Codimension one isometric immersions between Lorentz spaces

Codimension one immersions and the Kervaire invariant one problem

On Local Isometric Immersions into Complex and Quaternionic Projective Spaces

We will prove that if an open subset of CPn is isometrically immersed into CPm, withm < (4/3)n−2/3, then the image is totally geodesic. We will also prove that if an open subset of HPn isometrically immersed into HPm, with m < (4/3)n− 5/6, then the image is totally geodesic.

On isometric Lagrangian immersions

This article uses Cartan-Kähler theory to show that a small neighborhood of a point in any surface with a Riemannian metric possesses an isometric Lagrangian immersion into the complex plane (or by the same argument, into any Kähler surface). In fact, such immersions depend on two functions of a single variable. On the other hand, explicit examples are given of Riemannian three-manifolds which ...

Tiling spaces, codimension one attractors and shape

We establish a close relationship between, on the one hand, expanding, codimension one attractors of diffeomorphisms on closed manifolds (examples of so-called strange attractors), and, on the other, spaces which arise in the study of aperiodic tilings. We show that every such orientable attractor is homeomorphic to a tiling space of either a substitution or a projection tiling, depending on it...

Continuous Leafwise Harmonic Functions on Codimension One Transversely Isometric Foliations

Let F be a codimension one foliation on a closed manifold M which admits a transverse dimension one Riemannian foliation. Then any continuous leafwise harmonic functions are shown to be constant on leaves.