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Non-positive Curvature
Groups acting on non-positively curved spaces satisfy many interesting properties. One widely used concept of non-positive curvature for metric...
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Curvature
The notion of curvature is central to differential geometry and its numerous applications. The aim of this chapter is to offer a first introduction...
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The Uniqueness of Knieper Measure on Non-compact Rank 1 Manifolds of Non-positive Curvature
We study the Knieper measures of the geodesic flows on non-compact rank 1 manifolds of non-positive curvature. We construct the Busemann density on...
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Characteristic numbers, Jiang subgroup and non-positive curvature
By refining a result of Farrell and in terms of the Jiang subgroups of self-maps of smooth manifolds, we obtain a sufficient condition for the...
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A Non-spin Method to the Positive Weighted Mass Theorem for Weighted Manifolds
In this paper, we investigate the weighted mass for weighted manifolds. By establishing a version of density theorem and generalizing Geroch...
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Desingularizing positive scalar curvature 4-manifolds
We show that the bordism group of closed 3-manifolds with positive scalar curvature (psc) metrics is trivial by explicit methods. Our constructions...
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Positive Intermediate Ricci Curvature with Maximal Symmetry Rank
Generalizing the foundational work of Grove and Searle, the second author proved upper bounds on the ranks of isometry groups of closed Riemannian...
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Preserving Positive Intermediate Curvature
Consider a compact manifold N (with or without boundary) of dimension n . Positive m -intermediate curvature interpolates between positive Ricci...
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The Morse Index for Manifolds with Constant Sectional Curvature
We compute the Morse index of a critical submanifold of the energy functional on the loop space of a manifold with constant sectional curvature. The...
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Curvature estimates for a class of Hessian quotient type curvature equations
In this paper, we are concerned with the hypersurface that can be locally represented as a graph and satisfies a class of Hessian quotient type...
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Metrics and Curvature
We present a general perspective on the relation between geometry and topology, linking them through an abstract approach to the concept of...
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Compact Almost Hermitian Manifolds with Quasi-negative Curvature and the Almost Hermitian Curvature Flow
We show that along the almost Hermitian curvature flow, the non-positivity of the first Chern–Ricci curvature can be preserved if the initial almost...
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Three-Manifolds with Bounded Curvature and Uniformly Positive Scalar Curvature
In this note, we prove that for a complete noncompact three-dimensional Riemannian manifold with bounded sectional curvature, if it has uniformly...
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Complete Self-similar Hypersurfaces to the Mean Curvature Flow with Nonnegative Constant Scalar Curvature
As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise...
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Stretched non-positive Weyl connections on solvable Lie groups
We determine the structure of solvable Lie groups endowed with invariant stretched non-positive Weyl connections and find classes of solvable Lie...
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Ollivier Curvature of Random Geometric Graphs Converges to Ricci Curvature of Their Riemannian Manifolds
Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for...
