arXiv subjects
Publications and source records attributed to Anton Petrunin.
This is a collection of open questions in geometry that I think of as \textit{puzzles}: they stick to my brain --- I see grips, but no spare hands. Puzzle-charm is the only criterion for inclusion here; importance is ignored.
We solve John Milnor's problem: among all convex regions of a given area on the sphere, the round disk requires the greatest distortion in cartographic projections onto the plane.
The lectures discuss piecewise distance-preserving maps from a 2-dimensional polyhedral space into the plane.
We study the effects on length spaces imposed by quadratic inequalities on the six distances between the points in every quadruple.
An introduction to Alexandrov spaces with curvature bounded below. Topics include various comparison conditions, the globalization theorem, tangent spaces and spaces of directions, gradient flows, the splitting theorem, dimension and volume, Gromov's selection theorem, the boundary and the doubling theorem, and quotient spaces. We also give a brief overview of the two-dimensional theory, the main precursor to modern Alexandrov geometry.
We prove an isoperimetric inequality in terms of the mean shadow area.
We present a proof of Nash's smooth embedding theorem with an emphasis on accessibility and rigor.
Our goal is to show the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. The topics include 1. Reshetnyak's gluing theorem, 2. Estimates on the number of collisions in billiards, 3. Reshetnyak's majorization theorem, 4. Hadamard--Cartan globalization theorem, 5. Polyhedral spaces, 6. Construction of exotic aspherical manifolds, 7. The geometry of two-convex sets in Euclidean space, 8. Barycenters and dimension theory.
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the techniques that are absolutely essential for further study, keeping it problem-centered, elementary, visual, and virtually rigorous.
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the techniques that are absolutely essential for further study, keeping it problem-centered, elementary, visual, and virtually rigorous.
We consider the following question: When does a Riemannian manifold admit an embedding with a uniformly thick tubular neighborhood in another Riemannian manifold of large dimension?
We show that trees and their products meet octahedron comparison.
The book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalist. List of topics: Euclidean geometry: The Axioms / Half-planes / Congruent triangles / Perpendicular lines / Similar triangles / Parallel lines / Triangle geometry. Inversive geometry: Inscribed angles / Inversion. Non-Euclidean geometry: Neutral plane / Hyperbolic plane / Geometry of h-plane. Additional topics: Affine geometry / Projective geometry / Spherical insights / Projective model / Complex coordinates / Geometric constructions / Area.
Suppose M is a closed submanifold in a Euclidean ball of sufficiently large dimension. We give an optimal bound on the normal curvatures, guaranteeing that M is a sphere. The border cases consist of Veronese embeddings of the four projective planes.
We give a simplified proof of the generalized Kirszbraun theorem for Alexandrov spaces, which is due to Lang and Schroeder. We also discuss related questions, both solved and open.
Problems for the graduate students who want to improve problem-solving skills in geometry. Every problem has a short elegant solution -- this gives a hint which was not available when the problem was discovered.
We prove weak convergence of curvature tensors of Riemannian manifolds for converging noncollapsing sequences with a lower bound on sectional curvature.
A surface which does not admit a length nonincreasing deformation is called metric minimizing. We show that metric minimizing surfaces in CAT(0) spaces are locally CAT(0) with respect to their intrinsic metric.