51 research outputs found
Solvable extensions of negative Ricci curvature of filiform Lie groups
We give necessary and sufficient conditions of the existence of a
left-invariant metric of strictly negative Ricci curvature on a solvable Lie
group the nilradical of whose Lie algebra is a filiform Lie
algebra . It turns out that such a metric always exists, except
for in the two cases, when is one of the algebras of rank two,
or , and is a one-dimensional extension of
, in which cases the conditions are given in terms of certain
linear inequalities for the eigenvalues of the extension derivation.Comment: 15 page
Nilradicals of Einstein solvmanifolds
A Riemannian Einstein solvmanifold is called standard, if the orthogonal
complement to the nilradical of its Lie algebra is abelian. No examples of
nonstandard solvmanifolds are known. We show that the standardness of an
Einstein metric solvable Lie algebra is completely detected by its nilradical
and prove that many classes of nilpotent Lie algebras (Einstein nilradicals,
algebras with less than four generators, free Lie algebras, some classes of
two-step nilpotent ones) contain no nilradicals of nonstandard Einstein metric
solvable Lie algebras. We also prove that there are no nonstandard Einstein
metric solvable Lie algebras of dimension less than ten.Comment: 22 page
Totally geodesic hypersurfaces of homogeneous spaces
We show that a simply connected Riemannian homogeneous space M which admits a
totally geodesic hypersurface F is isometric to either (a) the Riemannian
product of a space of constant curvature and a homogeneous space, or (b) the
warped product of the Euclidean space and a homogeneous space, or (c) the
twisted product of the line and a homogeneous space (with the warping/twisting
function given explicitly). In the first case, F is also a Riemannian product;
in the last two cases, it is a leaf of a totally geodesic homogeneous
fibration. Case (c) can alternatively be characterised by the fact that M
admits a Riemannian submersion onto the universal cover of the group SL(2)
equipped with a particular left-invariant metric, and F is the preimage of the
two-dimensional solvable totally geodesic subgroup.Comment: 8 page
Harmonic homogeneous manifolds of nonpositive curvature
A Riemannian manifold is called harmonic if its volume density function
expressed in polar coordinates centered at any point is radial. Flat and
rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz
Conjecture) is true for manifolds of nonnegative scalar curvature and for some
other classes of manifolds, but is not true in general: there exists a family
of homogeneous harmonic spaces, the Damek-Ricci spaces, containing noncompact
rank-one symmetric spaces, as well as infinitely many nonsymmetric examples. We
prove that a harmonic homogeneous manifold of nonpositive curvature is either
flat, or is isometric to a Damek-Ricci space.Comment: 11 page
Einstein solvmanifolds attached to two-step nilradicals
A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous
Einstein space) is almost completely determined by the nilradical of its Lie
algebra. A nilpotent Lie algebra, which can serve as the nilradical of an
Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite
a substantial progress towards the understanding of Einstein nilradicals, there
is still a lack of classification results even for some well-studied classes of
nilpotent Lie algebras, such as the two-step ones. In this paper, we give a
classification of two-step nilpotent Einstein nilradicals in one of the rare
cases when the complete set of affine invariants is known: for the two-step
nilpotent Lie algebras with the two-dimensional center. Informally speaking, we
prove that such a Lie algebra is an Einstein nilradical, if it is defined by a
matrix pencil having no nilpotent blocks in the canonical form and no
elementary divisors of a very high multiplicity. We also discuss the connection
between the property of a two-step nilpotent Lie algebra and its dual to be an
Einstein nilradical.Comment: 16 page
Osserman manifolds of dimension 8
For a Riemannian manifold with the curvature tensor , the Jacobi
operator is defined by . The manifold is called
{\it pointwise Osserman} if, for every , the eigenvalues of the
Jacobi operator do not depend of a unit vector , and is
called {\it globally Osserman} if they do not depend of the point either.
R. Osserman conjectured that globally Osserman manifolds are flat or rank-one
symmetric. This Conjecture is true for manifolds of dimension .
Here we prove the Osserman Conjecture and its pointwise version for
8-dimensional manifolds.Comment: 18 pages, LaTE
Osserman manifolds and Weyl-Schouten Theorem for rank-one symmetric spaces
A Riemannian manifold is called Osserman (conformally Osserman,
respectively), if the eigenvalues of the Jacobi operator of its curvature
tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at
every point. Osserman Conjecture asserts that every Osserman manifold is either
flat or rank-one symmetric. We prove that both the Osserman Conjecture and its
conformal version, the Conformal Osserman Conjecture, are true, modulo a
certain assumption on algebraic curvature tensors in . As a
consequence, we show that a Riemannian manifold having the same Weyl tensor as
a rank-one symmetric space, is conformally equivalent to it.Comment: 25 page
Riemannian manifolds of dimension 7 whose skew-symmetric curvature operator has constant eigenvalues
A Riemannian manifold is called IP, if the eigenvalues of its skew-symmetric
curvature operator are pointwise constant. It was previously shown that for all
n\ge 4, except n=7, any IP manifold either has constant curvature, or is a
warped product, with some specific function, of a line and a space of constant
curvature. We extend this result to the case n = 7, and also study
3-dimensional IP manifolds.Comment: Corollary on p2 correcte
Einstein solvmanifolds with free nilradical
We classify solvable Lie groups with a free nilradical admitting an Einstein
left-invariant metric. Any such group is essentially determined by the
nilradical of its Lie algebra, which is then called an Einstein nilradical. We
show that among the free Lie algebras, there are very few Einstein nilradicals.
Except for the one-step (abelian) and the two-step ones, there are only six
others: f(2,3), f(2,4), f(2,5), f(3,3), f(4,3), f(5,3) (where f(m,p) is a free
p-step Lie algebra on m generators). The reason for that is the inequality-type
restrictions on the eigenvalue type of an Einstein nilradical obtained in the
paper.Comment: 14 pages, changes to introduction, one reference added, small changes
to the text and to the titl
Einstein solvmanifolds with a simple Einstein derivation
The structure of a solvable Lie groups admitting an Einstein left-invariant
metric is, in a sense, completely determined by the nilradical of its Lie
algebra. We give an easy-to-check necessary and sufficient condition for a
nilpotent algebra to be an Einstein nilradical whose Einstein derivation has
simple eigenvalues. As an application, we classify filiform Einstein
nilradicals (modulo known classification results on filiform graded Lie
algebras).Comment: 11 page
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