Geometric Realizations of Affine Kähler Curvature Models

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

doi:10.1007/s00025-011-0105-1

Cited by 11 publications

(31 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the following, by "the curvature tensor" we always mean the curvature tensor of type (1,3), if not stated otherwise.…”

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

“…Let (M, g) be a Riemannian manifold, p ∈ M a fixed point, and let R or R denote its curvature tensor field of type (0,4), or of type (1,3), respectively. Then the following two conditions are mutually equivalent:…”

Section: Proposition 23mentioning

confidence: 99%

“…Finally, a survey appeared as the last chapter of the monograph [2]. The extended study of pseudo-Riemannian curvature homogeneous spaces of arbitrary dimension and signature and for arbitrary orders was performed some years later by Gilkey and his collaborators (Brozos-Vásquez, Nikčević, and others) -see [3] and [7] for the surveys.…”

Section: Introductionmentioning

confidence: 99%

“…In the present article we initiate the study of the related topic, namely Riemannian curvature homogeneous spaces of type (1,3). As we shall see, "classical" curvature homogeneous spaces are also type (1,3)-curvature homogeneous but the latter spaces form a much bigger class.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On curvature‐homogeneous spaces of type (1,3)

Kowalski

Vanžurová

2011

Mathematische Nachrichten

View full text Add to dashboard Cite

Key words Riemannian manifold, curvature homogeneous manifold, locally homogeneous space MSC (2010) 53C21, 53C25, 53C30 Curvature homogeneous spaces have been studied by many authors. In this paper, we introduce and study a natural modification of this class, namely so-called curvature homogeneous spaces of type (1,3). We present a class of proper examples in every dimension and we prove a classification theorem in dimension 3 (for the generic case). We restrict ourselves to the Riemannian situation and to curvature homogeneity of order zero.

show abstract

“…In the following, by "the curvature tensor" we always mean the curvature tensor of type (1,3), if not stated otherwise.…”

Section: Definitions and Basic Propertiesmentioning

confidence: 99%

Section: Proposition 23mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On curvature‐homogeneous spaces of type (1,3)

Kowalski

Vanžurová

2011

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…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%

Transplanting geometrical structures

Euh

Gilkey

Park

et al. 2013

Differential Geometry and its Applications

Self Cite

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

(para)-Kähler Weyl Structures

Gilkey

Nikčević

2012

Recent Trends in Lorentzian Geometry

Self Cite

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

Geometric Realizations of Affine Kähler Curvature Models

Abstract: We show that every Kähler affine curvature model can be realized geometrically.Mathematics Subject Classification (2010). 53B20.

Cited by 11 publications

References 10 publications

On curvature‐homogeneous spaces of type (1,3)

On curvature‐homogeneous spaces of type (1,3)

Transplanting geometrical structures

(para)-Kähler Weyl Structures

Contact Info

Product

Resources

About