Related papers: An injectivity radius estimate in terms of metric …

We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive

The injectivity radius of a manifold is an important quantity, both from a theoretical point of view and in terms of numerical applications. It is the largest possible radius within which all geodesics are unique and length-minimizing. In…

Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzian manifolds under certain curvature and volume bounds. We establish several injectivity radius estimates at a point or…

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$,…

Arguably, geodesics are the most important geometric objects on a differentiable manifold. They describe candidates for shortest paths and are guaranteed to be unique shortest paths when the starting velocity stays within the so-called…

We prove for the first time a pointwise lower estimate of the normal injectivity radius of an embedded hypersurface in an arbitrary Riemannian manifold. Main applications include: (i) a pointwise lower estimate of the graphing radius of a…

Green's inequality shows that a compact Riemannian manifold with scalar curvature at least $n(n-1)$ has injectivity radius at most $\pi$, and that equality is achieved only for the radius 1 sphere. In this work we show how extra topological…

We exhibit conjugate points on the Stiefel manifold endowed with any member of the family of Riemannian metrics introduced by H\"uper et al. (2021). This family contains the well-known canonical and Euclidean metrics. An upper bound on the…

The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with…

We show that for any point $p$ in a closed Riemannian manifold $M$, there exists at least one point $q\in M$ such that $p$ is critical for the distance function from $q$. We also show that such a point $q$ cannot always be reached with…

We determine the cut locus of arbitrary non-simply connected, compact and irreducible Riemannian symmetric space explicitly, and compute injectivity radius and diameter for every type of them.

In this short note, we study the injectivity radius bound for three dimensional complete and non-compact Riemannian manifold with good leaf foliations and with bounded curvature up to first order. We obtain the injectivity bound by using…

Using Lie groupoids, we prove that the injectivity radius of a manifold with a Lie structure at infinity is positive.

This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \geq \frac{8}{\pi}R^2$. This inequality has long been conjectured for $R$ less…

In this note, we prove that for a complete noncompact three dimensional Riemannian manifold with bounded sectional curvature, if it has uniformly positive scalar curvature, then there is a uniform lower bound on the injectivity radius.

We prove that if a complete connected $n$-dimensional Riemannian manifold $M$ has radial sectional curvature at a base point $p\in M$ bounded from below by the radial curvature function of a two-sphere of revolution $\widetilde M$ belonging…

If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with…

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In…

In this paper we obtain a simple upper bound for the infimum of the Ricci curvatures of a complete Riemannian manifold with nonzero injectivity radius i(M) depending only on of the i(M). In case of rigidity the Riemannian manifold must be…

A lower-bound estimate of injectivity radius for complete Riemannian manifolds is discussed in a pure geometric viewpoint and is applied to study tangent cones at infinity of certain gradient Ricci solitons. We also study the asymptotic…