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Anton S. Galaev

Publications and source records attributed to Anton S. Galaev.

At least 37 records · Page 2Linked to original sources
arXiv2016

Covariant derivative of the curvature tensor of pseudo-Kählerian manifolds

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.

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arXiv2014

Classification of third-order symmetric Lorentzian manifolds

Third-order symmetric Lorentzian manifolds, i.e. Lorentzian manifold with zero third derivative of the curvature tensor, are classified. These manifolds are exhausted by a special type of pp-waves, they generalize Cahen-Wallach spaces and second-order symmetric Lorentzian spaces.

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arXiv2013

Note on the holonomy groups of pseudo-Riemannian manifolds

For an arbitrary subalgebra $\mathfrak{h}\subset\mathfrak{so}(r,s)$, a polynomial pseudo-Riemannian metric of signature $(r+2,s+2)$ is constructed, the holonomy algebra of this metric contains $\mathfrak{h}$ as a subalgebra. This result shows the essential distinction of the holonomy algebras of pseudo-Riemannian manifolds of index bigger or equal to 2 from the holonomy algebras of Riemannian and Lorentzian manifolds.

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arXiv2013

Pseudo-Riemannian manifolds with recurrent spinor fields

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold $(M,g)$ is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of $(M,g)$. We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their holonomy algebras: Riemannian manifolds; Lorentzian manifolds; pseudo-Riemannian manifolds with irreducible holonomy algebras; pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.

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arXiv2013

About the classification of the holonomy algebras of Lorentzian manifolds

The classification of the holonomy algebras of Lorentzian manifolds can be reduced to the classification of irreducible subalgebras $\mathfrak{h}\subset\mathfrak{so}(n)$ that are spanned by the images of linear maps from $\mathbb{R}^n$ to $\mathfrak{h}$ satisfying an identity similar to the Bianchi one. T. Leistner found all such subalgebras and it turned out that the obtained list coincides with the list of irreducible holonomy algebras of Riemannian manifolds. The natural problem is to give a simple direct proof to this fact. We give such proof for the case of semisimple not simple Lie algebras $\mathfrak{h}$.

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arXiv2013

Some applications of the Lorentzian holonomy algebras

It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given; the classification of two-symmetric Lorentzian manifolds is explained; conformally recurrent Lorentzian manifolds are classified; recurrent symmetric bilinear forms on Lorentzian manifolds are described.

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arXiv2013

Irreducible holonomy algebras of Riemannian supermanifolds

Possible irreducible holonomy algebras $\g\subset\osp(p,q|2m)$ of Riemannian supermanifolds under the assumption that $\g$ is a direct sum of simple Lie superalgebras of classical type and possibly of a one-dimensional center are classified. This generalizes the classical result of Marcel Berger about the classification of irreducible holonomy algebras of pseudo-Riemannian manifolds.

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arXiv2011

Two-symmetric Lorentzian manifolds

We classify two-symmetric Lorentzian manifolds using methods of the theory of holonomy groups. These manifolds are exhausted by a special type of pp-waves and, like the symmetric Cahen-Wallach spaces, they have commutative holonomy.

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arXiv2011

Examples of Einstein spacetimes with recurrent null vector fields

The Einstein Equation on 4-dimensional Lorentzian manifolds admitting recurrent null vector fields is discussed. Several examples of a special form are constructed. The holonomy algebras, Petrov types and the Lie algebras of Killing vector fields of the obtained metrics are found.

math.DG↗
arXiv2011

Irreducible holonomy algebras of odd Riemannian supermanifolds

Possible irreducible holonomy algebras $\g\subset\sp(2m,\Real)$ of odd Riemannian supermanifolds and irreducible subalgebras $\g\subset\gl(n,\Real)$ with non-trivial first skew-symmetric prolongations are classified. An approach to the classification of some classes of the holonomy algebras of Riemannian supermanifolds is discussed.

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arXiv2010

Holonomy of Einstein Lorentzian manifolds

The classification of all possible holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds is obtained. It is shown that each such algebra appears as the holonomy algebra of an Einstein (resp., vacuum Einstein) Lorentzian manifold, the direct constructions are given. Also the holonomy algebras of totally Ricci-isotropic Lorentzian manifolds are classified. The classification of the holonomy algebras of Lorentzian manifolds is reviewed and a complete description of the spaces of curvature tensors for these holonomies is given.

math.DG↗
arXiv2010

One component of the curvature tensor of a Lorentzian manifold

The holonomy algebra $\g$ of an $n+2$-dimensional Lorentzian manifold $(M,g)$ admitting a parallel distribution of isotropic lines is contained in the subalgebra $\simil(n)=(\Real\oplus\so(n))\zr\Real^n\subset\so(1,n+1)$. An important invariant of $\g$ is its $\so(n)$-projection $\h\subset\so(n)$, which is a Riemannian holonomy algebra. One component of the curvature tensor of the manifold belongs to the space $¶(\h)$ consisting of linear maps from $\Real^n$ to $\h$ satisfying an identity similar to the Bianchi one. In the present paper the spaces $¶(\h)$ are computed for each possible $\h$. This gives the complete description of the values of the curvature tensor of the manifold $(M,g)$. These results can be applied e.g. to the holonomy classification of the Einstein Lorentzian manifolds.

math.DG↗
arXiv2008

Irreducible complex skew-Berger algebras

Irreducible skew-Berger algebras $\g\subset\gl(n,\Co)$, i.e. algebras spanned by the images of the linear maps $R:\odot^2\Co^n\to\g$ satisfying the Bianchi identity, are classified. These Lie algebras can be interpreted as irreducible complex Berger superalgebras contained in $\gl(0|n,\Co)$.

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arXiv2008

Holonomy of supermanifolds

Holonomy groups and holonomy algebras for connections on locally free sheaves over supermanifolds are introduced. A one-to-one correspondence between parallel sections and holonomy-invariant vectors, and a one-to-one correspondence between parallel locally direct subsheaves and holonomy-invariant vector supersubspaces are obtained. As the special case, the holonomy of linear connections on supermanifolds is studied. Examples of parallel geometric structures on supermanifolds and the corresponding holonomies are given. For Riemannian supermanifolds an analog of the Wu theorem is proved. Berger superalgebras are defined and their examples are given.

math.DG↗
arXiv2006

Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2

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-Kählerian and special pseudo-Kählerian manifolds are classified. An example of metric for each possible holonomy algebra is given. This gives the classification of holonomy algebras for pseudo-Kählerian manifolds of index 2. Finally we consider some examples and applications. We describe examples of 4-dimensional Lie groups with left-invariant pseudo-Kählerian 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-Kählerian 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-Kählerian 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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