Results 111 to 120 of about 1,071 (142)

Bounded Ricci curvature and positive scalar curvature under Ricci flow

Pacific Journal of Mathematics, 2023
The authors consider a compact smooth Riemannian manifold \((M,g)\) without boundary, and its Ricci flow, which is a smooth family of metrics \( (g(t))_{t\geq 0}\) satisfying \(\partial _{t}g(t)=-2\mathrm{Ric}(g(t))\), where \(\mathrm{Ric}(g(t))\) denotes the Ricci curvature tensor of \(g(t)\), with the initial condition \(g(0)=g\).
Kröncke, Klaus, Marxen, Tobias, Vertman, Boris   +2 more
openaire   +1 more source

Weighted Ricci Curvature

2021
In Part I, we saw that the natural notions of Finsler curvatures (the flag and Ricci curvatures) can be introduced through the behavior of geodesics, and then several comparison theorems follow smoothly by similar arguments to the Riemannian case, or through the characterizations of these curvatures from the Riemannian geometric point of view.
openaire   +1 more source

Riemann Curvature and Ricci Curvature

2012
Curvatures are the central concept in geometry. The notion of curvature introduced by B. Riemann faithfully reveals the local geometric properties of a Riemann metric. This curvature is called the Riemann curvature in Riemannian geometry. The Riemann curvature can be extended to Finsler metrics as well as the sectional curvature.
Xinyue Cheng, Zhongmin Shen
openaire   +1 more source

The Pressure of Ricci Curvature

Geometriae Dedicata, 2003
If \(f\: SM \to \mathbb R\) is a continuous function on the sphere bundle of a closed Riemannian manifold \((M^n,g)\), the \textit{topological pressure} \(P(f)\) is defined by \(P(f) = \sup_{\mu \in M(\Phi)} \left(h_\mu + \int_{SM} f\, d\mu\right)\), where \(M(\Phi)\) is the set of all \(\Phi\)-invariant Borel probability measures (\(\Phi\) being the ...
Paternain, Gabriel P., Petean, Jimmy
openaire   +2 more sources

Ricci Curvature and Fundamental Group*

Chinese Annals of Mathematics, Series B, 2006
Let \(M\) be a compact Riemannian manifold with negative Ricci curvature. The author shows that if the universal cover of \(M\) has a pole and if any geodesic sphere centered at the pole is convex or concave, then the growth function of the fundamental group is at least exponential.
openaire   +1 more source

Metrics of Negative Ricci Curvature

The Annals of Mathematics, 1994
Using some deformation techniques the author is able to construct Riemannian metrics \(g\) of negative Ricci curvature \(r(g)\) and to prove in this way the following remarkable results: (i) For any \(n \geq 3\) there exist constants \(a(n) > b(n) > 0\) such that any manifold \(M\) with \(\dim M \geq 3\) admits a complete Riemannian metric \(g\) for ...
openaire   +1 more source

Graphs with Positive Ricci Curvature

Graphs and Combinatorics
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Qiqi Huang, Weihua He, Chaoqin Zhang
openaire   +1 more source

Ricci Curvature and Volume Convergence

The Annals of Mathematics, 1997
The author gives a new integral estimate of distances and angles on manifolds with a given lower Ricci curvature bound. He obtains this estimate via a Hessian estimate and states it in three different forms. Using this, he proves (among other things) the following conjectures: 1.
openaire   +1 more source

Ricci Curvature Comparison

1998
In this chapter we shall introduce some of the fundamental theorems for manifolds with lower Ricci curvature bounds. Two important techniques will be developed: relative volume comparison and weak upper bounds for the Laplacian of distance functions.
openaire   +1 more source