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Smooth metric measure spaces, quasi-Einstein metrics, and tractors

100%
Case J.
Open Mathematics
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2012
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tom 10
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nr 5
1733-1762
EN
We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.
2
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Curvatures of the diagonal lift from an affine manifold to the linear frame bundle

100%
Kowalski O., Sekizawa M.
Open Mathematics
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2012
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tom 10
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nr 3
837-843
EN
We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de León in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.
3
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Affine analogues of the Sasaki-Shchepetilov connection

100%
Robaszewska M.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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2016
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tom 15
EN
For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM ⊕ (ℝ × M) and two on TM ⊕ (ℝ2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝN, after suitable local affine embedding of (M,∇) into ℝN.
4
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Erratum to: “Smooth metric measure spaces, quasi-Einstein metrics, and tractors”

100%
Case J.
Open Mathematics
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2013
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tom 11
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nr 1
196-196
EN
The original version of the article was published in Central European Journal of Mathematics, 2012, 10(5), 1733–1762, DOI: 10.2478/s11533-012-0091-x. Unfortunately, it contains a typographical error in display of (27). Here we display the correct version of the equations.
5
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Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections

84%
Keller J., Tønnesen-Friedman C.
Open Mathematics
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2012
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tom 10
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nr 5
1673-1687
EN
We provide nontrivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple (M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold (M, L), generalizing the notions of both constant scalar curvature Kähler metric and Hermitian-Einstein metric.
6
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Hyperholomorphic connections on coherent sheaves and stability

69%
Verbitsky M.
Open Mathematics
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2011
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tom 9
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nr 3
535-557
EN
Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ▿ with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessary L 2-integrable. We show that such sheaves are polystable.
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