Showing papers on "Unit tangent bundle published in 1997"
TL;DR: In this article, the energy of a unit vector field on a Riemannian manifold M is defined to be the energy energy of the mapping M → T 1M, where the unit tangent bundle T1M is equipped with the restriction of the Sasaki metric.
Abstract: The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M → T1M, where the unit tangent bundle T1M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M=S5,S7,..., and an estimate on the index is obtained.
129 citations
Journal Article•
60 citations
TL;DR: In this paper, it was shown that conditionals on strong unstable Riemannian manifolds can be realized as Hausdorff measures, and that the cohomology classes of flip invariant cocycles are in a one-to-one correspondence to cross ratios on the space of four pairwise distinct points of the ideal boundary of the universal covering.
Abstract: Let $f$ be a flip-invariant Holder continuous function
on the
unit tangent bundle $T^1 M$ of a closed negatively curved Riemannian manifold
$M$. We show that conditionals on strong unstable manifolds
of the Gibbs equilibrium state defined by $f$
can be realized as Hausdorff measures. Moreover,
cohomology classes of flip invariant cocycles are in
one-to-one correspondence to cross ratios on the space of
four pairwise distinct points
of the ideal boundary of the universal covering $\tilde M$ of $M$.
28 citations
TL;DR: Theorem 4.2 as discussed by the authors states that the smooth projective family of curves is invariant to a change in the base of the curve family, i.e., the bresofthe family are invariant.
Abstract: MathematicsSubjectClassi cation(1991):14J10,14D99Itisawell-knownconsequenceoftheTorellitheoremthatasmoothprojec-tivefamilyofcurvesofgenusatleast2overaprojectiverationalorellipticcurveisisotrivial,thatis,the bresofthefamilyareisomorphic.Sincetheautomorphismgroupofacurveofgenusatleast2is nite,thisalsoimpliesthatthefamilybecomestrivialaftera nitebasechange.The above statement was generalized for smooth projective families ofminimalsurfacesofgeneraltypein[Migliorini95],andforsmoothprojec-tivefamiliesofvarieties(ofarbitrarydimension)withamplecanonicalbundlein[Kovacs96].Botharticlesstudiedfamiliesovercurves.Theaimofthisarticleistopresentafurthergeneralization,namelyletthebaseofthefamilyhavearbitrarydimension.0.1Theorem=4.2Theorem.Letf:X !S beasmoothmorphismofpro-jectivealgebraicvarietiessuchthatthecanonicalbundleofevery breoffisample.AssumethatS isbirationaltoanS
26 citations
TL;DR: In this article, it was shown that there exists a local product structure in the unit tangent bundle of a Riemannian manifold which is invariant under the geodesic flow.
Abstract: Let be a compact Riemannian manifold with no conjugate points such that its geodesic flow is expansive. We show that there exists a local product structure in the unit tangent bundle of the manifold which is invariant under the geodesic flow.
In particular, we have that the set of closed geodesics is dense and that the flow is topologically transitive.
20 citations
TL;DR: In this paper, it was proved that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.
Abstract: Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.
17 citations
TL;DR: In this article, the holomorphic wedge extendability of CR functions on an everywhere locally minimal generic submanifold M of C^n and having singularities contained in a sub manifold N of codimension 1, 2 or 3, assuming some transversality conditions about the relative disposition of N with respect to the complex tangent bundle to M was established.
Abstract: We establish the holomorphic wedge extendability of CR functions, defined on an everywhere locally minimal generic submanifold M of C^n and having singularities contained in a submanifold N of codimension 1, 2 or 3, assuming some transversality conditions about the relative disposition of N with respect to the complex tangent bundle to M. The statements hold in arbitrary codimension and are obtained by applying the theory of normal deformations of analytic discs, due to A.E. Tumanov in 1994 and also the FBI propagation of singularities phenomenon enjoyed by CR functions, due to J.-M. Trepreau in 1990. Related results in the hypersurface case were obtained simultaneously by B. Joricke in 1992-96 and by E. Porten in his thesis (1996).
16 citations
TL;DR: In this article, it was shown that the classical result of Riemannian geometry is in fact a simple consequence of a completely general theorem of the Calculus of Variations, which states that Jacobi fields along geodesics can be obtained as geodesies of the so-called complete lift of the metric g itself to the tangent bundle TQ.
Abstract: A classical result of Riemannian geometry states that Jacobi fields along geodesics of a Riemannian manifold (Q, g) can be obtained as geodesies of the so-called «complete lift» of the metric g itself to the tangent bundle TQ. We show that this classical result is in fact a very simple consequence of a completely general theorem of Calculus of Variations.
14 citations
TL;DR: The rotation number as mentioned in this paper is the asymptotic time average of the angular rotation of a given tangent vector under the action of the derivative flow in the tangent bundle over a Riemannian manifold.
Abstract: Rotation number is the asymptotic time average of the angular rotation of a given tangent vector under the action of the derivative flow in the tangent bundle over a Riemannian manifold M This angle in higher dimension is taken with respect to a reference given by the stochastic parallel transport along the trajectories and the canonical connection in the Stiefel bundle St2M So, these numbers give an angular complementary information to that given by the Lyapunov exponents. We lift the stochastic differential equation on M to a stochastic equation in the Stiefel bundle and we use Furstenberg-Khasminskii argument to prove the existence almost surely of the rotation numbers with respect to any invariant measure on this bundle. Finally we present some information on the dynamical system provided by the rotation number: rotation of the stable manifold (theorem 6.4)
11 citations
6 citations
TL;DR: In this paper, the authors used dynamical approach to study parabolic fixed points of Kleinian groups Γ ⊂ Iso(ℍn) with constant negative curvature.
Abstract: We use dynamical approach to study parabolic fixed points of Kleinian groups Γ ⊂ Iso(ℍn). Let ℋ be the horospherical foliation on the unit tangent bundle SM of manifold M = Γ\ℍn with constant negative curvature. We construct examples Γ ⊂ Iso(ℍ4) which show that horosphere based at parabolic fixed point w ∈ ∂ℍ4 can project to leaf ℋx ⊂ SM of complicated structure: it can be locally closed and not closed; not locally closed and non-dense in the non-wandering set Ω+ ⊂ SM of ℋ; dense in Ω+ (this is equivalent to w being a horospherical limit point). Using the natural duality, one gets the corresponding examples of Γ-orbits on the light cone. We give an elementary proof of the fact that conical limit point w ∈ ∂ℍn cannot be a parabolic fixed point.
01 Jan 1997
TL;DR: In this article, the authors show that the geo-desic on the unit tangent bundle of a two dimensional Riemannian manifold with constant negative curvature provides an example of a fast dynamo for the magnetic kinematic dynamo equation.
Abstract: C. CHICONE AND Y. LATUSHKIN(Communicated by Je rey B. Rauch)Abstract. We give elementary and explicit arguments to show that the geo-desic flow on the unit tangent bundle of a two dimensional Riemannian mani-fold with constant negative curvature provides an example of a \fast" dynamofor the magnetic kinematic dynamo equation.
Journal Article•
TL;DR: In this paper, it was shown that the normal bundle of a Lagrangian submanifold in a Khler manifold has a symplectic structure and provided the equivalent conditions for such a bundle to be Khler.
Abstract: We show that the normal bundle of a Lagrangian submanifold in a Khler manifold has a symplectic structure and provide the equivalent conditions for the normal bundle of such to be Khler.
Journal Article•
TL;DR: In this paper, a reformulation of the five-dimensional Kaluza-Klein theory is presented, taking into account the fact that these five dimensions of space-time are present on the tangent bundle of the Principal Bundle (B).
Abstract: This paper presents a reformulation of the five-dimensional Kaluza-Klein theory. taking into consideration the fact that these five dimensions of space-time are present on the tangent bundle of the Principal Bundle (B). Tb1s reformulation is possible due to the group U(l) of the fiber which endows B with the structure of a differential manifold. this allowing the reproduction of parallel transport notions on such a manifold. Through this we achieve a criterion to add extra dimensions to the theory. Once this criterion has been established. a metric is constructed on the tangent bundle of B. which is formed by a metric of space-time plus a connection, the a one-form defined in the fiber of B whose curvature 1s proportional to the electromagnetic field. This. the five-dimensional Einstein action is the sum of the four-dimensional Einstein action and the four-dimensional Maxwell action. This formulation yields the Einstein and Maxwell equations, as well as the Yang-Mills equations.
TL;DR: In this paper, the second order almost transverse structure of a differential manifold is defined and conditions for it to admit a real almost product and a generalised almost tangent structure of second order.
Abstract: The present work is based on a type of structures on a differential manifold $V$, called $G$-structures of the second kind, defined by endomorphism $J$ on the second order tangent bundle $T^2(V)$. Our objective is to give conditions for a differential manifold to admit a real almost product and a generalised almost tangent structure of second order. The concepts of the second order frame bundle $H^2(V)$, its structural group $L^2$ and its associated tangent bundle of second order $T^2(V)$ of a differentiable manifold $V$, are used from the point of view that is described in papers \cite{5} and \cite{6}. Also, the almost tangent structure of order two is mentioned and its generalisation, the second order almost transverse structure, is defined.
Posted Content•
TL;DR: In this article, the authors investigated the deformation space of Engel structures obtained by deforming certain canonical Engel structures on four-manifolds with boundary, and they showed that this subspace contains a subspace corresponding to Zoll metrics on the two-sphere modulo 'projective' equiv- alence.
Abstract: An Engel structure is a maximally non-integrable field of two-planes tangent to a four-manifold. Any two Engel structures are locally diffeomorphic. We investigate the deformation space of Engel structures obtained by deforming certain canonical Engel structures on four-manifolds with boundary. When the manifold is RP 3 × I where I is a closed interval, we show that this deformation space contains a subspace corresponding to Zoll metrics on the two-sphere (metrics all of whose geodesics are closed) modulo 'projective' equiv- alence. The main tool is a construction of an Engel manifold from a three-dimensional contact manifold and a method of reversing this construction. These are special instances of Cartan's method of pro- longation and deprolongation. The double prolongation of a surface X is an Engel manifold of the form SX ×S 1 where SX denotes the unit tangent bundle to X. The RP 3 × I example occurs in this way, since the unit tangent bundle of the two-sphere is RP 3 . Besides proving these new results, the article has the flavour of a review.