Einstein Metrics Induced by Natural Riemann Extensions (original) (raw)

Remarks on Generalized Einstein Manifolds

2008

The usual Einstein metrics are those for which the first Ricci contraction of the covariant Riemann curvature tensor is proportional to the metric. Assuming the same type of restrictions but instead on the different contractions of the generalized covariant Gauss-Kronecker tensors Rp, leads to several generalizations of Einstein’s condition. In this paper, we treat some properties of these metrics. Mathematics Subject Classification (2000). 53C25, 58E11.

Einstein extensions of Riemannian manifolds

Transactions of the American Mathematical Society, 2021

Given a Riemannian space N N of dimension n n and a field D D of symmetric endomorphisms on N N , we define the extension M M of N N by D D to be the Riemannian manifold of dimension n + 1 n+1 obtained from N N by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on N N and D D which imply that the extension M M is Einstein. In particular, we show that in this case, D D has constant eigenvalues; moreover, they are all integer (up to scaling) if det D ≠ 0 \det D \ne 0 . They must satisfy certain arithmetic relations which imply that there are only finitely many eigenvalue types of D D in every dimension (a similar result is known for Einstein solvmanifolds). We give the characterisation of Einstein extensions for particular eigenvalue types of D D , including the complete classification for the case when D D has two eigenvalues, one of which is multiplicity free. In the most interesting case, the extension is obtained, by an ex...

Properties of Modified Riemannian Extensions

Zurnal matematiceskoj fiziki, analiza, geometrii, 2015

Let M be an n−dimensional differentiable manifold with a symmetric connection ∇ and T * M be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension g ∇,c on T * M defined by means of a symmetric (0, 2)-tensor field c on M. We get the conditions under which T * M endowed with the horizontal lift H J of an almost complex structure J and with the metric g ∇,c is a Kähler-Norden manifold. Also curvature properties of the Levi-Civita connection and another metric connection of the metric g ∇,c are presented.

Riemannian manifolds with Einstein-like metrics

1985

1 n this thesis, we investigate propaties of manifolds with Riemannian metrics which satisfy conditions more general than those of J:'instein metrics, including the Lauer as special cases. Fhe /:"instein condition is weU known for being the l:"uln-l.agrange equation of a vw iational problem. '/'here is twt a great deal of difference between such metrics and melrics with Ricci tensor parallel for the Latter are locaUy Riemannian products of the former. More general classes of metrics considered include Ricci-Codazzi and Ricci cyclic parallel. Both of these are of constant scalar curvature. Our study is divided into thr•ee parts. We begin with certain metrics in 4-dimensions and conclude our results with three theorems, the first of which is equivalent to a result of Kasner /Kal] while the second and pan of the third is ktwwn to Derdzinski / Del,2]. Next we construct the metrics mentioned above on spheres of odd dimension.'J'he construction is similar to Jensen's /Jell but more direct and is due essentiaUy to Gray and Vanhecke /GV}. In this way we obtain ,beside the standard metric, the second l:.'instein metric of Jensen. As for the Ricci-Codazzi metrics, they are essentiaUy Jiinstein, but the Ricci cyclic parallel mell ics seem to form a larger class. FinaUy,we consider subalgebras of the exceptional Lie algebra g2. Making use of computer programmes in 'reduce' we compute aU the corresponding metrics on the quotient spaces associated with G2.

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

Journal of Geometry, 2018

Let M be an n−dimensional differentiable manifold equipped with a torsion-free linear connection ∇ and T * M its cotangent bundle. The present paper aims to study a metric connection ∇ with nonvanishing torsion on T * M with modified Riemannian extension g ∇,c. First, we give a characterization of fibre-preserving projective vector fields on (T * M, g ∇,c) with respect to the metric connection ∇. Secondly, we study conditions for (T * M, g ∇,c) to be semi-symmetric, Ricci semi-symmetric, Z semi-symmetric or locally conharmonically flat with respect to the metric connection ∇. Finally, we present some results concerning the Schouten-Van Kampen connection associated to the Levi-Civita connection ∇ of the modified Riemannian extension g ∇,c. Mathematics subject classification 2010. 53C07, 53C35, 53A45.

On semi-Riemannian manifolds satisfying some generalized Einstein metric conditions

arXiv (Cornell University), 2023

The difference tensor R • C − C • R of a semi-Riemannian manifold (M, g), dim M ≥ 4, formed by its Riemann-Christoffel curvature tensor R and the Weyl conformal curvature tensor C, under some assumptions, can be expressed as a linear combination of (0, 6)-Tachibana tensors Q(A, T), where A is a symmetric (0, 2)-tensor and T a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.

On the geometry of gradient Einstein-type manifolds

In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold varrg\varrgvarrg, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen.

On a class of Einstein space-time manifolds

Publicationes Mathematicae Debrecen

We deal with a general space-time (M, g) with usual differentiability conditions and hyperbolic metric g of index 1, which carries 3 skewsymmetric Killing vector fields X, Y , Z having as generative the unit time-like vector field e of the hyperbolic metric g. It is shown that such a space-time (M, g) is an Einstein manifold of curvature −1, which is foliated by space-like hypersurfaces M s normal to e and the immersion x : M s → M is pseudo-umbilical. In addition, it is proved that the vector fields X, Y , Z and e are exterior concurrent vector fields and X, Y , Z define a commutative Killing triple, M admits a Lorentzian transformation which is in an orthocronous Lorentz group and the distinguished spatial 3-form of M is a relatively integral invariant of the vector fields X, Y and Z.

On the Scalar Curvature of Einstein Manifolds

1997

We show that there are high-dimensional smooth compact manifolds which admit pairs of Einstein metrics for which the scalar curvatures have opposite signs. These are counter-examples to a conjecture considered by Besse. The proof hinges on showing that the Barlow surface has small deformations with ample canonical line bundle.