Hamiltonian structures on foliations

We discuss hamiltonian structures of the Gelfand-Dorfman complex of projectable vector fields and differential forms on a foliated manifold. Such a structure defines a Poisson structure on the algebra of foliated functions, and embeds the given foliation into a larger, generalized foliation with presymplectic leaves. In a so-called tame case, the structure is induced by a Poisson structure of the manifold. Cohomology spaces and classes relevant to geometric quantization are also considered.Comment: LaTex, 18 page

1850 Spiritual Directory of the Congregation of the Holy Ghost and of the Immaculate Heart of Mary

Contents -- Part I. -- Rules Concerning Certain General dispositions Which Ought to Animate the Members of the Congregation -- (p. 1) -- Chapter I. -- Of the love and esteem we out to have for our holy vocation -- (p. 1) -- Chapter II. -- Of the love and devotedness we ought to have for our Congregation in particular. -- (p. 2) -- Chapter III. -- Of the esteem we should have for our rules and constitutions and of the fidelity with which we should observe them. -- (p. 3) -- Chapter IV. -- Of the devotion which ought to animate the members of the Congregation. -- (p. 4) -- Chapter V. -- Of the spirit that ought to direct the members of the Congregation in their general conduct. -- (p. 5) -- Chapter VI. -- Of the spirit of humility that ought to animate the members of the congregation in their general conduct. -- (p. 6) -- Chapter VII. -- Of the spirit of simplicity that ought to animate the members of the Congregation. -- (p. 8) -- Chapter VIII. -- Of the interior spirit that ought to animate the members of the Congregation. -- (p. 9) -- Chapter IX. -- Of the gravity and religious modesty which ought to distinguish the members of the Congregation. -- (p. 10) -- Part II. -- The Spirit Which Ought to Animate the Members of the Congregation in the Accomplishment of their Duties as Religious. -- (p. 12) -- Chapter I. -- Of the religious spirit in general. -- (p. 12) -- Chapter II. -- Of the spirit that ought to animate the members in regard to Holy Poverty. -- (p. 13) -- Chapter III. -- Of the spirit which ought to animate the members in the practice of Chastity. -- (p. 15) -- Chapter IV. -- Of the spirit that ought to animate the members in the practice of obedience. -- (p. 17) -- Part III. -- Of the Spirit that Should Animate the Members of the Congregation in the Accomplishment of their Duties as Members of a Community -- (p. 20) -- Chapter I. -- Of the community spirit in general. -- (p 20) -- Chapter II. -- Of the spirit of union and charity that should exist among the different parts or communities of the Congregation. -- (p. 20) -- Chapter III. -- Of the spirit of union and charity that should distinguish the members in their relations with one another. -- (p. 21) -- Chapter IV. -- Of the dispositions that should animate inferiors in respect to Superiors. -- (p. 23) -- Part IV. -- Of the Spirit and Dispositions that Should Animate the Missionaries in the Exercise of their Apostolate and in Their Dealings with Souls. -- (p. 25) -- Chapter I. -- Of the general dispositions that should distinguish the members as true apostles of Jesus Christ. -- (p. 25) -- Chapter II. -- Of the spirit that should animate the missionaries in particular -- (p. 28) -- Chapter III. -- Of the general rules that should govern the conduct of missionaries in their relations with souls whom they evangelize. -- (p. 31) -- Chapter IV. -- Of the spirit in which they should administer the Sacraments and perform their other sacred functions. -- (p. 33) -- Chapter V. -- Of the manner in which they should fulfill their duties as preachers of the Word of God. -- (p. 34)https://dsc.duq.edu/spiritan-rc/1005/thumbnail.jp

Lie algebroid structures on a class of affine bundles

We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over the real numbers. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown that the affine Lie algebroid structure gives rise to a coboundary operator on such forms. The concept of admissible curves and dynamical systems whose integral curves are admissible, brings an associated affine bundle into the picture, on which one can define in a natural way a prolongation of the original affine Lie algebroid structure.Comment: 28 page

Lie algebroid structures and Lagrangian systems on affine bundles

As a continuation of previous papers, we study the concept of a Lie algebroid structure on an affine bundle by means of the canonical immersion of the affine bundle into its bidual. We pay particular attention to the prolongation and various lifting procedures, and to the geometrical construction of Lagrangian-type dynamics on an affine Lie algebroid.Comment: 28 pages, Late

Clifford-Finsler Algebroids and Nonholonomic Einstein-Dirac Structures

We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal metrics and linear and nonlinear connections define different types of Finsler, Lagrange and/or Riemann-Cartan spaces. A generalization to spinor fields and Dirac operators on nonholonomic manifolds motivates the theory of Clifford algebroids defined as Clifford bundles, in general, enabled with nonintegrable distributions defining the nonlinear connection. In this work, we elaborate the algebroid spinor differential geometry and formulate the (scalar, Proca, graviton, spinor and gauge) field equations on Lie algebroids. The paper communicates new developments in geometrical formulation of physical theories and this approach is grounded on a number of previous examples when exact solutions with generic off-diagonal metrics and generalized symmetries in modern gravity define nonholonomic spacetime manifolds with uncompactified extra dimensions.Comment: The manuscript was substantially modified following recommendations of JMP referee. The former Chapter 2 and Appendix were elliminated. The Introduction and Conclusion sections were modifie

Adapted connections on metric contact manifolds

In this paper, we describe the space of adapted connections on a metric contact manifold through the space of their torsion tensors. The torsion tensor is an element of the space of TM-valued two-forms, which splits into various subspaces. We study the parts of the torsion tensor according to this splitting to completely describe the space of adapted connections. We use this description to obtain characterizations of the generalized Tanaka-Webster connection and to describe the Dirac operators of adapted connections.Comment: 25 pages; some remarks added, minor correction

Instructions for Missionaries

Unfinished by Libermann, Instructions for Missionaries was first published in Directoire Spirituel. Table of Contents -- Introduction -- (p. 1) -- Chapter I Our Obligation of aiming at Holiness -- (p. 3) -- Chapter II Holiness is the Life of God in us, imparted through sanctifying Grace -- (p. 9) -- Chapter III Degrees of Holiness in our Works -- (p. 12) -- Chapter IV Holiness requires that We act habitually through Grace -- (p. 16) -- Chapter V Holiness of the Missionary -- (p. 21) -- Chapter VI Obstacles to Holiness -- (p. 25) -- Chapter VII Means of Growth in Holiness -- (p. 29) -- Chapter VIII Further Means of Sanctification -- (p.33) -- Chapter IX Practical Union -- (p.40) -- Chapter X Mental Prayer and Recollection -- (p.45) -- Chapter XI Mental Prayer and Steadiness of Attention -- (p. 50) -- Chapter XII Mental Prayer and our Approach to God in Love -- (p. 53) -- Chapter XIII Different Kinds of Mental Prayer -- (p. 56) --Chapter XIV Subject Matter of Mental Prayer -- (p.61) -- Conclusion -- (p. 63)https://dsc.duq.edu/spiritan-rc/1021/thumbnail.jp

Poisson brackets with prescribed Casimirs

We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved by using a convenient almost cosymplectic structure. Several examples and applications are presented.Comment: 24 page

Wave packet evolution in non-Hermitian quantum systems

The quantum evolution of the Wigner function for Gaussian wave packets generated by a non-Hermitian Hamiltonian is investigated. In the semiclassical limit $\hbar\to 0$ this yields the non-Hermitian analog of the Ehrenfest theorem for the dynamics of observable expectation values. The lack of Hermiticity reveals the importance of the complex structure on the classical phase space: The resulting equations of motion are coupled to an equation of motion for the phase space metric---a phenomenon having no analog in Hermitian theories.Comment: Example added, references updated, 4 pages, 2 figure

Generalized Reduction Procedure: Symplectic and Poisson Formalism

We present a generalized reduction procedure which encompasses the one based on the momentum map and the projection method. By using the duality between manifolds and ring of functions defined on them, we have cast our procedure in an algebraic context. In this framework we give a simple example of reduction in the non-commutative setting.Comment: 39 pages, Latex file, Vienna ESI 28 (1993