Volume 16, Issue 2

Let C be a generic smooth curve of genus g ≥ 4.We study the normal functions and the infinitesimal invariants associated to Ceresa cycles W k − W − k , k = 2, . . . , g − 2. We show how they can be obtained from the normal function associated to the basic cycle C − C− and, for k = 2, we also explicitly determine the zero locus of the infinitesimal invariant. If C is general hyperelliptic of genus g = 3, we define the K-theoretic counterpart of W 2 − W − 2 , generalizing a construction of A. Collino, and we prove that it is indecomposable.

Let K be a global field or , let F be a nonzero quadratic form on K N with N ≥ 2, and let V be a subspace of K N .We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of (V, F) such that each such family spans V as a K-vector space. This result generalizes and extends a well known theorem of Vaaler [16] and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels’ theorem on small zeros of quadratic forms. All bounds on the height are explicit.

Let (M, g, ∇) be a pseudo-Riemannian manifold with a torsion free linear connection and let J g be the generalized complex structure on M defined by g; see [13], [14]. We prove that in the case J g is ∇- integrable the ±i-eigenbundles of J g , E 1,0 Jg and E 0,1 Jg are complex Lie algebroids.Moreover E 0,1 Jg and (E 1,0 Jg )* are canonically isomorphic, thus we define the concept of generalized ∂̅-operator of (M, g, ∇) and we describe a class of generalized holomorphic sections of T(M) ⊕ T*(M). Also we relate the Lie bialgebroid property of (E 1,0 Jg , (E 1,0 Jg )*) to conditions on the metric g in the case of affine Hessian manifolds.

We study the geodesic orbit property for nilpotent Lie groups N endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When N acts on itself by left-translations we show that it is a geodesic orbit space if and only if the metric is bi-invariant. Assuming N is 2-step nilpotent and with non-degenerate center we give algebraic conditions on the Lie algebra n of N which imply that every geodesic is the orbit of a one-parameter subgroup of N.Auto(N). In addition we present an example of an almost g.o. space such that for null homogeneous geodesics, the natural parameter of the orbit is not always the affine parameter of the geodesic.

We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.

We study complete noncompact 1-minimal stable hypersurfaces M in S n+1 . We obtain that there is no stable complete noncompact 1-minimal hypersurface in S 4 with nonzero Gauss-Kronecker curvature and _nite total curvature. We show that there is no stable complete noncompact 1-minimal hypersurface in S 4 with polynomial volume growth, nonzero Gauss-Kronecker curvature and sup M H 2 < 5/2. At last, we discuss stable complete noncompact hypersurfaces M with constant scalar curvature R > 1 in S n+1 and show another nonexistence result. These results are generalised versions of results of Alencar, Santos, Zhou and Silva Neto on stable hypersurfaces in R n+1 with constant scalar curvature.

The tangent space of a Lie loop, a non-associative Lie group, carries the structure of a Sabinin algebra, an algebraic concept generalizing Lie algebras. Alternatively a Sabinin algebra can be interpreted as the universal local covariant of a flat affine connection. Combining this interpretation of Sabinin algebras with a version of the parallel transport equation we prove that every Sabinin algebra structure determines and is determined by the one-sided linear terms in the Baker-Campbell-Hausdorff formula for the multiplication in the underlying loop. Based on this resultwe prove that every abstract Sabinin algebra over ℝ is the tangent algebra of a global Lie loop.

We show that certain divisors of Brill-Noether and Gieseker-Petri type span extremal rays of the effective cone in the moduli space of stable genus one curves with n ordered marked points. In particular, they are different from the infinitely many extremal rays found in [3].

Chen and Lue (2007) initiated the study of spherical submanifolds with finite type spherical Gauss map. In this paper, we firstly prove that a submanifold M n of the unit sphere S m−1 has non-mass-symmetric 1-type spherical Gauss map if and only if Mn is an open part of a small n-sphere of a totally geodesic (n + 1)- sphere S n+1 . S m−1 . Thenwe show that a non-totally umbilical hypersurfaceM of S n+1 with nonzero constant mean curvature in S n+1 has mass-symmetric 2-type spherical Gauss map if and only if the scalar curvature curvature of M is constant. Finally, we classify constant mean curvature surfaces in S 3 with mass-symmetric 2-type spherical Gauss map.

AnErdős-Ko-Rado set of generators of a hyperbolic quadric is a set of generatorswhich are pairwise not disjoint. In this article we classify the second largest maximal Erdos-Ko-Rado set of generators of the hyperbolic quadrics Q + (4n + 1, q), q ≥ 3.