Equivalence of Dynamics for Nonholonomic Systems with Transverse Constraints

This paper is concerned with the dynamics of a mechanical system subject to nonintegrable constraints. In the first part, we prove the equivalence between the classical nonholonomic equations and those derived from the nonholonomic variational formulation, proposed by Kozlov in [10–12], for a class of constrained systems with constraints transverse to a foliation. This result extends the equivalence between the two formulations, proved for holonomic constraints, to a class of linear nonintegrable ones. In the second part, we derive the nonholonomic variational reduced equations for a constrained system with symmetry and constraint transverse to a principal bundle fibration, using a reduction procedure similar to the one developed in [5]. The resulting equations are compared with the nonholonomic reduced ones through mechanical examples.     

Hermitian Metric Rigidity for Foliated Vector Bundles

We show that the canonical isometric imbedding of the symplectic group Sp(n) into R⁴n² gives the least-dimensional isometric imbedding into the Euclidean space, even in the local standpoint. We prove this result by calculating the quantity p_G determined by the curvature of Sp(n), which serves as an obstruction to the existence of local isometric imbeddings. We also exhibit the estimates on the value p_G for the remaining compact classical simple Lie groups, and improve the previous results on the codimension of local isometric imbeddings of these groups.     

Volume-Minimizing Foliations on Spheres

The volume of a k-dimensional foliation in a Riemannian manifold Mⁿ is defined as the mass of the image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing the construction by Gluck and Ziller (Comment. Math. Helv. 61 (1986), 177–192), ‘singular’ foliations by 3-spheres are constructed on round spheres S⁴ⁿ⁺³, as well as a singular foliation by 7-spheres on S¹⁵, which minimize volume within their respective relative homology classes. These singular examples, even though they are not homologous to the graph of a foliation, provide lower bounds for volumes of regular three-dimensional foliations of S⁴ⁿ⁺³ and regular seven-dimensional foliations of S¹⁵, since the double of these currents will be homologous to twice the graph of any smooth foliation by 3-manifolds.The second author was supported during this research by grants from the Universidade de Sāo Paulo, FAPESP Proc. 1999/02684-5, and Lehigh University, and thanks those institutions for enabling the collaboration involved in this work.Mathematics Subject Classifications (2000). 53C12, 53C38.     

Del Pezzo foliations with log canonical singularities

In this paper, we classify del Pezzo foliations of rank at least 3 on projective manifolds and with log canonical singularities in the sense of McQuillan.     

Stein compacts in Levi-flat hypersurfaces

We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface and holomorphic convexity of compact sets in , or bounded in part by . Applications include extendability of Cauchy-Riemann functions, solvability of the -equation, approximation of Cauchy-Riemann and holomorphic functions, and global regularity of the -Neumann operator.

     

Growth of leaves in transversely affine foliations

In this note we give estimates for the growth of leaves in transversely affine foliations which depend on the properties of the affine holonomy group.

     

Gradient flows within plane fields

We consider the dynamics of vector fields on three-manifolds which are constrained to lie within a plane field, such as occurs in nonholonomic dynamics. On compact manifolds, such vector fields force dynamics beyond that of a gradient flow, except in cases where the underlying manifold is topologically simple (i.e., a graph-manifold). Furthermore, there are strong restrictions on the types of gradient flows realized within plane fields: such flows lie on the boundary of the space of nonsingular Morse-Smale flows. This relationship translates to knot-theoretic obstructions for the link of singularities in the flow. In the case of an integrable plane field, the restrictions are even finer, forcing taut foliations on surface bundles. The situation is completely different in the case of contact plane fields, however: it is easy to realize gradient fields within overtwisted contact structures (the nonintegrable analogue of a foliation with Reeb components). Received: December 9, 1997.     

Optimal estimates for the uncoupling of differential equations

Our aim in this note is to give optimal conditions on the spectral gap for the existence of an uncoupling of a differential equation of the form = Cz + H(=) into a system ofuncoupled equations of the form (x, y) = (Ax, By) + (F(x, (x)), G((y),y)), whereC=A×B is a bounded linear operator on a Banach spaceZ=X×Y satisfying a spectral gap condition, andH=(F,G) is a Lipschitz function withH(0) = 0. We also give optimal conditions for the regularity of the manifoldsgraph andgraph , and optimal conditions for the regularity of the leaves of two foliations of the phase space associated to the uncoupling. Sharp estimates for the Lipschitz constant of and and for the Hölder exponent of the uncoupling homeomorphism and its inverse are also given.     

Center manifold and stability for skew-product flows

We study the existence and smoothness of global center, center-stable, and center-unstable manifolds for skew-product flows. Smooth invariant foliations to the center stable and center unstable manifolds are also discussed.     

Manifolds modelled over free modules over the double numbers

Manifolds over the algebra of double numbers, which include the case of manifolds equipped with a pair of equidimensional supplementary foliations, are studied. To this end, B-holomorphic functions and B-analytic functions on B ⁿ, where B denotes the algebra of double numbers, are defined and studied. This revised version was published online in June 2006 with corrections to the Cover Date.