17,614 research outputs found
Mathematical Properties of a Class of Four-dimensional Neutral Signature Metrics
While the Lorenzian and Riemanian metrics for which all polynomial scalar
curvature invariants vanish (the VSI property) are well-studied, less is known
about the four-dimensional neutral signature metrics with the VSI property.
Recently it was shown that the neutral signature metrics belong to two distinct
subclasses: the Walker and Kundt metrics. In this paper we have chosen an
example from each of the two subcases of the Ricci-flat VSI Walker metrics
respectively.
To investigate the difference between the metrics we determine the existence
of a null, geodesic, expansion-free, shear-free and vorticity-free vector, and
classify these spaces using their infinitesimal holonomy algebras. The
geometric implications of the holonomy algebras are further studied by
identifying the recurrent or covariantly constant null vectors, whose existence
is required by the holonomy structure in each example. We conclude the paper
with a simple example of the equivalence algorithm for these pseudo-Riemannian
manifolds, which is the only approach to classification that provides all
necessary information to determine equivalence.Comment: 18 page
A family of complex nilmanifolds with infinitely many real homotopy types
We find a one-parameter family of non-isomorphic nilpotent Lie algebras
, with , of real dimension eight with
(strongly non-nilpotent) complex structures. By restricting to take
rational values, we arrive at the existence of infinitely many real homotopy
types of -dimensional nilmanifolds admitting a complex structure. Moreover,
balanced Hermitian metrics and generalized Gauduchon metrics on such
nilmanifolds are constructed.Comment: 15 page
Six dimensional solvmanifolds with holomorphically trivial canonical bundle
We determine the 6-dimensional solvmanifolds admitting an invariant complex
structure with holomorphically trivial canonical bundle. Such complex
structures are classified up to isomorphism, and the existence of strong
K\"ahler with torsion (SKT), generalized Gauduchon, balanced and strongly
Gauduchon metrics is studied. As an application we construct a holomorphic
family of compact complex manifolds such that satisfies the
-Lemma and admits a balanced metric for any ,
but the central limit neither satisfies the -Lemma nor
admits balanced metrics.Comment: 32 pages; to appear in IMR
Ricci-flat and Einstein pseudoriemannian nilmanifolds
This is partly an expository paper, where the authors' work on
pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new
criterion is given for the existence of a diagonal Einstein metric on a nice
nilpotent Lie group. Classifications of special classes of Ricci-flat metrics
on nilpotent Lie groups of dimension are obtained. Some related open
questions are presented.Comment: 30 pages, 1 figure. v2: added a comment on a recent example of an
Einstein nilpotent Lie algebra of dimension 7; added a remark and a question
concerning the characteristically nilpotent case; replaced the
"\sigma-compatible" condition with the more general "\sigma-diagonal"; added
3 reference
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