One of the classical problems in differential geometry is the investigation of closed manifolds which admit Riemannian metrics with given lower bounds for the sectional or the Ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold. Despite many efforts during the past decades, this problem is still far from being...

As the first step in the direction of the Hopf conjecture on the non-existence of metrics with positive sectional curvature on S2 × S2 the authors of [GT] suggested the following (Weak Hopf) conjecture (on the rigidity of non-negatively curved metrics on S2 × R3): ”The boundary S2 × S2 of the S2 × B3 ⊂ S2 × R3 with an arbitrary complete metric of non-negative sectional curvature contains a poin...

Quantitative and objective methods to evaluate the morphology of the reconstructed breast may help plastic surgeons improve their surgical practice, and thus ultimately help breast cancer survivors derive the intended psychosocial benefits of reconstruction. Recently, we developed a quantitative and objective way to measure the curvature of the breast on standard clinical photographs. Here we c...

Based on the ideas of Bessa-Jorge-Montenegro [4] we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 are proper, (compact if N is compact). In addition, if N is Hadamard then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realized as subma...

Let (M, g) be a simply connected complete Kähler manifold with nonpositive sectional curvature. Assume that g has constant negative holomorphic sectional curvature outside a compact set. We prove that M is then biholomorphic to the unit ball in Cn, where dimCM = n. Résumé. Soit (M, g) une variété kählérienne complète et simplement connexe à courbure sectionnelle non-positive. Supposons que g ai...

We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collection of 2-orbifolds satisfying lower bounds on sectional curvature and volume, ...

In this paper, we prove that if every totally real bisectional curvature of an n(≥ 3)-dimensional complete Kähler submanifold of a complex projective space of constant holomorphic sectional curvature c is greater than c 4(n2−1)n(2n− 1), then it is totally geodesic. Mathematics Subject Classifications: 53C50, 53C55, 53C56.

In this paper we consider perturbed harmonic map equations for maps between closed Riemannian manifolds. In the case where the target manifold has negative sectional curvature we prove among other results that for a large class of semilinear and quasilinear perturbations, the perturbed harmonic map equations have solutions in any homotopy class of maps for which the Euler characteristic of the ...

We construct examples of 4-dimensional manifolds M supporting a locally CAT(0)metric, whose universal covers Q M satisfy Hruska’s isolated flats condition, and contain 2-dimensional flats F with the property that @1F Š S ,! S Š @1 Q M are nontrivial knots. As a consequence, we obtain that the group 1.M/ cannot be isomorphic to the fundamental group of any compact Riemannian manifold of nonposit...

Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We consider two conformally equivalent metrics for which the distances between curves are nontrivial. We show that in the case of the simpler of the two metrics, the only minimal geodesics are those corresponding to curve evolution in which the points o...