Minimal sphere bundles in Euclidean spheres

The purpose of this paper is to pursue to work initiated by Hsiang-Lawson and study cohomogeneity 1 minimal hypersurfaces in Euclidean spheres which are equivariant under the linear isotropy representation of a rank 3 compact symmetric space.Supported by the grant NSF DMS 90-01089 and by CNPq (Brazil)     

Harmonicity of sections of sphere bundles

We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold equipped with the Sasaki metric and discuss the characterising condition for critical points. Furthermore, we provide a useful method for computing the tension field in some particular situations. Such a method is shown to be adequate for many tensor fields defined on manifolds M equipped with a G-structure compatible with . This leads to the construction of several new examples of differential forms which are harmonic sections or determine a harmonic map from into its sphere bundle.     

Homotheties and topology of tangent sphere bundles

We prove a Theorem on homotheties between two given tangent sphere bundles S _r M of a Riemannian manifold M, g of \({{\rm dim}\geq3}\) , assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constant negative scalar curvature which are not Einstein. Recalling results on the associated almost complex structure I ^G and symplectic structure \({\omega^G}\) on the manifold TM, generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel–Whitney characteristic classes of the manifolds TM and S _r M.     

Minimal triangulations of sphere bundles over the circle

For integers d?2 and ε=0 or 1, let S1,d−1(ε) denote the sphere product S¹×Sd−1 if ε=0 and the twisted sphere product if ε=1. The main results of this paper are: (a) if then S1,d−1(ε) has a unique minimal triangulation using 2d+3 vertices, and (b) if then S1,d−1(ε) has minimal triangulations (not unique) using 2d+4 vertices. In this context, a minimal triangulation of a manifold is a triangulation using the least possible number of vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than spheres) for which the minimal triangulation is unique. Actually, we show that while S1,d−1(ε) has at most one (2d+3)-vertex triangulation (one if , zero otherwise), in sharp contrast, the number of non-isomorphic (2d+4)-vertex triangulations of these d-manifolds grows exponentially with d for either choice of ε. The result in (a), as well as the minimality part in (b), is a consequence of the following result: (c) for d?3, there is a unique (2d+3)-vertex simplicial complex which triangulates a non-simply connected closed manifold of dimension d. This amazing simplicial complex was first constructed by Kühnel in 1986. Generalizing a 1987 result of Brehm and Kühnel, we prove that (d) any triangulation of a non-simply connected closed d-manifold requires at least 2d+3 vertices. The result (c) completely describes the case of equality in (d). The proofs rest on the Lower Bound Theorem for normal pseudomanifolds and on a combinatorial version of Alexander duality.     

Geodesic Ricci solitons on unit tangent sphere bundles

In the paper, (Abbassi and Kowalski, Ann Glob Anal Geom, 38: 11–20, 2010) the authors study Einstein Riemannian $g$ natural metrics on unit tangent sphere bundles. In this study, we equip the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$ with an arbitrary Riemannian $g$ natural metric $\tilde{G}$ and we show that if the geodesic flow $\tilde{\xi }$ is the potential vector field of a Ricci soliton $(\tilde{G},\tilde{\xi },\lambda )$ on $T_1M,$ then $(T_1M,\tilde{G})$ is Einstein. Moreover, we show that the Reeb vector field of a contact metric manifold is an infinitesimal harmonic transformation if and only if it is Killing. Thus, we consider a natural contact metric structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ over $T_1 M$ and we show that the geodesic flow $\tilde{\xi }$ is an infinitesimal harmonic transformation if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi },\tilde{\xi })$ is Sasaki $\eta $ -Einstein. Consequently, we get that $(\tilde{G},\tilde{\xi }, \lambda )$ is a Ricci soliton if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ is Sasaki-Einstein with $\lambda = 2(n-1) >0.$ This last result gives new examples of Sasaki–Einstein structures.     

Holomorphic Hermitian vector bundles over the Riemann sphere

We give a complete classification of isomorphism classes of all SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹. We construct a canonical bijective correspondence between the isomorphism classes of SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹ and the isomorphism classes of pairs ({Hn}_n∈Z,T), where each Hn is a finite dimensional Hilbert space with Hn=0 for all but finitely many n, and T is a linear operator on the direct sum _⊕n∈ZHn such that T(Hn)⊂Hn+2 for all n.     

H-contact unit tangent sphere bundles of Riemannian manifolds

A contact metric manifold is said to be H-contact, if its characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold M equipped with the standard contact metric structure is H-contact if and only if M is 2-stein.     

A model for embedding finite coverings defined by principal bundles into bundles of manifolds

For any finite group G we construct a canonical model for embedding a principal G-bundle fibrewise into a given locally trivial fibration with a connected manifold M of dimension n⩾2 as fibre. The construction uses configuration spaces. We apply the construction to obtain a canonical model for the class of principal G-bundles which are polynomial when considered as covering maps. Finally, we give an algebraic characterization of the polynomial principal G-bundles in terms of homomorphisms into braid groups.     

The curvature of the Sasaki metric of tangent sphere bundles

We study the sectional curvaturesK of the Sasaki metric of tangent sphere bundles over spaces of constant curvatureK(T ₁(M ⁿ, K)). We give precise bounds on the variation of the Ricci curvature and a bound on the scalar curvature ofT ₁ (M ⁿ, K) that is uniform onK. In an appendix we calculate and give lower bounds for the lengths of closed geodesics onT ₁ S ⁿ. titles.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 132–145.     

A Borsuk-Ulam type theorem for sphere bundles over stunted projective spaces

A CW complex B is described as I-trivial if there does not exist a Z₂-map from Si−1 to S(α) for any vector bundle α over B and any integer i with i>dimα. For n>1, we determine all positive integers m for which the stunted projective space is I-trivial, where F=R,C or H.     

Holomorphic vector bundles on the Riemann sphere and the 21 st Hilbert problem

This report was presented at the first French-Russian colloquium on Geometry in Luminy in May 1992.