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1.
Stuart Armstrong 《Annals of Global Analysis and Geometry》2008,33(1):47-69
The aim of this paper and its sequel is to introduce and classify the holonomy algebras of the projective Tractor connection.
After a brief historical background, this paper presents and analyses the projective Cartan and Tractor connections, the various
structures they can preserve, and their geometric interpretations. Preserved subbundles of the Tractor bundle generate foliations
with Ricci-flat leaves. Contact- and Einstein-structures arise from other reductions of the Tractor holonomy, as do U(1) and bundles over a manifold of smaller dimension. 相似文献
2.
For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of
the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of
this construction is to realise the normal conformal Tractor holonomy as affine holonomy of such a connection. We give an
example of an ambient connection for which this is the case, and which is torsion free if we start the construction with a
C-space, and in addition Ricci-flat if we start with an Einstein manifold. Thus, for a C-space this example leads to an ambient metric in the weaker sense of Čap and Gover, and for an Einstein space to a Ricci-flat
ambient metric in the sense of Fefferman and Graham.
Current address for first author: Erwin Schr?dinger International Institute for Mathematical Physics (ESI), Boltzmanngasse
9, 1090 Vienna, Austria
Current address for second author: Department of Mathematics, University of Hamburg, Bundesstra?e 55, 20146 Hamburg, Germany 相似文献
3.
Graham Hall 《Central European Journal of Mathematics》2012,10(5):1763-1770
This paper discusses the connection between projective relatedness and conformal flatness for 4-dimensional manifolds admitting
a metric of signature (+,+,+,+) or (+,+,+,−). It is shown that if one of the manifolds is conformally flat and not of the
most general holonomy type for that signature then, in general, the connections of the manifolds involved are the same and
the second manifold is also conformally flat. Counterexamples are provided which place limitations on the potential strengthening
of the results. 相似文献
4.
Quo-Shin Chi 《Annals of Global Analysis and Geometry》1991,9(3):197-204
Using the twistor theory on quaternionic Kaehler manifolds and some recent results on Blaschke manifolds and compact manifolds whose holonomy group is Spin (7), we prove that a Blaschke manifold of nonnegative scalar curvature whose holonomy group is exceptional is isometric to a projective space. 相似文献
5.
The classification of restricted holonomy groups of \(n\) -dimensional Lorentzian manifolds was obtained about ten years ago. However, up to now, not much is known about the structure of the full holonomy group. In this paper we study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. Based on the classification of the restricted holonomy groups of such manifolds, we prove several structure results about the full holonomy. We establish a construction method for manifolds with disconnected holonomy starting from a Riemannian manifold and a properly discontinuous group of isometries. This leads to a variety of examples, most of them being quotients of pp-waves with disconnected holonomy, including a non-flat Lorentzian manifold with infinitely generated holonomy group. Furthermore, we classify the full holonomy groups of solvable Lorentzian symmetric spaces and of Lorentzian manifolds with a parallel null spinor. Finally, we construct examples of globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces, disconnected full holonomy and a parallel spinor. 相似文献
6.
7.
8.
Robert Clancy 《Annals of Global Analysis and Geometry》2011,40(2):203-222
We find new examples of compact Spin(7)-manifolds using a construction of Joyce (J. Differ. Geom., 53:89–130, 1999; Compact manifolds with special holonomy. Oxford University Press, Oxford, 2000). The essential ingredient in Joyce’s construction is a Calabi–Yau 4-orbifold with particular singularities admitting an
antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces in
weighted projective spaces for suitable Calabi–Yau 4-orbifolds. 相似文献
9.
Kwan Hui Nam 《Geometriae Dedicata》2012,157(1):205-216
As first defined by Smillie, an affine manifold with diagonal holonomy is a manifold equipped with an atlas such that the
changes of charts are restrictions of elements of the subgroup of Aff (
\mathbbRn{\mathbb{R}^n}) formed by diagonal matrices. Refining Smillie’s theorem, Benoist proved that if a compact manifold M is split into manifolds with corners corresponding to complete simplicial fans of a fixed frame by its hypersurfaces with
normal crossing, then the product of M and a torus of suitable dimension is a finite cover of an affine manifold with diagonal holonomy, and conversely. Motivated
by the result of Benoist, we introduce a “Benoist manifold” and a natural definition of complexity for them. In particular,
we study some properties of “Benoist manifolds”. 相似文献
10.
Thomas Leistner 《Differential Geometry and its Applications》2006,24(5):458-478
The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves. 相似文献
11.
Carlos Currás-Bosch 《Israel Journal of Mathematics》1986,53(3):315-320
In this paper we give a generalisation of Kostant’s Theorem about theA
x
-operator associated to a Killing vector fieldX on a compact Riemannian manifold. Kostant proved (see [6], [5] or [7]) that in a compact Riemannian manifold, the (1, 1)
skew-symmetric operatorA
x
=L
x
−≡
x
associated to a Killing vector fieldX lies in the holonomy algebra at each point. We prove that in a complete non-compact Riemannian manifold (M, g) theA
x
-operator associated to a Killing vector field, with finite global norm, lies in the holonomy algebra at each point. Finally
we give examples of Killing vector fields with infinite global norms on non-flat manifolds such thatA
x
does not lie in the holonomy algebra at any point. 相似文献
12.
This paper is a continuation of Math. Res. Lett. 12 (2005), 493–512. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of Sn by looking at the conormal bundle of appropriate submanifolds of Sn. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey–Lawson
for submanifolds in Rn in their pioneering paper, Acta Math. 148 (1982), 47–157. We also construct calibrated submanifolds in complete metrics with special holonomy G2 and Spin(7) discovered by Bryant and Salamon (Duke Math. J. 58 (1989), 829–850) on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are
constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal
in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make
some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds
in a compact manifold with special holonomy.
Mathematics Subject Classification (2000): 53-XX, 58-XX. 相似文献
13.
In this article, we survey recent work on the construction and geometry of representations of a quiver in the category of
coherent sheaves on a projective algebraic manifold. We will also prove new results in the case of the quiver • ← • → •. 相似文献
14.
Anton S. Galaev 《Annals of Global Analysis and Geometry》2017,51(3):245-265
It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-Kählerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-Kählerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group. 相似文献
15.
Summary The ideas of holonomy group fixing an m-dimensional plane in a Finsler space were given by one of the present authors[1]. In that paper the deformation properties of the space admitting such holonomy group were of main consideration and, indeed,
the decomposition characteristics of the space were not touched upon. In the present paper we consider the decomposition of
the space due to the existence of holonomy group. The geometry is constructed on the decomposed metric of the space. The decomposition
properties of various entities such as the connection parameters, the covariant derivatives, the curvature tensors, and the
projective curvature tensors have been studied. In all there are six articles in the paper. The first of these is introductory.
The next three articles are dealt with the Cartan's approach to Finsler space whereas the fifth one is dealt with Berwald's
approach. The last article is devoted to the theory of decomposition in the projective curvature tensors.
Entrata in Redazione il 18 ottobre 1969. 相似文献
16.
The aim of this paper is to present some results about spin structures on flat manifolds. We prove that any finite group can
be the holonomy group of a flat spin manifold. Moreover, we shall give some methods of constructing spin structures related
to the holonomy representation. 相似文献
17.
S. E. Stepanov 《Mathematical Notes》1995,58(1):752-756
We consider the theory of constant rank projective mappings of compact Riemannian manifolds from the global point of view.
We study projective immersions and submersions. As an example of the results, letf:(M, g) → (N, g′) be a projective submersion of anm-dimensional Riemannian manifold (M, g) onto an (m−1)-dimensional Riemannian manifold (N, g′). Then (M, g) is locally the Riemannian product of the sheets of two integrable distributions Kerf
* and (Kerf
*)⊥ whenever (M, g) is one of the two following types: (a) a complete manifold with Ric ≥ 0; (b) a compact oriented manifold with Ric ≤ 0.
Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 111–118, July, 1995.
This work was partially supported by the Russian Foundation for Basic Research grant No. 94-01-0195. 相似文献
18.
The aim of this paper is to present some results about spin structures on flat manifolds. We prove that any finite group can
be the holonomy group of a flat spin manifold. Moreover, we shall give some methods of constructing spin structures related
to the holonomy representation. 相似文献
19.
We extend the Gallot–Tanno theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over a manifold
admits a parallel symmetric (0, 2)-tensor then it is Riemannian. Applications of this result to the existence of metrics with
distinct Levi-Civita connections but having the same unparametrized geodesics and to the projective Obata conjecture are given.
We also apply our result to show that the holonomy group of a closed (O(p + 1, q), S
p,q
)-manifold does not preserve any nondegenerate splitting of
\mathbb Rp+1,q{\mathbb {R}^{p+1,q}}. 相似文献
20.
In this paper, we prove that the natural metric on the connected component of the unit in the (Lie) motion group of a compact
Finsler manifold supplied with its inner metric generates a bi-invariant inner Finsler metric. The latter is defined by the
invariant Chebyshev norm on the Lie algebra of generators of 1-parameter motion subgroups on the manifold. This norm is equal
to the maximal value of the generator’s length. A δ-homogeneous manifold is characterized by the condition that the canonical
projection of the component onto the manifold is a submetry with respect to their inner metrics. The Chebyshev norms for the
Euclidean spheres, the Berger spheres, and homogeneous Riemannian metrics on the 3-dimensional complex projective space are
found. This gives interesting examples of invariant norms on Lie algebras and a new method for the separating of delta-homogeneous
but not normal metrics.
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra,
2008. 相似文献