Nonpositive Extrinsic Curvature Research Articles

Complete submanifolds with relative nullity in space forms

We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive extrinsic curvature and submanifolds with integrable relative conullity. We show that no complete submanifold of hyperbolic space with sufficiently high index of relative nullity has extrinsic geometry bounded away from zero. As an application of these results, we derive an interesting corollary for complete submanifolds of hyperbolic space with nonpositive extrinsic curvature and discourse on their relation to Milnor’s conjecture about complete surfaces with second fundamental form bounded away from zero. Finally, we also prove that every complete Euclidean submanifold with integrable relative conullity is a cylinder over the relative conullity.

Submanifolds with nonpositive extrinsic curvature

We prove that complete submanifolds, on which the weak Omori-Yau maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the larger such curvatures are. This work unifies and generalizes several previous results on submanifolds with nonpositive extrinsic curvature.

Extremal surface barriers

We present a generic condition for Lorentzian manifolds to have a barrier that limits the reach of boundary-anchored extremal surfaces of arbitrary dimension. We show that any surface with nonpositive extrinsic curvature is a barrier, in the sense that extremal surfaces cannot be continuously deformed past it. Furthermore, the outermost barrier surface has nonnegative extrinsic curvature. Under certain conditions, we show that the existence of trapped surfaces implies a barrier, and conversely. In the context of AdS/CFT, these barriers imply that it is impossible to reconstruct the entire bulk using extremal surfaces. We comment on the implications for the firewall controversy.

Warped product structure of submanifolds with nonpositive extrinsic curvature in space forms

By means of a simple warped product construction we obtain examples of submanifolds with nonpositive extrinsic curvature and minimal index of relative nullity in any space form. We then use this to extend to arbitrary space forms four known splitting results for Euclidean submanifolds with nonpositive sectional curvature.

On submanifolds with nonpositive extrinsic curvature

On submanifolds with nonpositive extrinsic curvature

Riemannian Hilbert submanifolds of nonpositive extrinsic curvature

We consider strongly parabolic, Hilbert submanifolds in Riemannian Hilbert manifolds. We prove that their properties are analogous to the known properties in the finite-dimensional case. The main geometric result consists of Theorem 3: a complete, Riemannian, Hilbert submanifold of nonpositive extrinsic curvature and finite codimension in a Hilbert sphere is a great sphere.

SURFACES OF NONPOSITIVE EXTRINSIC CURVATURE IN SPACES OF CONSTANT CURVATURE

This paper investigates surfaces of nonpositive extrinsic curvature in a pseudo- Riemannian space of curvature 1, Kahlerian submanifolds of complex projective space , and saddle surfaces in spherical space . It is determined under what conditions a surface is a totally geodesic submanifold.Bibliography: 14 titles.

COMPLETE $ l$-DIMENSIONAL SURFACES OF NONPOSITIVE EXTRINSIC CURVATURE IN A RIEMANNIAN SPACE

This article studies complete -dimensional surfaces of nonpositive extrinsic 2-dimensional sectional curvature and nonpositive -dimensional curvature (for even) in Euclidean space , in the sphere , in the complex projective space , and in a Riemannian space . If the embedding codimension is sufficiently small, then a compact surface in or is a totally geodesic great sphere or complex projective space, respectively. If is a compact surface of negative extrinsic 2-dimensional curvature in a Riemannian space , then there are restrictions on the topological type of the surface. It is shown that a compact Riemannian manifold of nonpositive -dimensional curvature cannot be isometrically immersed as a surface of small codimension. The order of growth of the volume of complete noncompact surfaces of nonpositive -dimensional curvature in Euclidean space is estimated; it is determined when such surfaces are cylinders. A question about surfaces in which are homeomorphic to a sphere and which have nonpositive extrinsic curvature is looked at.Bibliography: 25 titles.