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A synthetic characterization of the hemisphere
Christopher B. Croke
We show that round hemispheres are the only compact 2 dimensional Riemannian manifolds (with or without boundary) such that almost every pair of complete geodesics intersect once a…
Boundary case of equality in optimal Loewner-type inequalities
Victor Bangert, Christopher Croke, Sergei V. Ivanov +1
We prove certain optimal systolic inequalities for a closed Riemannian manifold (X,g), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its firs…
Filling area conjecture and ovalless real hyperelliptic surfaces
Victor Bangert, Christopher Croke, Sergei V. Ivanov +1
We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus…
Universal volume bounds in Riemannian manifolds
Christopher B. Croke, Mikhail G. Katz
In this survey article we will consider universal lower bounds on the volume of a Riemannian manifold, given in terms of the volume of lower dimensional objects (primarily the leng…
Conjugacy rigidity for nonpositively curved graph manifolds
Christopher B. Croke
We show that the metric of nonpositively curved graph manifolds is determined by its geodesic flow. More precisely we show that if the geodesic flows of two nonpositively curved gr…
The volume and lengths on a three sphere
Christopher B. Croke
We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the vo…