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Holonomy groups and special geometric structures of pseudo-K\"ahlerian manifolds of index 2
Differential Geometry · Submitted 14 Dec 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This gives the classification of holonomy algebras for pseudo-K\"ahlerian manifolds of index 2.
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The abstract, sentence by sentence
The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreducible not irreducible subalgebras of $\su(1,n+1)$ ($n\geq 0$) are classified. Weakly-irreducible not irreducible holonomy algebras of pseudo-K\"ahlerian and special pseudo-K\"ahlerian manifolds are classified. An example of metric for each possible holonomy algebra is given. This gives the classification of holonomy algebras for pseudo-K\"ahlerian manifolds of index 2. Finally we consider some examples and applications. We describe examples of 4-dimensional Lie groups with left-invariant pseudo-K\"ahlerian metrics and determine their holonomy algebras. We use our classification of holonomy algebras to give a new proof for the classification of simply connected pseudo-K\"ahlerian symmetric spaces of index 2 with weakly-irreducible not irreducible holonomy algebras. We consider time-like cones over Lorentzian Sasaki manifolds. These cones are also pseudo-K\"ahlerian manifolds of index 2. We describe the local DeRham-Wu decomposition of the cone in terms of the initial Lorentzian Sasaki manifold and we describe all possible weakly-irreducible not irreducible holonomy algebras of such cones.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. | background 0.52 | 0.04 | 0.02 | asserted 1.00 |
| s1 | The classification of Riemannian holonomy algebras is a classical result. | background 1.00 | 0.03 | 0.01 | asserted 1.00 |
| s2 | The classification of Lorentzian holonomy algebras was obtained recently. | background 0.99 | 0.05 | 0.02 | asserted 1.00 |
| s3 | In the present paper weakly-irreducible not irreducible subalgebras of $\su(1,n+1)$ ($n\geq 0$) are classified. | contribution 0.87 | 0.92 | 0.16 | asserted 1.00 |
| s4 | Weakly-irreducible not irreducible holonomy algebras of pseudo-K\"ahlerian and special pseudo-K\"ahlerian manifolds are classified. | result 0.54 | 0.94 | 0.55 | asserted 1.00 |
| s5 | An example of metric for each possible holonomy algebra is given. | contribution 0.64 | 0.94 | 0.06 | asserted 1.00 |
| s6 | This gives the classification of holonomy algebras for pseudo-K\"ahlerian manifolds of index 2. | result 0.64 | 0.93 | 0.81 | asserted 1.00 |
| s7 | Finally we consider some examples and applications. | contribution 0.93 | 0.74 | 0.02 | asserted 0.84 |
| s8 | We describe examples of 4-dimensional Lie groups with left-invariant pseudo-K\"ahlerian metrics and determine their holonomy algebras. | contribution 0.92 | 0.96 | 0.03 | asserted 1.00 |
| s9 | We use our classification of holonomy algebras to give a new proof for the classification of simply connected pseudo-K\"ahlerian symmetric spaces of index 2 with weakly-irreducible not irreducible holonomy algebras. | contribution 0.68 | 0.95 | 0.09 | asserted 1.00 |
| s10 | We consider time-like cones over Lorentzian Sasaki manifolds. | contribution 1.00 | 0.70 | 0.02 | asserted 0.90 |
| s11 | These cones are also pseudo-K\"ahlerian manifolds of index 2. | contribution 0.58 | 0.50 | 0.04 | asserted 1.00 |
| s12 | We describe the local DeRham-Wu decomposition of the cone in terms of the initial Lorentzian Sasaki manifold and we describe all possible weakly-irreducible not irreducible holonomy algebras of such cones. | result 0.58 | 0.94 | 0.07 | asserted 1.00 |
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Original abstract
The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreducible not irreducible subalgebras of $\su(1,n+1)$ ($n\geq 0$) are classified. Weakly-irreducible not irreducible holonomy algebras of pseudo-K\"ahlerian and special pseudo-K\"ahlerian manifolds are classified. An example of metric for each possible holonomy algebra is given. This gives the classification of holonomy algebras for pseudo-K\"ahlerian manifolds of index 2. Finally we consider some examples and applications. We describe examples of 4-dimensional Lie groups with left-invariant pseudo-K\"ahlerian metrics and determine their holonomy algebras. We use our classification of holonomy algebras to give a new proof for the classification of simply connected pseudo-K\"ahlerian symmetric spaces of index 2 with weakly-irreducible not irreducible holonomy algebras. We consider time-like cones over Lorentzian Sasaki manifolds. These cones are also pseudo-K\"ahlerian manifolds of index 2. We describe the local DeRham-Wu decomposition of the cone in terms of the initial Lorentzian Sasaki manifold and we describe all possible weakly-irreducible not irreducible holonomy algebras of such cones.
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How the literature uses it
5 arXiv papers cite this in 11 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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This gives the classification of holonomy algebras for pseudo-K\"ahlerian manifolds of index 2
Classification of connected holonomy groups of pseudo-K\"ahlerian manifolds of index 2 · 0.98
Papers citing it · 6 sentences without a confident label
Compares with it 2 papers
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In addition, the same techniques are successfully applied to a comparison of Cahen-Wallach spaces and one of the possible pseudo-Kähler symmetric spaces of index $2$ in the classification given in [ref] , giving a more general picture of complex plane waves.
Strongly degenerate homogeneous pseudo-K\"ahler structures of linear type and complex plane waves · Introduction · 0.90
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These results were obtained in much more complicated way in the thesis [ref] .
Holonomy classification of Lorentz-K\"ahler manifolds · Introduction · 0.96
Cites it as background 2 papers
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For pseudo-Riemannian manifolds of signature different from Riemannian and Lorentzian ones only some partial case are considered [ref] .
Holonomy algebras of Einstein pseudo-Riemannian manifolds · Introduction · 0.81
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Galaev classified holonomy algebras of pseudo-Kählerian manifolds of index 2 [ref] .
Holonomy groups of $G_2^*$-manifolds · Introduction · 0.90
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