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1.
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
2.
IfM
2 is a nondegenerate surface in a 4-dimensional Riemannian manifold
, then there is a natural affine metricg defined onM
2. It is shown that this affine metricg is conformal to the induced Riemannian metric onM
2 if and only ifM
2 is a minimal submanifold of
in the usual Riemannian sense. If the conformal factor is a constant, then the two metrics are said to be homothetic. It is shown that there does not exist a nondegenerate surface in Euclidean space 4 or hyperbolic spaceH
4 whose affine metric is homothetic to the induced Riemannian metric. Furthermore, ifM
2 is a nondegenerate surface in the standard 4-sphereS
4 whose affine metric is homothetic to the induced Riemannian metric, thenM
2 is a Veronese surface.T. Cecil was supported by NSF Grant No. DMS-9101961. 相似文献
3.
Suhyoung Choi 《Geometriae Dedicata》2003,97(1):81-92
An affine manifold is a manifold with a flat affine structure, i.e. a torsion-free flat affine connection. We slightly generalize the result of Hirsch and Thurston that if the holonomy of a closed affine manifold is isomorphic to amenable groups amalgamated or HNN-extended along finite groups, then the Euler characteristic of the manifold is zero confirming an old conjecture of Chern. The technique is from Kim and Lee's work using the combinatorial Gauss–Bonnet theorem and taking the means of the angles by amenability. We show that if an even-dimensional manifold is obtained from a connected sum operation from K(, 1)s with amenable fundamental groups, then the manifold does not admit an affine structure generalizing a result of Smillie. 相似文献
4.
Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure. 相似文献
5.
Brian Clarke 《Annals of Global Analysis and Geometry》2011,39(2):131-163
We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate
the topology on the manifold of metrics induced by the distance function of the L
2 Riemannian metric—so-called because it induces an L
2 topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L
1-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically
with the completion of the L
2 metric. We also give a user-friendly criterion for convergence (with respect to the L
2 metric) in the manifold of metrics. 相似文献
6.
Rolf Walter 《Results in Mathematics》1995,27(1-2):194-197
For submanifolds of the affine space Rn, it is very important to derive a riemannian or pseudo-riemannian metric on the manifold just from affine data of the configuration. It is in this way that the equi-affine hypersurface theory is initiated by the so called Blaschke-Berwald metric (for the most recent state of affine hypersurface theory see the book of Li-Simon-Zhao [1993] and the vast literature given there). The same is true for the centro-affine geometry of codimension-two submanifolds (cf. Walter [1988], [1991 a]). Another instance where such a metric has been constructed from affine data are the (two-dimensional) surfaces of R4 (Burstin-Mayer [1927]). Recently, the geometry of these surfaces has been taken up by Nomizu-Vrancken [1993] with respect to the construction of a new transversal plane bundle. In the present note, we deal with the existence and, in particular, non-existence of elliptic points of the Burstin-Mayer metric from a local and global viewpoint. 相似文献
7.
Oliver Baues Vicente Corté s 《Proceedings of the American Mathematical Society》2001,129(8):2403-2407
We prove that any simply connected special Kähler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any nondegenerate holomorphic function. We also show that a classical result of Calabi and Pogorelov on parabolic spheres implies Lu's theorem on complete special Kähler manifolds with a positive definite metric.
8.
Takeshi Sasaki 《Geometriae Dedicata》1995,57(3):317-333
We formulate an affine theory of immersions of ann-dimensional manifold into the Euclidean space of dimensionn+n(n+1)/2 and give a characterization of critical immersions relative to the induced volume functional in terms of the affine shape operator. 相似文献
9.
Amalendu Ghosh 《Annals of Global Analysis and Geometry》2009,35(4):431-441
In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete
K-contact manifold admitting both the Einstein-Weyl structures W
± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove
that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f
η) are presented.
相似文献
10.
We study a 2-dimensional manifold that admits a homogeneous action of a 3-dimensional Lie group G, and has a 2-form invariant under G. We show that such a manifold can be realized as a surface in the affine 3-space, and list such realizations.
相似文献
11.
Tom Willmore 《Results in Mathematics》1988,13(3-4):403-408
In order to define an affine immersion of manifolds in affine differential geometry, it is necessary to choose a set of normal planes to the immersed manifold. The theory is then developed after this choice has been made. However, it was shown by A.G.Walker [WA1] that a torsion-free affine connexion on a manifold determines canonically a pseudo-Riemannian metric on the cotangent bundle, called the Riemann-extension of the affine connexion. By making use of this pseudo-Riemannian metric it is possible to define an affine immersion without making a suitable choice of normal planes. 相似文献
12.
Otto Röschel 《Journal of Geometry》1990,38(1-2):165-170
At a pointk
0 aC
4-curve k of an affine Cayley-Klein-plane (CK-plane) has a unique hyperosculating logarthmic spiral. We give a construction of the pole p of this spiral, which consists of an affine and a metric part. This metric part is a similar one in the three CK — planes. It is shown that this result is connected with results dealing with the center of the osculating circle given by R.Bereis in [2,p.248].
Herrn Prof.Dr.K.STRUBECKER zum 85.Geburtstag gewidmet 相似文献
Herrn Prof.Dr.K.STRUBECKER zum 85.Geburtstag gewidmet 相似文献
13.
Fa En Wu 《数学学报(英文版)》2010,26(10):2003-2014
Some of the variation formulas of a metric were derived in the literatures by using the local coordinates system, In this paper, We give the first and the second variation formulas of the Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using the moving frame method. We establish a relation between the variation of the volume of a metric and that of a submanifold. We find that the latter is a consequence of the former. Finally we give an application of these formulas to the variations of heat invariants. We prove that a conformally flat metric g is a critical point of the third heat invariant functional for a compact 4-dimensional manifold M, then (M, g) is either scalar flat or a space form. 相似文献
14.
Tatsuya Kawabe 《manuscripta mathematica》1998,95(1):529-543
The fundamental groupΓ of a compact complete affine manifold is represented as an affine crystallographic subgroup of Aff(n). L.S.Auslander conjectured thatΓ is virtually solvable. Our purpose is to find the algebraic condition onΓ which leads affirmative answer to the conjecture. 相似文献
15.
Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold 总被引:1,自引:0,他引:1
Marian Ioan Munteanu 《Mediterranean Journal of Mathematics》2008,5(1):43-59
In this paper we study a Riemannian metric on the tangent bundle T(M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger–Gromoll metric and a compatible almost complex structure which confers a structure
of locally conformal almost K?hlerian manifold to T(M) together with the metric. This is the natural generalization of the well known almost K?hlerian structure on T(M). We found conditions under which T(M) is almost K?hlerian, locally conformal K?hlerian or K?hlerian or when T(M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find
an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from
T(M). Moreover, we found that this map preserves also the natural contact structures obtained from the almost Hermitian ambient
structures on the unit tangent bundle and the tangent sphere bundle, respectively.
This work was also partially supported by Grant CEEX 5883/2006–2008, ANCS, Romania. 相似文献
16.
The energy of an oriented q-distribution ? in a compact oriented manifold M is defined to be the energy of the section of the Grassmannian manifold of oriented q-planes in M induced by ?. In the Grassmannian, the Sasaki metric is considered. We show here a condition for a distribution to be a critical point of the energy functional. In the spheres, we see that Hopf fibrations are critical points. Later, we prove the instability for these fibrations. 相似文献
17.
A pair of points in a riemannian manifold M is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of
points is insecure. A manifold is secure if all pairs of points in M are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure.
Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds.
We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with
any riemannian metric and a closed surface of genus one with generic metric are totally insecure. 相似文献
18.
Hossein Abbaspour 《Topology》2005,44(5):1059-1091
Let M be a closed, oriented and smooth manifold of dimension d. Let LM be the space of smooth loops in M. In [String topology, preprint math.GT/9911159] Chas and Sullivan introduced the loop product, a product of degree -d on the homology of LM. We aim at identifying the three manifolds with “nontrivial” loop product. This is an application of some existing powerful tools in three-dimensional topology such as the prime decomposition, torus decomposition, Seifert fiber space theorem, torus theorem. 相似文献
19.
Brian Clarke 《Calculus of Variations and Partial Differential Equations》2010,39(3-4):533-545
We prove that the L 2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L 2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold. 相似文献
20.
We investigate the holonomy group of a linear metric connection with skew-symmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any non-degenerated 2-form or any spinor. Suitable integral formulas allow us to prove similar properties in case of a compact Riemannian manifold equipped with a metric connection of skew-symmetric torsion. On the Aloff-Wallach space N(1,1) we construct families of connections admitting parallel spinors. Furthermore, we investigate the geometry of these connections as well as the geometry of the underlying Riemannian metric. Finally, we prove that any 7-dimensional 3-Sasakian manifold admits 2-parameter families of linear metric connections and spinorial connections defined by 4-forms with parallel spinors.Mathematics Subject Classification (2000):53 C 25, 81 T 30We thank Andrzej Trautman for drawing our attention to these papers by Cartan – see [27]. 相似文献