Related papers: On the geodesic diameter of surfaces with involuti…

This is an English translation of the following paper, published several years ago: Nikonorov Yu.G., Nikonorova Yu.V. Generalized Popoviciu's problem (Russian), Tr. Rubtsovsk. Ind. Inst., 7, 229-232 (2000), Zbl. 0958.51021. All inserted…

This is an English translation of the following paper, published several years ago: Nikonorov Yu.G. On a characterization of critical points of the scalar curvature functional (Russian), Tr. Rubtsovsk. Ind. Inst., 7, 211-217 (2000), Zbl.…

This is an expository article for Elsevier's Encyclopedia of Mathematical Physics on the subject in the title. Comments/corrections welcome.

This is a corrected version of my paper "Application of integral geometry to minimal surfaces" appeared in International J. Math. vol. 4 Nr. 1 (1993), 89-111. The correction concerns Proposition 3.5. We discuss this correction in Appendix…

The results of this paper have been greatly superseded by those in the paper "Contact geometry and isosystolic inequalities" (arXiv:1109.4253) by the same authors.

We give a brief survey of some of the geometry of mirror symmetry, written in 2004 for the "Encyclopaedia of Mathematical Physics". Probably a little bit out of date now in a few places, but hey.

Survey article on the geometry of spherical varieties. Invited survey for Transformation Groups.

This paper is an exposition of the major results of P. Eberlein's paper, "When is a geodesic flow of Anosov type? I," in the special case when the manifold $M$ is a surface. We follow Eberlein's coverage closely, adding details when…

This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…

An expository article on Turaev surfaces written for "A Concise Encyclopedia of Knot Theory," to appear.

We study the geodesics on an invariant surface of a three dimensional Riemannian manifold. The main results are: the characterization of geodesic orbits; a Clairaut's relation and its geometric interpretation in some remarkable three…

We describe the gauge-theoretic approach to transformations in integrable geometry through discussion of two classical examples: surfaces of constant negative Gauss curvature and isothermic surfaces. These are purely expository notes…

This PhD dissertation is concerned with integral geometric inverse problems. The geodesic ray transform is an operator that encodes the line integrals of a function along geodesics. The dissertation establishes many conditions when such…

As shown by Masur in 80s, for any translation surface there exists a periodic geodesic of bounded length, the directions of periodic geodesics are dense in the unit circle, and the number of cylinders of periodic geodesics of length at most…

This paper is a review of recent and classical results on integrable geodesic flows on Riemannian manifolds and topological obstructions to integrability. We also discuss some open problems.

We show that spheres in all dimensions $\geq3$ can be deformed to have diameter larger than the distance between any pair of antipodal points. This answers a question of Yurii Nikonorov.

The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…

This is a recent conference report on the Kobayashi Problem on hyperbolicity of generic projective hypersurfaces. As an appendix, a (non-updated) author's survey article of 1992 on the same subject, published in an edition with a limited…

In this revised version, we add some expository material and references and make some minor corrections.

In this article we explore the relationship between the systole and the diameter of closed hyperbolic orientable surfaces. We show that they satisfy a certain inequality, which can be used to deduce that their ratio has a (genus dependent)…