138 research outputs found
Manifolds with large isotropy groups
We classify all simply connected Riemannian manifolds whose isotropy groups
act with cohomogeneity less than or equal to two.Comment: 21 page
Natural Diagonal Riemannian Almost Product and Para-Hermitian Cotangent Bundles
We obtain the natural diagonal almost product and locally product structures
on the total space of the cotangent bundle of a Riemannian manifold. We find
the Riemannian almost product (locally product) and the (almost) para-Hermitian
cotangent bundles of natural diagonal lift type. We prove the characterization
theorem for the natural diagonal (almost) para-K\"ahlerian structures on the
total spaces of the cotangent bundle.Comment: 10 pages, will appear in Czechoslovak Mathematical Journa
Pseudo-K\"ahler Lie algebras with abelian complex structures
We study Lie algebras endowed with an abelian complex structure which admit a
symplectic form compatible with the complex structure. We prove that each of
those Lie algebras is completely determined by a pair (U,H) where U is a
complex commutative associative algebra and H is a sesquilinear hermitian form
on U which verifies certain compatibility conditions with respect to the
associative product on U. The Riemannian and Ricci curvatures of the associated
pseudo-K\"ahler metric are studied and a characterization of those Lie algebras
which are Einstein but not Ricci flat is given. It is seen that all
pseudo-K\"ahler Lie algebras can be inductively described by a certain method
of double extensions applied to the associated complex asssociative commutative
algebras
Covariant derivative of the curvature tensor of pseudo-K\"ahlerian 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-K\"ahlerian 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-K\"ahlerian 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.Comment: the final version accepted to Annals of Global Analysis and Geometr
On paraquaternionic submersions between paraquaternionic K\"ahler manifolds
In this paper we deal with some properties of a class of semi-Riemannian
submersions between manifolds endowed with paraquaternionic structures, proving
a result of non-existence of paraquaternionic submersions between
paraquaternionic K\"ahler non locally hyper paraK\"ahler manifolds. Then we
examine, as an example, the canonical projection of the tangent bundle, endowed
with the Sasaki metric, of an almost paraquaternionic Hermitian manifold.Comment: 13 pages, no figure
A Triplectic Bi-Darboux Theorem and Para-Hypercomplex Geometry
We provide necessary and sufficient conditions for a bi-Darboux Theorem on
triplectic manifolds. Here triplectic manifolds are manifolds equipped with two
compatible, jointly non-degenerate Poisson brackets with mutually involutive
Casimirs, and with ranks equal to 2/3 of the manifold dimension. By definition
bi-Darboux coordinates are common Darboux coordinates for two Poisson brackets.
We discuss both the Grassmann-even and the Grassmann-odd Poisson bracket case.
Odd triplectic manifolds are, e.g., relevant for Sp(2)-symmetric
field-antifield formulation. We demonstrate a one-to-one correspondence between
triplectic manifolds and para-hypercomplex manifolds. Existence of bi-Darboux
coordinates on the triplectic side of the correspondence translates into a flat
Obata connection on the para-hypercomplex side.Comment: 31 pages, LaTeX. v2: Changed title; Added references. v3: Minor
reorganization of pape
A connection with parallel totally skew-symmetric torsion on a class of almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics
The subject of investigations are the almost hypercomplex manifolds with
Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is
introduced such that the structure of these manifolds is parallel with respect
to D and its torsion is totally skew-symmetric. The class of the nearly Kaehler
manifolds with respect to the first almost complex structure is of special
interest. It is proved that D has a D-parallel torsion and is weak if it is not
flat. Some curvature properties of these manifolds are studied.Comment: 18 page
Completeness in supergravity constructions
We prove that the supergravity r- and c-maps preserve completeness. As a
consequence, any component H of a hypersurface {h=1} defined by a homogeneous
cubic polynomial such that -d^2 h is a complete Riemannian metric on H defines
a complete projective special Kahler manifold and any complete projective
special Kahler manifold defines a complete quaternionic Kahler manifold of
negative scalar curvature. We classify all complete quaternionic Kahler
manifolds of dimension less or equal to 12 which are obtained in this way and
describe some complete examples in 16 dimensions.Comment: 29 page
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