Related papers: Riemannian Rigidity of the Parallel Postulate in T…

Euclidean geometry consists of straightedge-and-compass constructions and reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. We consider three versions of…

We show that submanifolds of Euclidean space which are calibrated by a constant-coefficient differential form and have flat normal bundles are planes. In fact, in a Riemannian manifold equipped with a parallel calibration, a calibrated…

We prove flatness of complete Riemannian planes and cylinders without conjugate points under optimal conditions on the area growth.

In this paper we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces.In particular, using the moving frame method, we prove that $\mathbb{CP}^3$ is the only twistor space whose Bochner tensor is…

We prove that any complete, uniformly elliptic Weingarten surface in Euclidean $3$-space whose Gauss map image omits an open hemisphere is a cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman and Schoen for…

As a much later addition to the original Euclidean geometry, the parallel postulate distinguishes non-Euclidean geometries from Euclidean geometry. This paper will show that the parallel postulate is unnecessary because the 4th Euclidean…

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic…

We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to the complex plane in any complete orientable four-dimensional Riemannian manifold with uniformly positive isotropic curvature. We also…

We show that the classical equivalence of Euclid's parallel postulate and Playfair's axiom collapses in the absence of triangle congruence. In particular, we construct a non-SAS geometry that models the Playfair axiom but not the parallel…

We prove that the isoperimetric inequalities in the euclidean and hyperbolic plane hold for all euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems…

We prove that the only surfaces in $3$-dimensional Euclidean space $\R^3$ with constant Gaussian curvature $K$ and constructed by the sum of two space curves are cylindrical surfaces, in particular, $K=0$.

In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder as the only self-shrinkers satisfying a classic geometric assumption, namely the union of all tangent affine submanifolds of a complete…

In this note we prove that every two-dimensional entire Willmore graph in $R^3$ with square integrable mean curvature is a plane.

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle,…

We discuss the rigidity (or lack thereof) imposed by different notions of having an abundance of zero curvature planes on a complete Riemannian 3-manifold. We prove a rank rigidity theorem for complete 3-manifolds, showing that having…

The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian…

We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean…

A conjecture of Burns and Knieper asks whether a 2-plane with a metric without conjugate points, and with a geodesic foliation whose lines are at bounded Hausdorff distance, is necessarily flat. We prove this conjecture in two cases: under…

It follows from a general theorem of Bonk and Eremenko that closed plane curves which are contractible in the complement to the integral lattice satisfy a linear isoperimetric inequality. We give an alternative proof of this fact. Our…

Roughly, a conformal tiling of a Riemann surface is a tiling where each tile is a suitable conformal image of a Euclidean regular polygon. In 1997, Bowers and Stephenson constructed an edge-to-edge conformal tiling of the complex plane…