Related papers: Nonnegatively and Positively curved manifolds
We provide techniques for studying the nonnegatively curved left-invariant metrics on a compact Lie group. For "straight" paths of left-invariant metrics starting at bi-invariant metrics and ending at nonnegatively curved metrics, we deduce…
We describe a construction of Riemannian metrics of nonnegative sectional curvature on a closed smooth nonorientable 4-manifold with fundamental group of order two that realizes a homotopy class that was not previously known to contain…
We give a systematic construction of uniruled surfaces in positive characteristic. Using this construction, we find surfaces of general type with non-trivial vector fields, surfaces with arbitrarily non-reduced Picard schemes as well as…
The purpose of the paper is to review a variety of recent developments in the theory of positive solutions of general linear elliptic and parabolic equations of second-order on noncompact Riemannian manifolds, and to point out a number of…
The symmetry-rank of a riemannian manifold is by definition the rank of its isometry group. We determine precisely which smooth closed manifolds admit a positively curved metric with maximal symmetry-rank.
This is a survey article on recent results on vector bundles on symmetric product of non-singular projective curves.
Given a perverse sheaf on the moduli stack of principally polarized abelian varieties or the moduli stack of smooth curves with n marked points over a field of characteristic zero, we prove that the (orbifold) Euler characteristic is…
The aim of this paper is to give some characterizations for N-Legendre and N-slant curves in the unit tangent bundles of surfaces endowed with natural diagonal lifted structures.
These are lecture notes supporting a minicourse taught at the Summer School in Total Positivity and Quantum Field Theory at CMSA Harvard in June 2025. We give an introduction to positive geometries and their canonical forms. We present the…
In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of…
This is a survey of the inverse spectral problem on (mainly compact) Riemannian manifolds, with or without boundary. The emphasis is on wave invariants: on how wave invariants have been calculated and how they have been applied to concrete…
In this paper, we study the geometry of the manifolds of geodesics of a Zoll surface of positive Gauss curvature, show how these metrics induce Finsler metrics of constant flag curvature and give some explicit constructions.
We classify closed, simply-connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism.
We prove a result on equivariant deformations of flat bundles, and as a corollary, we obtain two ``splitting in a finite cover'' theorems for isometric group actions on Riemannian manifolds with infinite fundamental groups, where the…
We collect a few guesses on possible implications of a lower bound on the scalar curvature of a Riemannian manifold on the size and shape of this manifold.
In this note we prove an effective characterization of when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations, weakening the…
In this paper we announce the following result: ``Every manifold of dimension $\ge3$ admits a complete negatively Ricci curved metric.'' Furthermore we describe some sharper results and sketch proofs.
In this paper, an n-dimensional complete open manifold with nonnegative Ricci curvature and collapsing volume has been investigated. If its radial sectional curvature bounded from below, it shows that such a manifold is of finite…
In the lecture notes, the author will survey the development of conformal geometry on four dimensional manifolds. The topic she chooses is one on which she has been involved in the past twenty or more years: the study of the integral…
We give an exposition of graded and microformal geometry, and the language of $Q$-manifolds. $Q$-manifolds are supermanifolds endowed with an odd vector field of square zero. They can be seen as a non-linear analogue of Lie algebras (in…