Moving constraints as stabilizing controls in classical mechanics

The paper analyzes a Lagrangian system which is controlled by directly assigning some of the coordinates as functions of time, by means of frictionless constraints. In a natural system of coordinates, the equations of motions contain terms which are linear or quadratic w.r.t.time derivatives of the control functions. After reviewing the basic equations, we explain the significance of the quadratic terms, related to geodesics orthogonal to a given foliation. We then study the problem of stabilization of the system to a given point, by means of oscillating controls. This problem is first reduced to the weak stability for a related convex-valued differential inclusion, then studied by Lyapunov functions methods. In the last sections, we illustrate the results by means of various mechanical examples.Comment: 52 pages, 4 figure

Modified differentials and basic cohomology for Riemannian foliations

We define a new version of the exterior derivative on the basic forms of a Riemannian foliation to obtain a new form of basic cohomology that satisfies Poincar\'e duality in the transversally orientable case. We use this twisted basic cohomology to show relationships between curvature, tautness, and vanishing of the basic Euler characteristic and basic signature.Comment: 20 pages, references added, minor corrections mad

Energy properness and Sasakian-Einstein metrics

In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition that admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type inequality. At the end of this paper, we also obtain a Miyaoka-Yau type inequality in Sasakian geometry.Comment: 27 page

Riemannian Gauge Theory and Charge Quantization

In a traditional gauge theory, the matter fields \phi^a and the gauge fields A^c_\mu are fundamental objects of the theory. The traditional gauge field is similar to the connection coefficient in the Riemannian geometry covariant derivative, and the field-strength tensor is similar to the curvature tensor. In contrast, the connection in Riemannian geometry is derived from the metric or an embedding space. Guided by the physical principal of increasing symmetry among the four forces, we propose a different construction. Instead of defining the transformation properties of a fundamental gauge field, we derive the gauge theory from an embedding of a gauge fiber F=R^n or F=C^n into a trivial, embedding vector bundle F=R^N or F=C^N where N>n. Our new action is symmetric between the gauge theory and the Riemannian geometry. By expressing gauge-covariant fields in terms of the orthonormal gauge basis vectors, we recover a traditional, SO(n) or U(n) gauge theory. In contrast, the new theory has all matter fields on a particular fiber couple with the same coupling constant. Even the matter fields on a C^1 fiber, which have a U(1) symmetry group, couple with the same charge of +/- q. The physical origin of this unique coupling constant is a generalization of the general relativity equivalence principle. Because our action is independent of the choice of basis, its natural invariance group is GL(n,R) or GL(n,C). Last, the new action also requires a small correction to the general-relativity action proportional to the square of the curvature tensor.Comment: Improved the explanations, added references, added 3 figures and an appendix, corrected a sign error in the old figure 4 (now figure 5). Now 33 pages, 7 figures and 2 tables. E-mail Serna for annimation

Cohomology of Horizontal Forms

The complex of s-horizontal forms of a smooth foliation F on a manifold M is proved to be exact for every s = 1, . . . , n = codim F, and the cohomology groups of the complex of its global sections, are introduced. They are then compared with other cohomology groups associated to a foliation, previously introduced. An explicit formula for an s-horizontal primitive of an s-horizontal closed form, is given. The problem of representing a de Rham cohomology class by means of a horizontal closed form is analysed. Applications of these cohomology groups are included and several specific examples of explicit computation of such groups-even for non-commutative structure groups-are also presented