Moebius geometry of submanifolds in ? n

In this paper we define a Moebius invariant metric and a Moebius invariant second fundamental form for submanifolds in ? ⁿ and show that in case of a hypersurface with n≥ 4 they determine the hypersurface up to Moebius transformations. Using these Moebius invariants we calculate the first variation of the moebius volume functional. We show that any minimal surface in ? ⁿ is also Moebius minimal and that the image in ? ⁿ of any minimal surface in ℝ ⁿ unter the inverse of a stereographic projection is also Moebius minimal. Finally we use the relations between Moebius invariants to classify all surfaces in ?³ with vanishing Moebius form. Received: 18 November 1997     

Affine differential geometry of surfaces in ℝ4

Employing the method of moving frames, i.e. Cartan's algorithm, we find a complete set of invariants for nondegenerate oriented surfacesM ² in ⁴ relative to the action of the general affine group on ⁴. The invariants found include a normal bundle, a quadratic form onM ² with values in the normal bundle, a symmetric connection onM ² and a connection on the normal bundle. Integrability conditions for these invariants are also determined. Geometric interpretations are given for the successive reductions to the bundle of affine frames overM ², obtained by using the method of moving frames, that lead to the aforementioned invariants. As applications of these results we study a class of surfaces known as harmonic surfaces, finding for them a complete set of invariants and their integrability conditions. Further applications involve the study of homogeneous surfaces; these are surfaces which are fixed by a group of affine transformations that act transitively on the surface. All homogeneous harmonic surfaces are determined.     

Surfaces with Vanishing Moebius Form in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

An important Moebius invariant in the theory of Moebius surfaces in S^n is the so-called Moebius form. In this paper,we give a complete classification of surfaces in S^n with vanishing Moebius form under the Moebius transformation group.     

Transgression on hyperkähler manifolds and generalized higher torsion forms

We propose a generalization of the Hodge dd_c-lemma to the case of hyperk?hler manifolds. As an application we derive a global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperk?hler manifolds. In Section 3 we consider the fourth order transgression for the infinite-dimensional bundle arising from local families of hyperk?hler manifolds. We propose a local construction of the fourth order transgression of the Chern character form. We derive an explicit expression for the arising hypertorsion differential form. Its zero-degree part may be expressed in terms of the Laplace operators defined on the fibres of the local family.     

The Gauss-Bonnet theorem for vector bundles

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles. Dedicated to the memory of Philip Bell Research partially supported by NSF grant DMS-9703852.     

Sn中Moebius形式为零的曲面

本文证明了S^n中Moebius形式为零且法丛平坦的曲面的余维数约化定理，并且给出了这类曲面的分类．在此基础上，进一步给出了S^n中Moebius形式为零的曲面的分类．     

Number of Singularities of Stable Maps

Sharp estimates for the Ricci curvature of a submanifold M ⁿ of an arbitrary Riemannian manifold N ^n+p are established. It is shown that the equality in the lower estimate of the Ricci curvature of M ⁿ in a space form N ^n+p (c) is achieved only when M ⁿ is quasiumbilical with a flat normal bundle. In the case when the codimension p satisfies 1 ≤ p ≤ n − 3, the only submanifolds in N ^n+p (c) on which the Ricci curvature is minimal are the conformally flat ones with a flat normal bundle.      

The geometry of the bundle of connections

A generalized symplectic structure on the bundle of connections of an arbitrary principal G-bundle is defined by means of a -valued differential 2-form on C(P), which is related to the generalized contact structure on . The Hamiltonian properties of are also analyzed. Received August 31, 1999; in final form January 4, 2000 / Published online February 5, 2001     

Harmonic morphisms as unit normal bundles¶of minimal surfaces

Let be an isometric immersion between Riemannian manifolds and be the unit normal bundle of f. We discuss two natural Riemannian metrics on the total space and necessary and sufficient conditions on f for the projection map to be a harmonic morphism. We show that the projection map of the unit normal bundle of a minimal surface in a Riemannian manifold is a harmonic morphism with totally geodesic fibres. Received: 6 February 1999     

On variational approach to differential invariants of rank two distributions

We construct differential invariants for generic rank 2 vector distributions on n-dimensional manifolds, where n?5. Our method for the construction of invariants is completely different from the Cartan reduction-prolongation procedure. It is based on the dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the theory of unparameterized curves in the Lagrange Grassmannian, developed in [A. Agrachev, I. Zelenko, Geometry of Jacobi curves I, J. Dynam. Control Syst. 8 (1) (2002) 93-140; II, 8 (2) (2002) 167-215]. In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n?5. In the next paper [I. Zelenko, Fundamental form and Cartan's tensor of (2,5)-distributions coincide, J. Dynam. Control. Syst., in press, SISSA preprint, Ref. 13/2004/M, February 2004, math.DG/0402195] we show that in the case n=5 our fundamental form coincides with the Cartan covariant biquadratic binary form, constructed in 1910 in [E. Cartan, Les systemes de Pfaff a cinque variables et les equations aux derivees partielles du second ordre, Ann. Sci. Ecole Normale 27 (3) (1910) 109-192; reprinted in: Oeuvres completes, Partie II, vol. 2, Gautier-Villars, Paris, 1953, pp. 927-1010]. Therefore first our approach gives a new geometric explanation for the existence of the Cartan form in terms of an invariant degree four differential on an unparameterized curve in Lagrange Grassmannians. Secondly, our fundamental form provides a natural generalization of the Cartan form to the cases n>5. Somewhat surprisingly, this generalization yields a rational function on the fibers of the appropriate vector bundle, as opposed to the polynomial function occurring when n=5. For n=5 we give an explicit method for computing our invariants and demonstrate the method on several examples.     

Minimal and Harmonic Unit Vector Fields in and Its Dual Space

 The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results hold for the dual space . (Received 27 August 1999; in revised form 18 November 1999)     

HORIZONTAL DISTRIBUTIONS ON SPACES OF FIBERS

Abstract

Geometric methods for systems of partial differential equations and multiple integral problems in the calculus of variations lead naturally to differentiable manifolds that resemble fiber bundles but do not possess a structure group; in terms of local coordinates, π:B→M_n|(xⁱ, q^α)→(xⁱ), dim(B) = N + n, dim(M_n) = n. The standard notions of horizontal distributions, horizontal and vertical subspaces of T(B), T(B) = V(B) ⊕ H(B), horizontal lifts of curves in M_n, and horizontal and vertical dual subspaces with Λ¹(B) = V*(B) ⊕ H*(B) are shown to be well defined in B. The absence of a structure group is compensated for by an analysis based on the homogeneous ideals V and H that are generated by the canonical bases of V*(B) and H*(B), respectively. The differential system constructed from the generators of the horizontal ideal is shown to lead to a unique system of connection 1-forms and torsion 2-forms under the requirements that they have vacuous intersections with the horizontal ideal. The horizontal ideal is shown to be completely integrable if and only if the torsion 2-forms vanish throughout B, in which case the curvature 2-forms are congruent to zero mod H, and the curvature 2-forms are shown to have a vacuous intersection with H if and only if the horizontal distribution is affine. The paper concludes with a study of the mapping properties of the connection, torsion and curvature. These are significantly more general than those of a fiber bundle since the absence of a structure group allows mappings of the form 'xⁱ = φⁱ(x,q), 'q^α = φ^α (x,q).     

Conformally Kähler base metrics for Einstein warped products

A Riemannian metric g with Ricci curvature r is called nontrivial quasi-Einstein, in a sense given by Case, Shu and Wei, if it satisfies (−a/f)∇df+r=λg, for a smooth nonconstant function f and constants λ and a>0. If a is a positive integer, it was noted by Besse that such a metric appears as the base metric for certain warped Einstein metrics. This equation also appears in the study of smooth metric measure spaces. We provide a local classification and an explicit construction of Kähler metrics conformal to nontrivial quasi-Einstein metrics, subject to the following conditions: local Kähler irreducibility, the conformal factor giving rise to a Killing potential, and the quasi-Einstein function f being a function of the Killing potential. Additionally, the classification holds in real dimension at least six. The metric, along with the Killing potential, form an SKR pair, a notion defined by Derdzinski and Maschler. It implies that the manifold is biholomorphic to an open set in the total space of a CP¹ bundle whose base manifold admits a Kähler-Einstein metric. If the manifold is additionally compact, it is a total space of such a bundle or complex projective space. Additionally, a result of Case, Shu and Wei on the Kähler reducibility of nontrivial Kähler quasi-Einstein metrics is reproduced in dimension at least six in a more explicit form.     

Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

In this paper we study a Riemannian metric on the tangent bundle T(M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger–Gromoll metric and a compatible almost complex structure which confers a structure of locally conformal almost K?hlerian manifold to T(M) together with the metric. This is the natural generalization of the well known almost K?hlerian structure on T(M). We found conditions under which T(M) is almost K?hlerian, locally conformal K?hlerian or K?hlerian or when T(M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from T(M). Moreover, we found that this map preserves also the natural contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively. This work was also partially supported by Grant CEEX 5883/2006–2008, ANCS, Romania.     

A characterization of Moebius isoparametric hypersurfaces of the sphere

We consider a proper, umbilic-free immersion of an n-dimensional manifold M in the sphere S ⁿ⁺¹. We show that M is a Moebius isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Moebius curvature.     

Some remarks on Legendrian rectifiable currents

A rectifiable current of dimension n−1 in the sphere bundle Sℝⁿ≃ℝⁿ×S ^{n −1} for euclidean space is Legendrian if it annihilates the contact 1-form α (i.e. T(α∧φ)=0 for all forms φ of degree n−2). Such a current may be naturally associated to any convex set or to any singular real analytic variety, and induces the curvature measures of such a set. We prove that the projection to ℝⁿ of a carrier of a general such T is C ²-rectifiable in the sense of Anzellotti–Serapioni. We deduce that the boundary of a set with positive reach, as well as its singular skeleta, are C ²-rectifiable. In case ∂T= 0 we prove also that the curvature measures associated to T satisfy the analogues of the classical variational formulas for curvature integrals. It follows that such formulas are valid for the curvature measures of subsets of space forms. Received: 3 December 1997/ Revised version: 25 May 1998     

On the volume of unit vector fields on spaces of constant sectional curvature

A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields.     

S~n中具Moebius平坦法丛的子流形

本文研究S~n中不含脐点、Moebius形式为零且具Moebius平坦法丛的子流形的Moebius特性。分别利用子流形的Moebius Ricci曲率与Blaschke张量、Moebius标准数量曲率以及Blaschke张量与Moebius标准数量曲率之间所满足的某种内蕴关系刻画了S~n中子流形的Moebius特性,得到了S~n中法丛平坦子流形的两个分类定理。     

Hypersurfaces with isotropic para-Blaschke tensor

Let Mnbe an n-dimensional submanifold without umbilical points in the(n + 1)-dimensional unit sphere Sn+1.Four basic invariants of Mnunder the Moebius transformation group of Sn+1are a 1-form Φ called moebius form,a symmetric(0,2) tensor A called Blaschke tensor,a symmetric(0,2) tensor B called Moebius second fundamental form and a positive definite(0,2) tensor g called Moebius metric.A symmetric(0,2) tensor D = A + μB called para-Blaschke tensor,where μ is constant,is also an Moebius invariant.We call the para-Blaschke tensor is isotropic if there exists a function λ such that D = λg.One of the basic questions in Moebius geometry is to classify the hypersurfaces with isotropic para-Blaschke tensor.When λ is not constant,all hypersurfaces with isotropic para-Blaschke tensor are explicitly expressed in this paper.