl geometric structure is trivial for m ≥ 6. By contrast, the 4-dimensional setting is, as always, rather special as it turns out that there are (para)-Kähler Weyl algebraic curvature tensors which are not Riemannian if m = 4. Since every (para)-Kähler Weyl algebraic curvature tensor is geometrically realizable and since every 4-dimensional Hermitian manifold admits a unique (para)-Kähler Weyl structure, there are also non-trivial 4-dimensional Hermitian (para)-Kähler Weyl manifolds. MSC: 53B05, 15A72, 53A15, 53B10, 53C07, 53C25. IntroductionLet ∇ be a torsion free connection on a pseudo-Riemannian manifold (M, g) of even dimension m = 2m ≥ 4. The triple (M, g, ∇) is said to be a Weyl structure if there exists a smooth 1-form φ so that ∇g = −2φ ⊗ g. Such a geometric structure was introduced by Weyl [37] in an attempt to unify gravity with electromagnetism. Although this approach failed for physical reasons, these geometries are still studied for their intrinsic interest [2,10,21,27,28]; they also appear in the mathematical physics literature [12,20,26]. Weyl geometry is relevant to submanifold geometry [25] and to contact geometry [15]. The pseudo-Riemannian setting also is important [1,24,32] as are para-complex geometries [11,13]. See also [9,22,30,31] for related results. The literature in the field is vast and we can only give a flavor of it for reasons of brevity. We shall be primarily interested in the Hermitian setting. However since there are applications to higher signature geometry, we include the pseudo-Hermitian context as well; similarly we treat para-Hermitian geometries as they can be studied with little additional effort. Section 1.1 of the Introduction deals with the real setting. In Theorem 1.1, we recall the basic theorems of geometric realizability for affine, Riemannian, and Weyl curvature models and in Theorem 1.2 provide various characterizations of the notion of a trivial Weyl structure. Section 1.2 treats the (para)-Kähler setting. In Theorem 1.3 we recall geometric realizibility results for (para)-Kähler affine and (para)-Kähler Riemannian curvature models. Theorem 1.4 presents results in the geometric setting for (para)-Kähler Weyl manifolds. Theorem 1.5 is one of the two main results of this paper: every (para)-Kähler curvature model is geometrically realizable. The proof of Theorem 1.5 relies on a curvature decomposition result; the second main result of the paper, Theorem 1.6, discusses the space of (para)-Kähler Weyl algebraic curvature tensors.1.1. Riemannian, Affine, and Weyl geometry. Let (V, ·, · ) be an inner product space of signature (p, q) and dimension m = p+ q; an inner product of signatur...

show abstract