C2 densely the 2-sphere has an elliptic closed geodesic

In this short note, we give simple proof of the Ricci flow's local existence and uniqueness on closed Einstein manifolds. We suggest a new setting for studying the space of Riemannian metrics on a compact manifold.

We present a survey on generic singularities of geodesic flows in smooth signature changing metrics (often called pseudo-Riemannian) in dimension 2. Generically, a pseudo-Riemannian metric on a 2-manifold $S$ changes its signature (degenerates) along a curve $S_0$, which locally separates $S$ into a Riemannian ($R$) and a Lorentzian ($L$) domain. The geodesic flow does not have singularities over $R$ and $L$, and for any point $q \in R \cup L$ and every tangential direction $...

The question of whether or not the set of Zoll metrics on the 2-sphere is connected is still open. Here we show that a naive application of the Ricci flow is not sufficient to answer this problem.

We prove that the isometric embedding of any metric of differentiability class C1 in E3 exists. We use simplified notation for the given metric, namely geodesic parameters, and level parameters for the embedded surface in E3. Central to our discussion will be solutions of initial value problems for two first order non-linear partial differential equations. We also make use of the classical theory of linear algebraic systems. We will prove local isometric embedding. An example...

For metric spaces with curvature less than or equal to x, x<0, it is shown that a recurrent geodesic can be approximated by closed geodesics. A counter example is provided for the converse.

Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics. In the context of G_2 geometry, there are several geometric flows which arise. Each flow provides a potential means to study the geometry and topology associated with a given class of G_2 structures. We will introduce these flows, and describe some of t...

We construct a concrete example of constant Gauss curvature $K = 1$ on the 2-sphere having all geodesics closed and of same length.

The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.

The elastic flow, which is the $L^2$-gradient flow of the elastic energy, has several applications in geometry and elasticity theory. We present stable discretizations for the elastic flow in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk, the elliptic plane as well as any conformal parameterization of a two-dimensional manifold in ${\mathbb R}^d$, ...

A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.