52 research outputs found
Tanaka structures (non holonomic G-structures) and Cartan connections
Let \gh = \gh_{-k}\oplus \cdots \oplus \gh_{l} (k >0, l \geq 0) be a finite
dimensional real graded Lie algebra, with a Euclidian metric \langle \cdot ,
\cdot \rangle adapted to the gradation. The metric \langle\cdot , \cdot \rangle
is called admissible if the codifferentials \partial^{*} : C^{k+1}(\gh_{-}, \gh
) \ra C^{k} (\gh_{-}, \gh) (k\geq 0) are Ad_{Q}-invariant (Lie(Q) =
\gh_{0}\oplus \gh_{+}). We find necessary and sufficient conditions for a
Euclidian metric, adapted to the gradation, to be admissible, and we develop a
theory of normal Cartan connections, when these conditions are satisfied. We
show how the treatment by A. Cap and J. Slovak (Parabolic Geometry I,
Mathematical Surveys and Monographs, vol. 154, 2009), about normal Cartan
connections of semisimple type, fits into our theory. We also consider in some
detail the case when \gh = t^{*} (\gg ) is the cotangent Lie algebra of a
non-positively graded Lie algebra \gg.Comment: 24 pages; minor corrections with respect to the previous versio
Spin structures on compact homogeneous pseudo-Riemannian manifolds
We study spin structures on compact simply-connected homogeneous
pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G.
We classify flag manifolds F = G/H of a compact simple Lie group which are
spin. This yields also the classification of all flag manifolds carrying an
invariant metaplectic structure. Then we investigate spin structures on
principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently
simply-connected homogeneous complex manifolds M=G/L of a compact semisimple
Lie group G. We study the topology of M and we provide a sufficient and
necessary condition for the existence of an (invariant) spin structure, in
terms of the Koszul form of F. We also classify all C-spaces which are fibered
over an exceptional spin flag manifold and hence they are spin
Homogeneous almost K\"ahler manifolds and the Chern-Einstein equation
Given a non compact semisimple Lie group we describe all homogeneous
spaces carrying an invariant almost K\"ahler structure . When
is abelian and is of classical type, we classify all such spaces which
are Chern-Einstein, i.e. which satisfy for some
, where is the Ricci form associated to the Chern
connection.Comment: Some sentences have been added in the Introductio
Prolongation of Tanaka structures: an alternative approach
The classical theory of prolongation of G-structures was generalized by N.
Tanaka to a wide class of geometric structures (Tanaka structures), which are
defined on a non-holonomic distribution. Examples of Tanaka structures include
subriemannian, subconformal, CR-structures, structures associated to second
order differential equations and structures defined by gradings of Lie algebras
(in the setting of parabolic geometry). Tanaka's prolongation procedure
associates to a Tanaka structure of finite order a manifold with an absolute
parallelism. It is a very fruitful method for the description of local
invariants, investigation of the automorphism group and the equivalence
problem. In this paper we develop an alternative constructive approach for
Tanaka's prolongation procedure, based on the theory of quasi-gradations in
filtered vector spaces, G-structures and their torsion functions.Comment: 28 pages, minor changes, one reference adde
Shortest and Straightest Geodesics in Sub-Riemannian Geometry
There are many equivalent definitions of Riemannian geodesics. They are
naturally generalised to sub-Riemannian manifold, but become non-equivalent. We
give a review of different definitions of geodesics of a sub-Riemannian
manifold and interrelation between them. We recall three variational
definitions of geodesics as (locally) shortest curves (Euler-Lagrange,
Pontyagin and Hamilton) and three definitions of geodesics as straightest
curves (d'Alembert , Levi-Civita-Schouten and Cartan-Tanaka ), used in
nonholonomic mechanics and discuss their interrelations. We consider a big
class of sub-Riemannian manifolds associated with principal bundle over a
Riemannian manifolds, for which shortest geodesics coincides with straightest
geodesics. Using the geometry of flag manifolds, we describe some classes of
compact homogeneous sub-Riemannian manifolds (including contact sub-Riemannian
manifolds and symmetric sub-Riemannian manifolds) where straightest geodesics
coincides with shortest geodesics. Construction of geodesics in these cases
reduces to description of Riemannian geodesics of the Riemannian homogeneous
manifold or left-invariant metric on a Lie group.Comment: 35 page
Differential geometry of Cartan connections
For a more general notion of Cartan connection we define characteristic
classes, we investigate their relation to usual characteristic classes
Homogeneous para-K\"ahler Einstein manifolds
A para-K\"ahler manifold can be defined as a pseudo-Riemannian manifold
with a parallel skew-symmetric para-complex structures , i.e. a
parallel field of skew-symmetric endomorphisms with or,
equivalently, as a symplectic manifold with a bi-Lagrangian
structure , i.e. two complementary integrable Lagrangian distributions.
A homogeneous manifold of a semisimple Lie group admits an
invariant para-K\"ahler structure if and only if it is a covering of
the adjoint orbit of a semisimple element We give a
description of all invariant para-K\"ahler structures on a such
homogeneous manifold. Using a para-complex analogue of basic formulas of
K\"ahler geometry, we prove that any invariant para-complex structure on defines a unique para-K\"ahler Einstein structure with given
non-zero scalar curvature. An explicit formula for the Einstein metric is
given.
A survey of recent results on para-complex geometry is included.Comment: 44 page
Partially-flat gauge fields on manifolds of dimension greater than four
We describe two extensions of the notion of a self-dual connection in a
vector bundle over a manifold M from dim M=4 to higher dimensions. The first
extension, Omega-self-duality, is based on the existence of an appropriate
4-form Omega on the Riemannian manifold M and yields solutions of the
Yang-Mills equations. The second is the notion of half-flatness, which is
defined for manifolds with certain Grassmann structure T^C M \cong E \otimes H.
In some cases, for example for hyper-Kaehler manifolds M, half-flatness implies
Omega-self-duality. A construction of half-flat connections inspired by the
harmonic space approach is described. Locally, any such connection can be
obtained from a free prepotential by solving a system of linear first order
ODEs.Comment: 8 page
Conification of K\"ahler and hyper-K\"ahler manifolds
Given a K\"ahler manifold endowed with a Hamiltonian Killing vector field
, we construct a conical K\"ahler manifold such that is
recovered as a K\"ahler quotient of . Similarly, given a
hyper-K\"ahler manifold endowed with a Killing vector field
, Hamiltonian with respect to the K\"ahler form of and satisfying
, we construct a hyper-K\"ahler cone such
that is a certain hyper-K\"ahler quotient of . In this way, we
recover a theorem by Haydys. Our work is motivated by the problem of relating
the supergravity c-map to the rigid c-map. We show that any hyper-K\"ahler
manifold in the image of the c-map admits a Killing vector field with the above
properties. Therefore, it gives rise to a hyper-K\"ahler cone, which in turn
defines a quaternionic K\"ahler manifold. Our results for the signature of the
metric and the sign of the scalar curvature are consistent with what we know
about the supergravity c-map.Comment: conjecture replaced by referenc
Maximally homogeneous para-CR manifolds of semisimple type
An almost para-CR structure on a manifold is given by a distribution together with a field of involutive
endomorphisms of . If satisfies an integrability condition, then
is called a para-CR structure. The notion of maximally homogeneous
para-CR structure of a semisimple type is given. A classification of such
maximally homogeneous para-CR structures is given in terms of appropriate
gradations of real semisimple Lie algebras
- …