Showing papers in "Differential Geometry and Its Applications in 1992"
TL;DR: In this article, the authors studied the calculus of forms along the tangent bundle projection τ, initiated in a previous paper with the same title, and provided a list of commutators of important derivations and special attention paid to degree zero derivations having a Leibnitz-type duality property.
Abstract: The study of the calculus of forms along the tangent bundle projection τ, initiated in a previous paper with the same title, is continued. The idea is to complete the basic ingredients of the theory up to a point where enough tools will be available for developing new applications in the study of second-order dynamical systems. A list of commutators of important derivations is worked out and special attention is paid to degree zero derivations having a Leibnitz-type duality property. Various ways of associating tensor fields along τ to corresponding objects on TM are investigated. When the connection coming from a given second-order system is used in this process, two important concepts present themselves: one is a degree zero derivation called the dynamical covariant derivative; the other one is a type (1, 1) tensor field along τ, called the Jacobi endomorphism. It is illustrated how these concepts play a crucial role in describing many of the interesting geometrical features of a given dynamical system, which have been dealt with in the literature.
91 citations
TL;DR: In this article, the authors describe the geometry and topology of a compact simply connected positively curved Riemannian 6-manifold F′ which is related to the flag manifold F over C P2, and an infinite series of simply connected circle bundles over F′ with positive sectional curvature.
Abstract: We describe the geometry and the topology of a compact simply connected positively curved Riemannian 6-manifold F′ which is related to the flag manifold F over C P2, and an infinite series of simply connected circle bundles over F′, also with positive sectional curvature. All of these spaces are biquotients of the Lie group SU (3) and they are not homeomorphic to a homogeneous space of positive curvature.
80 citations
TL;DR: A geometric framework for constrained dynamical systems that allows to describe in a unified way a general type of first order singular differential equations on a manifold and a stabilization algorithm for the lagrangian formalism is obtained.
Abstract: A geometric framework for constrained dynamical systems is presented. It allows to describe in a unified way a general type of first order singular differential equations on a manifold; these equations can not be written in normal form since the derivatives appear multiplied by a linear operator, therefore we call them linearly constrained systems. The concepts of constraints and morphisms between linearly constrained systems are defined, and their relationships studied. Finally, a stabilization algorithm is devised and carefully discussed in order to solve the equation of motion. Our formalism includes the presymplectic and the lagrangian formalisms, as well as higher order lagrangians, and we give several applications of it; in particular, a stabilization algorithm for the lagrangian formalism is obtained.
59 citations
TL;DR: In this paper, the authors studied two classes of Riemannian manifolds which extend the class of locally symmetric spaces: manifolds all of whose Jacobi operators Rγ have constant eigenvalues (C -spaces) or parallel eigenspaces along geodesics γ.
Abstract: One studies two classes of Riemannian manifolds which extend the class of locally symmetric spaces: manifolds all of whose Jacobi operators Rγ have constant eigenvalues ( C -spaces) or parallel eigenspaces ( B -spaces) along geodesics γ. One gives several examples, derives equivalent characterizations and treats classifications for the two- and the three-dimensional case.
47 citations
TL;DR: In this article, the triplyperiodic minimal P surface of Schwarz is investigated and it is shown that this surface is stable under periodic volume-preserving variations and the conjugate D surface is also stable.
Abstract: The triply-periodic minimal P surface of Schwarz is investigated. It is shown that this surface is stable under periodic volume-preserving variations. As a consequence, the conjugate D surface is also stable. Other notions of stability are discussed.
46 citations
TL;DR: In this paper, the asymptotics of Tr L 2 ( Pe - tP 2 ) are computed for the boundary value problem on a compact manifold with smooth boundary, where P is a first order operator of Dirac type on M with suitable local boundary conditions.
Abstract: Let M be a compact manifold with smooth boundary. Let P be a first order operator of Dirac type on M with suitable local boundary conditions. We compute the asymptotics of Tr L 2 ( Pe - tP 2 ). This is equivalent to evaluating the residues of the eta function for the corresponding boundary value problem.
39 citations
TL;DR: In this paper, a complex ∂ : Λ i ( e ) ⊗ D V ( e ), i = 0, 1,…, where ∂ is associated with the Cartan connection on e (∞) and is constructed using Frolicher-Nijenhuis bracket.
Abstract: For any differential equation e a complex ∂ : Λ i ( e ) ⊗ D V ( e ) → Λ i +1 ( e ) ⊗ D V ( e ), i = 0, 1,…, is introduced, where Λ i ( e ) is the module of i -forms, D V ( e ) is the algebra of vertical derivations, while ∂ is associated with the Cartan connection on e (∞) and is constructed using Frolicher-Nijenhuis bracket. 0-cohomology of this complex coincides with the Lie algebra of (higher) symmetries for e , 1-cohomology is the set of equivalence classes of nontrivial deformations for the equation structure. Three spectral sequences are associated with this complex. Using one of them cohomology super algebra for the “empty” (= J ∞ (π)) and evolution equations is described. The notion of evolutionary super derivation is introduced. Two examples are considered to illustrate the general theory—those of the heat and of the Burgers equations.
38 citations
TL;DR: In this paper, a gravitational analogue of Min-Oo's gap theorem for Yang-Mills fields is given, where the gap is defined as the distance from the Earth's center.
Abstract: A gravitational analogue is given of Min-Oo's gap theorem for Yang-Mills fields.
31 citations
TL;DR: In this paper, it was shown that the above Kurose's result still remains valid when (M, g, ▿) is assumed only to be ▿-isotropic.
Abstract: A statistical manifold ( M , g , ▿) is a Riemannian manifold ( M , g ) equipped with torsion-free affine connections ▿, ▿ ∗ which are dual with respect to g . A point p \te M is said to be ▿-isotropiv if the sectional curvatures have the same value k ( p ), and ( M , g , ▿) is said to be ▿-isotropic when M consists entirely of ▿-isotropic points. When the difference tensor α of ▿ and the Levi-Civita connection ▿ 0 of g is “apolar” with respect to g , Kurose has shown that α ≡ 0, and hence ▿ = ▿ ∗ = ▿ 0 , provided that k ( p ) = k (constant). His proof relies on the existence of affine immersion which may no longer hold when k ( p ) is not constant. One objective of this paper is to show that the above Kurose's result still remains valid when ( M , g , ▿) is assumed only to be ▿-isotropic. We also discuss the case where ( M , g ) is complete Riemannian.
30 citations
TL;DR: In this article, the authors give a characterization of graded symplectic forms by studying the module of derivations of a graded sheaf, which gives rise to canonical Poisson brackets on the graded manifold.
Abstract: We give a characterization of graded symplectic forms by studying the module of derivations of a graded sheaf. When the graded sheaf is the sheaf of differentiable forms on the underlying manifold M, we find canonical liftings from metrics on TM to odd symplectic forms, and from symplectic forms on M and metrics on TM to even symplectic forms. These graded symplectic forms give rise to canonical Poisson brackets on the graded manifold.
23 citations
TL;DR: In this paper, the minimal interaction principle for a charged particle in an external electromagnetic field describes in symplectic language the orbit of the Hamiltonian formalism on the cotangent bundle T * E under the affine action on the canonical two-form d p ∧ d q of the additive group ⋀ 1 (E ).
Abstract: In classical mechanics, the minimal interaction principle for a charged particle in an external electromagnetic field describes in symplectic language the orbit of the Hamiltonian formalism on the cotangent bundle T * E under the affine action on the canonical two-form d p ∧ d q of the additive group ⋀ 1 ( E ). Local description of this orbit amounts to the Poincare Lemma for differential two-forms. We present an infinite-dimensional analog of these where the manifold E is replaced by a bundle π : E → M . Nonabelian case, corresponding to the external field being of Yang-Mills rather than electromagnetic type, is also discussed. In this case the orbit turns out to be much simpler and “smaller” than the abelian one.
TL;DR: The natural vector valued l-forms Q on the natural bundles associated with product-preserving functors (including the tangent bundle, the bundle of first order k-velocities, the bundles of second order l-vectors and the bundling of linear frames) and on the cotangent bundle are classified in this paper.
Abstract: The natural vector valued l-forms Q on the natural bundles associated with product preserving functors (including the tangent bundle, the bundle of first order k-velocities, the bundle of second order l-velocities and the bundle of linear frames) and on the cotangent bundle are classified. Then, these forms Q are used to study the torsion r = (I', Q) of connections P on the above bundles, where ( 1, -1 is the Frolicher-Nijenhuis bracket.
TL;DR: Using the methods of twistor geometry, the authors classified compact conformally flat hermitian surfaces by associating to the universal covering of such a surface an open set of a quadric in CP 3 (the twistor space of S4).
Abstract: Using the methods of twistor geometry we classify compact conformally flat hermitian surfaces. This is accomplished by first associating to the universal covering of such a surface an open set of a quadric in CP 3 (the twistor space of S4) and then applying some classical projective geometry.
TL;DR: In this article, it was shown that the integration by separation of variables of the geodesic Hamilton-Jacobi equation always occurs in orthogonal coordinates on Riemannian and Lorentzian manifolds.
Abstract: Coordinates which allow the integration by separation of variables of the geodesic Hamilton-Jacobi equation are called separable. Particular interest is placed on orthogonal separable coordinates. In this paper it is proved that on a Riemannian manifold with constant curvature and on a Lorentzian manifold with constant positive curvature every system of separable coordinates has an orthogonal equivalent, i.e. that in these manifolds the integration by separation of variables of the geodesic Hamilton-Jacobi equation always occurs in orthogonal coordinates. Proofs of this property concerning strictly-Riemannian manifolds of positive, negative and zero constant curvature (and also for conformally flat manifolds) were firstly given by Kalnins and Miller (1982–1986). The proof presented here is based on elementary properties of Killing vectors of an affine space and on a geometrical characterization of the equivalence classes of separable coordinates.
TL;DR: In this paper, a natural correspondence between null holomorphic curves in P3 ≃ C3 ∪ P2 and holomorphic curve in P∗3 that lie on C (Q1), the projective cone over a certain quadric curve Q1.
Abstract: There exist a natural correspondence, determined by classical osculation duality, between null holomorphic curves in P3 ≃ C3 ∪ P2 and holomorphic curves in P∗3 that lie on C (Q1), the projective cone over a certain quadric curve Q1. This facilitates the study of minimal surfaces in R3 in terms of holomorphic curves on C(Q1). Algebraic curves on C(Q1) generate complete branched minimal surfaces of finite total Gaussian curvature. The ‘end’ structure, branch points and total Gaussian curvature of the minimal surface are determined by features of the corresponding algebraic curve. Natural compactifications of the moduli spaces of null meromorphic curves in C3 are given by linear systems on the Hirzebruch surface S2.
TL;DR: In this paper, a local classification of all three-dimensional Riemannian manifolds whose Ricci tensor satisfies the equation ▿(ric-1 4 sg) = 1 20 ds ⊙ g is given.
Abstract: One derives a local classification of all three-dimensional Riemannian manifolds whose Ricci tensor satisfies the equation ▿(ric– 1 4 sg) = 1 20 ds ⊙ g.
TL;DR: In this paper, it was shown that naturality is equivalent to commuting with Lie differentiation and generalized this result to the case of gauge-natural operators and showed that nonlinear geometric operations (e.g., the exterior differential) commute with the Lie derivative.
Abstract: It is well known that linear geometric operations (like the exterior differential) commute with the Lie derivative. A detailed analysis of both the concepts of geometric operations and of Lie differentiation leads to the proof of a converse implication even in the nonlinear case. So naturality is equivalent to commuting with Lie differentiation. We also generalize this result to the case of gauge-natural operators.
TL;DR: Infinitesimal natural and gauge-natural lifts are defined as special "systems" of vector fields on a fibred manifold p : E -+ B as discussed by the authors, which are defined via the commutativity with Lie derivatives.
Abstract: Infinitesimal natural and gauge-natural lifts are defined as special "systems" of vector fields on a fibred manifold p : E -+ B. (Infinitesimally) natural and gauge-natural operators are defined via the commutativity with Lie derivatives.
TL;DR: In this article, the maximum 2-rank almost Grassmannizable 6-webs AGW (6, 3, 2) of codimension two given on a six-dimensional differentiable manifold X 6 is described.
Abstract: The maximum 2-rank almost Grassmannizable 6-webs AGW (6, 3, 2) of codimension two given on a six-dimensional differentiable manifold X 6 are described.
TL;DR: For a transversely oriented foliation on an oriented Riemannian manifold, an evaluation of the exterior derivative of the mean curvature one-form on basic transverse fields is recalled in this paper.
Abstract: For a transversely oriented foliation on an oriented Riemannian manifold, an evaluation of the exterior derivative of the mean curvature one-form on basic transverse fields is recalled (2). This formula has rich geometric properties which reside in the (basic) cohomology class of the mean curvature one-form provided it is closed. These properties are explored. In the case the foliation is bundle-like, additional pleasant geometric properties obtain.
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