Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We consider the Carnot–Carathéodory distance canonically associated with an almost-Riemannian structure and study the problem of Lipschitz equivalence between two such distances on the same compact oriented surface. We analyze the generic case, allowing in particular for the presence of tangency points, i.e., points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labeled graph associated with it.     

Normal forms and invariants for 2-dimensional almost-Riemannian structures

2-Dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket.In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are “complete” in the sense that they permit to recognize locally isometric structures.The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution.For Riemannian points such that the gradient of the Gaussian curvature K is different from zero, we use the level set of K as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel.Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization.     

A Poisson manifold of strong compact type

We construct a corank one Poisson manifold which is of strong compact type, i.e., the associated Lie algebroid structure on its cotangent bundle is integrable, and the source 1-connected (symplectic) integration is compact. The construction relies on the geometry of the moduli space of marked K3 surfaces.     

Symmetries of Contact Metric Manifolds

We study the Lie algebra of infinitesimal isometries on compact Sasakian and K-contact manifolds. On a Sasakian manifold which is not a space form or 3-Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K-contact structure, we prove that there exists a Killing vector field of constant length which is not an infinitesimal automorphism of the structure if and only if the manifold is obtained from the Konishi bundle of a compact pseudo-Riemannian quaternion-Kähler manifold after changing the sign of the metric on a maximal negative distribution. We also prove that nonregular Sasakian manifolds are not homogeneous and construct examples with cohomogeneity one. Using these results we obtain in the last section the classification of all homogeneous Sasakian manifolds.     

Symmetry reduction of sh-Lie structures and of local functionals

Reduced sh-Lie structures have been studied for the case when a Lie group acts on the fibers of a vector bundle while preserving the base space of the bundle. In this paper we investigate how one obtains a reduced sh-Lie structure using the ideas of symmetry reduction where the action of the Lie group is transversal to the fibers of the bundle. We also show how local functionals are reduced using these ideas.     

Cotangent bundles of 4-dimensional hypercomplex Lie groups

 We study geometrical structures on the cotangent bundle T ^* G of a Lie group G which are left-invariant with respect to the Lie group structure on T ^* G determined by a left-invariant affine structure ∇ on G. In particular, we investigate the existence of conformally hyper-K?hler metrics and hyper-K?hler with torsion (HKT) structures on the cotangent bundle of hypercomplex 4-dimensional Lie groups. By applying In?nü-Wigner contractions to compact semisimple Lie algebras we obtain non semisimple Lie algebras endowed with invariant HKT structures. Received: 4 February 2002 / Revised version: 20 August 2002 Research partially supported by MURST and GNSAGA (Indam) of Italy Mathematics Subject Classification (2000): 53C26, 22E25     

Some remarks on the global structure of proper Lie groupoids in low codimensions

We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation, i.e. one whose kernel consists only of ineffective arrows, on a smooth vector bundle. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as an extension, by some bundle of compact Lie groups, of some action groupoid G?X with G compact.     

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [2] and of the anomaly flow [3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.     

Commutators of flow maps of nonsmooth vector fields

Relying on the notion of set-valued Lie bracket introduced in an earlier paper, we extend some classical results valid for smooth vector fields to the case when the vector fields are just Lipschitz. In particular, we prove that the flows of two Lipschitz vector fields commute for small times if and only if their Lie bracket vanishes everywhere (i.e., equivalently, if their classical Lie bracket vanishes almost everywhere). We also extend the asymptotic formula that gives an estimate of the lack of commutativity of two vector fields in terms of their Lie bracket, and prove a simultaneous flow box theorem for commuting families of Lipschitz vector fields.     

Witten deformation and the equivariant index

Let M be a compact Riemannian manifold endowed with an isometric action of a compact, connected Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on M. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associated to the isolated singular points of an equivariant bundle map that is locally Clifford multiplication by a Killing vector field near these points.      

A cohomology for vector valued differential forms

A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Frölicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of traceless vector valued differential forms, equipped with a new natural differential concomitant as graded Lie bracket. We find two graded Lie algebra structures on the space of differential forms. Some consequences and related results are also discussed.     

Combinatorial Lie bialgebras of curves on surfaces

Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6) (1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a variant of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we found by programming the algorithm which answer negatively Turaev's question about the characterization of simple curves in terms of the cobracket. Further computations suggest an alternative characterization of simple curves in terms of the bracket of a curve and its inverse. Turaev's question is still open in genus zero.     

Fano contact manifolds and nilpotent orbits

A contact structure on a complex manifold M is a corank 1 subbundle F of T_M such that the bilinear form on F with values in the quotient line bundle L = T_M/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.?If is a simple Lie algebra, the unique closed orbit in (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.?In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M P(H ₀(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant. Received: July 28, 1997     

Bihermitian Structures on Complex Surfaces

Bihermitian complex surfaces are oriented conformal four-manifoldsadmitting two independent compatible complex structures. Non-anti-self-dualbihermitian structures on R⁴ and the four-dimensional torusT⁴ have recently been discovered by P. Kobak. We show that anoriented compact 4-manifold, admitting a non-anti-self-dualbihermitian structure, is a torus or K3 surface in the stronglybihermitian case (when the two complex structures are independentat each point) or, otherwise, must be obtained from the complexprojective plane or a minimal ruled surface of genus less than2 by blowing up points along some anti-canonical divisor (butthe actual existence of bihermitian structures in the lattercase is still an open question). The paper includes a generalmethod for constructing non-anti-self-dual bihermitian structureson tori, K3 surfaces and S¹ x S³. Further properties of compactbihermitian surfaces are also investigated. 1991 MathematicsSubject Classification: 53C12, 53C55, 32J15.     

The space of harmonic Prym differentials on a variable compact Riemann surface

The harmonic Prym differentials and their period classes play an important role in the modern theory of functions on compact Riemann surfaces [1–7]. We study the harmonic Prym bundle, whose fibers are the spaces of harmonic Prym differentials on variable compact Riemann surfaces and find its connection with Gunning’s cohomological bundle over the Teichmüller space for two important subgroups of the inessential and normalized characters on a compact Riemann surface. We study the periods of holomorphic Prym differentials for essential characters on variable compact Riemann surfaces.     

Poisson-Nijenhuis structures and the Vinogradov bracket

We express the compatibility conditions that a Poisson bivector and a Nijenhuis tensor must fulfil in order to be a Poisson-Nijenhuis structure by means of a graded Lie bracket. This bracket is a generalization of Schouten and Frölicher-Nijenhuis graded Lie brackets defined on multivector fields and on vector valued differential forms respectively.Partially supported by Fundació Caixa Castelló.Partially supported by the Spanish DGICYT grant #P B91-0324.     

Proper groupoids and momentum maps: Linearization, affinity, and convexity

We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural integral affine structure, which makes it into an affine orbifold with locally convex polyhedral boundary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps.     

The Goldman bracket characterizes homeomorphisms

We show that a homotopy equivalence between compact, connected, oriented surfaces with non-empty boundary is homotopic to a homeomorphism if and only if it commutes with the Goldman bracket.     

Group actions on algebras and the graded Lie structure of Hochschild cohomology

Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms. We examine the Gerstenhaber bracket with a view toward deformations and developing bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We give some general results describing when brackets are zero for polynomial skew group algebras, which allow us in particular to find noncommutative Poisson structures. For abelian groups, we express the bracket using inner products of group characters. Lastly, we interpret results for graded Hecke algebras.