Showing papers on "Unit tangent bundle published in 2005"

On natural metrics on tangent bundles of Riemannian manifolds

Abstract: There is a class of metrics on the tangent bundle T M of a Rie- mannian manifold (M; g) (oriented , or non-oriented, respectively), which are 'naturally constructed' from the base metric g (15). We call them \g-natural metrics" on T M. To our knowledge, the geometric properties of these general metrics have not been studied yet. In this paper, generalizing a process of Musso-Tricerri (cf. (18)) of nding Riemannian metrics on T M from some quadratic forms on OM R m to nd metrics (not necessary Riemannian) on T M, we prove that all g-natural metrics on T M can be obtained by Musso- Tricerri's generalized scheme. We calculate also the Levi-Civita connection of Riemannian g-natural metrics on T M. As application, we sort out all Riemannian g-natural metrics with the following properties, respectively: 1) The b ers of T M are totally geodesic. 2) The geodesic o w on T M is incom- pressible. We shall limit ourselves to the non-oriented situation.

On Riemannian g-natural Metrics of the Form a.g s + b.g h + c.g v on the Tangent Bundle of a Riemannian Manifold (M, g)

Abstract: Let (M, g) be a Riemannian manifold and TM its tangent bundle. In [5] we have investigated the family of all Riemannian g-natural metrics G on TM (which depends on 6 arbitrary functions of the norm of a vector u ∈ TM). In this paper, we continue this study under some additional geometric properties, and then we restrict ourselves to the subfamily {G=a.g s + b.g h + c.g v , a, b and c are constants satisfying a > 0 and a(a + c) − b2 > 0}. It is known that the Sasaki metric g s is extremely rigid in the following sense: if (TM, g s ) is a space of constant scalar curvature, then (M, g) is flat. Here we prove, among others, that every Riemannian g-natural metric from the subfamily above is as rigid as the Sasaki metric.

On Bernstein type theorems in Finsler spaces with the volume form induced from the projective sphere bundle

Abstract: By using the volume form induced from the projective sphere bundle of the Finsler manifold, we study the Finsler minimal submanifolds. It is proved that such a volume form for the Randers metric F = α+β in a Randers space is just that for the Riemannian metric a, and therefore the Bernstein type theorem in the special Randers space of dimension ≤ 8 is true. Moreover, a Bernstein type theorem in the 3-dimensional Minkowski space is established by considering the volume form induced from the projective sphere bundle.

Non)regularity of Projections of Measures Invariant Under Geodesic Flow

Abstract: We show that, unlike in the 2-dimensional case [LL], the Hausdorff dimension of a measure invariant under the geodesic flow is not necessarily preserved under the projection from the unit tangent bundle onto the base manifold if the base manifold is at least 3-dimensional. In the 2-dimensional case we reprove the preservation theorem due to Ledrappier and Lindenstrauss [LL] using the general projection formalism of Peres and Schlag [PS]. The novelty of our proof is that it illustrates the reason behind the failure of the preservation in the higher dimensional case. Finally, we show that the projected measure has fractional derivatives of order γ for all γ 2 and the base manifold has dimension 2.

The projective tangent bundles of a complex three-fold

Abstract: The points of X are the ordered triples of one 2-dimensional and two 1dimensional linear subspaces of the standard hermitian space C4 which are pairwise hermitian orthogonal. The manifold X carries two homogenous complex structures, namely the 5-dimensional complex manifold X1 consisting of the flags in C4 of type (0) ⊂ (1) ⊂ (3) ⊂ (4), i. e. the origin contained in a one-dimensional linear subspace, contained in a three-dimensional linear subspace, contained in C4, and the complex manifold X2 consisting of flags of type (0) ⊂ (1) ⊂ (2) ⊂ (4). Borel and I prove for the first Chern class c1 and the Chern number c1

Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane

Abstract: SAMUEL BOISSI`EREAbstract. The cohomology of the Hilbert schemes of points on smooth pro-jective surfaces can be approached both with vertex algebra tools and equi-variant tools. Using the first tool, we study the existence and the structure ofuniversal formulas for the Chern classes of the tangent bundle over the Hilbertscheme of points on a projective surface. The second tool leads then to nicegenerating formulas in the particular case of the Hilbert scheme of points onthe affine plane.

Lines on complex contact manifolds, II

Abstract: Let X be a complex-projective contact manifold with is a general point, then all contact lines through x are smooth, no two of them share a common tangent direction at x, and the union of all contact lines through x forms a cone over an irreducible, smooth base. As a corollary, we obtain that the tangent bundle of X is stable.

Algebraic Mapping-Class Groups of Orientable Surfaces with Boundaries

Abstract: Let Sg,b,p denote a surface which is connected, orientable, has genus g, has b boundary components, and has p punctures. Let Σg,b,p denote the fundamental group of Sg,b,p.

Contact Metric Geometry of the Unit Tangent Sphere Bundle

Abstract: We review some of the most interesting results concerning the interactions between the geometry of a Riemannian manifold and the one of its unit tangent sphere bundle, equipped with its natural contact metric structure.

Transverse totally geodesic submanifolds of the tangent bundle

Abstract: It is well-known that ifis a smooth vector field on a given Rie- mannian manifold M n thennaturally defines a submanifold �(M n ) transverse to the fibers of the tangent bundle TM n with Sasaki metric. In this paper, we are interested in transverse totally geodesic subman- ifolds of the tangent bundle. We show that a transverse submanifold N l of TM n (1 ≤ l ≤ n) can be realized locally as the image of a sub- manifold F l of M n under some vector fielddefined along F l . For such images �(F l ), the conditions to be totally geodesic are presented. We show that these conditions are not so rigid as in the case of l = n, and we treat several special cases (� of constant length, � normal to F l , M n of constant curvature, M n a Lie group anda left invariant vector field).

On the torsion-free connections on higher order frame bundles

Abstract: Using the r-jets of flows of vector fields, we show that every torsion-free linear r-th order connection on the tangent bundle of a manifold M determines a reduction of the (r+1)-st order frame bundle of M to the general linear group We deduce that this reduction coincides with another reduction constructed earlier

Manifolds with an $SU(2)$-action on the tangent bundle

Abstract: We study manifolds arising as spaces of sections of complex manifolds fibering over CP 1 with the normal bundle of each section isomorphic to O(k)⊗C n

Conformal vector fields on tangent bundle of a riemannian manifold

Abstract: Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. Theconformal and fiber preserving vector fields on TM have well-known physical interpretations and havebeen studied by physicists and geometricians. Here we define a Riemannian or pseudo-Riemannian liftmetric g􀀄 on TM , which is in some senses more general than other lift metrics previously defined onTM , and seems to complete these works. Next we study the lift conformal vector fields on (TM,g􀀄) andprove among the others that, every complete lift conformal vector field on TM is homothetic, andmoreover, every horizontal or vertical lift conformal vector field on TM is a Killing vector.

Higher order Hessian structures on manifolds

Abstract: In this paper we define nth order Hessian structures on manifolds and study them. In particular, whenn = 3, we make a detailed study and establish a one-to-one correspondence betweenthird-order Hessian structures and acertain class of connections on the second-order tangent bundle of a manifold. Further, we show that a connection on the tangent bundle of a manifold induces a connection on the second-order tangent bundle. Also we define second-order geodesics of special second-order connection which gives a geometric characterization of symmetric third-order Hessian structures.

On the stability of the tangent bundle of a hypersurface in a Fano variety

Abstract: Let M be a complex projective Fano manifold whose Picard group is isomorphic to Z and the tangent bundle TM is semistable. Let Z ⊂ M be a smooth hypersurface of degree strictly greater than degree(TM)(dimC Z−1)/(2 dimC Z−1) and satisfying the condition that the inclusion of Z in M gives an isomorphism of Picard groups. We prove that the tangent bundle of Z is stable. A similar result is proved also for smooth complete intersections in M . The main ingredient in the proof is a vanishing result for the top cohomology of the twisted holomorphic differential forms on Z.

A topological minorization for the volume of vector fields on 5-manifolds

Abstract: A vector field X on a Riemannian manifold determines a submanifold in the tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. When M is compact, the volume is well defined and, usually, this functional is studied for unit fields. Parallel vector fields are trivial minima of this functional. For manifolds of dimension 5, we obtain an explicit result showing how the topology of a vector field with constant length influences its volume. We apply this result to the case of vector fields that define Riemannian foliations with all leaves compact.

On extrinsic geometry of unit normal vector fields of Riemannian hyperfoliations.

Abstract: We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a totally umbilic hyperfoliation. A corresponding example shows the non-triviality of this condition. In the 2-dimensional case, we give a complete description of Riemannian manifolds admitting a geodesic unit vector field with totally geodesic property.

On special types of minimal and totally geodesic unit vector fields

Abstract: We present a new equation with respect to a unit vector field on Riemannian manifold $M^n$ such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and prove that within this class the Hopf vector field is a unique global one with totally geodesic property. For the wider class of geodesic unit vector fields on a sphere we give a new necessary and sufficient condition to generate a totally geodesic submanifold in $T_1S^n$.

Random $k$-surfaces

Abstract: Invariant measures for the geodesic flow on the unit tangent bundle of a negatively curved Riemannian manifold are a basic and well-studied subject. This paper continues an investigation into a 2-dimensional analog of this flow for a 3-manifold N. Namely, the article discusses 2-dimensional surfaces immersed into N whose product of principal curvature equals a constant k between 0 and 1, surfaces which are called k-surfaces. The ?2-dimensional? analog of the unit tangent bundle with the geodesic flow is a ?space of pointed k-surfaces?, which can be considered as the space of germs of complete k-surfaces passing through points of N. Analogous to the 1-dimensional lamination given by the geodesic flow, this space has a 2-dimensional lamination. An earlier work [1] was concerned with some topological properties of chaotic type of this lamination, while this present paper concentrates on ergodic properties of this object. The main result is the construction of infinitely many mutually singular transversal measures, ergodic and of full support. The novel feature compared with the geodesic flow is that most of the leaves have exponential growth.

Full description of totally geodesic unit vector fields on 2-dimensional Riemannian manifolds

Abstract: We give a full geometrical description of local totally geodesic unit vector field on Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its unit tangent bundle with the Sasaki metric.

Explicit geodesic flow-invariant distributions using SL2(ℝ)-representation ladders

Abstract: An explicit construction of a geodesic flow-invariant distribution lying in the discrete series of weight 2k isotopic component is found, using techniques from representation theory of SL2(ℝ). It is found that the distribution represents an AC measure on the unit tangent bundle of the hyperbolic plane minus an explicit singular set. Finally, via an averaging argument, a geodesic flow-invariant distribution on a closed hyperbolic surface is obtained.

Linear transports along paths in vector bundles. III. Curvature and torsion

Abstract: Curvature and torsion of linear transports along paths in, respectively, vector bundles and the tangent bundle to a differentiable manifold are defined and certain their properties are derived

Vector bundles over Grassmannians and the skew-symmetric curvature operator

Abstract: A pseudo-Riemannian manifold is said to be spacelike Jordan IP if the Jordan normal form of the skew-symmetric curvature operator depends upon the point of the manifold, but not upon the particular spacelike 2-plane in the tangent bundle at that point. We use methods of algebraic topology to classify connected spacelike Jordan IP pseudo-Riemannian manifolds of signature ( p , q ) , where q ⩾ 11 , p ⩽ q − 6 4 and where the set { q , … , q + p } does not contain a power of 2.

On the symmetrisation of nonholonomic jets

Abstract: The canonical involution in the second order iterated tangent bundle is generalised for an arbitrary order and transfered to jet spaces. The classification of all symmetrised nonholonomic jets of the third order is given.

Legendre transformation and lifting of multi-vectors

Abstract: This paper has three objectives. First to recall the link between the classical Legendre-Fenschel transformation and a useful isomorphism between 1-jets of functions on a vector bundle and on its dual. As a particular consequence we obtain the classical isomorphism between the cotangent bundle of the tangent bundle $T^*TM$ and the tangent bundle of the cotangent bundle $TT^*M$ of any manifold $M.$ Secondly we show how to use this last isomorphism to construct the lifting of any contravariant tensor field on a manifold $M$ to the tangent bundle $TM$ which generalizes the classical lifting of vector fields. We also show that, in the antisymmetric case, this lifting respects the Schouten bracket. This gives a new proof of a recent result of Crainic and Moerdijk. Finally we give an application to the study of the stability of singular points of Poisson manifold and Lie algebroids.

Uniruled varieties with split tangent bundle

Abstract: Beauville asked if a compact Kahler manifold with split tangent bundle has a universal covering that is a product of manifolds. We use Mori theory and elementary results about holomorphic foliations to study this problem for projective uniruled varieties. In particular we obtain an affirmative answer for rationally connected varieties in any dimension and uniruled varieties in dimension 4.

Geometric analysis of tangent measures

Abstract: The authors study the tangent measures of a Radon measure μ and obtain the rectifia-bility of tangent measures and the existence of the micro tangent set of a continuous function by using the duality and the blow-up technique. The authors also give out the properties of flatness of tangent measures. As an application of tangent measures, the authors, with the help of Energy of μ, prove further the Marstrand's theorem.