Topic
Splitting theorem
About: Splitting theorem is a research topic. Over the lifetime, 389 publications have been published within this topic receiving 7298 citations.
Papers published on a yearly basis
Papers
1 Jan 1994
TL;DR: Ossanna and Bonaccorsi as mentioned in this paper gave a lecture on comparison theory in Riemannian geometry, where they considered the case when the Riccati ODE is not constant.
Abstract: The subject of these lecture notes is comparison theory in Riemannian geometry: What can be said about a complete Riemannian manifold when (mainly lower) bounds for the sectional or Ricci curvature are given? Starting from the comparison theory for the Riccati ODE which describes the evolution of the principal curvatures of equidistant hypersurfaces, we discuss the global estimates for volume and length given by Bishop-Gromov and Toponogov. An application is Gromov’s estimate of the number of generators of the fundamental group and the Betti numbers when lower curvature bounds are given. Using convexity arguments, we prove the ”soul theorem” of Cheeger and Gromoll and the sphere theorem of Berger and Klingenberg for nonnegative curvature. If lower Ricci curvature bounds are given we exploit subharmonicity instead of convexity and show the rigidity theorems of Myers-Cheng and the splitting theorem of Cheeger and Gromoll. The Bishop-Gromov inequality shows polynomial growth of finitely generated subgroups of the fundamental group of a space with nonnegative Ricci curvature (Milnor). We also discuss briefly Bochner’s method. The leading principle of the whole exposition is the use of convexity methods. Five ideas make these methods work: The comparison theory for the Riccati ODE, which probably goes back to L.Green [15] and which was used more systematically by Gromov [20], the triangle inequality for the Riemannian distance, the method of support function by Greene and Wu [16],[17],[34], the maximum principle of E.Hopf, generalized by E.Calabi [23], [4], and the idea of critical points of the distance function which was first used by Grove and Shiohama [21]. We have tried to present the ideas completely without being too technical. These notes are based on a course which I gave at the University of Trento in March 1994. It is a pleasure to thank Elisabetta Ossanna and Stefano Bonaccorsi who have worked out and typed part of these lectures. We also thank Evi Samiou and Robert Bock for many valuable corrections.
891 citations
776 citations
TL;DR: In this article, the splitting theorem for complete manifolds with Ricci curvature was extended to manifolds of nonnegative or positive Ricci curve curvature. But the results of these results are restricted to complete manifold with Ricmf > 0 and Euclidean volume growth.
Abstract: The basic rigidity theorems for manifolds of nonnegative or positive Ricci curvature are the "volume cone implies metric cone" theorem, the maximal diameter theorem, [Cg], and the splitting theorem, [CG]. Each asserts that if a certain geometric quantity (volume or diameter) is as large as possible relative to the pertinent lower bound on Ricci curvature, then the metric on the manifold in question is a warped product metric of a particular type. In this paper we provide quantitative generalizations of the above mentioned results. Among the applications are the splitting theorem for GromovHausdorff limit spaces X, where Mn -* X, Ricmn ? -i see [FY]. Other applications include the assertion that for complete manifolds, M', with Ricmf > 0 and Euclidean volume growth, all tangent cones at infinity are metric cones; compare [BKN], [CT], [P1]. Via resealing arguments, there are also strong consequences for the local structure of manifolds whose Ricci curvature satisfies a fixed lower bound and for their Gromov-Hausdorff limits. Some of these are announced in [CCol]; for a more detailed discussion see [CCo2], [CCo3], [CCo4]. Our work further develops and significantly extends techniques which were introduced in [Col], [Co2] and significantly extended in [Co3], in order to prove certain "stability" conjectures of Anderson-Cheeger, Gromov and Perelman. The results of [Col]-[Co3] were announced in [Co4]. We briefly review some of those results. Let dGH denote the Gromov-Hausdorff distance between metric spaces; see [GLP]. Let S' denote the unit sphere and recall that S' is the unique complete
753 citations
Book•
1 Sep 1996
TL;DR: In this paper, the Laplace transform and its complex inversion were studied in the context of positive semigroups and boundedness of resolvability of the resolvent.
Abstract: 1 Spectral bound and growth bound- 11 C0-semigroups and the abstract Cauchy problem- 12 The spectral bound and growth bound of a semigroup- 13 The Laplace transform and its complex inversion- 14 Positive semigroups- Notes- 2 Spectral mapping theorems- 21 The spectral mapping theorem for the point spectrum- 22 The spectral mapping theorems of Greiner and Gearhart- 23 Eventually uniformly continuous semigroups- 24 Groups of non-quasianalytic growth- 25 Latushkin - Montgomery-Smith theory- Notes- 3 Uniform exponential stability- 31 The theorem of Datko and Pazy- 32 The theorem of Rolewicz- 33 Characterization by convolutions- 34 Characterization by almost periodic functions- 35 Positive semigroups on Lp-spaces- 36 The essential spectrum- Notes Ill- 4 Boundedness of the resolvent- 41 The convexity theorem of Weis and Wrobel- 42 Stability and boundedness of the resolvent- 43 Individual stability in B-convex Banach spaces- 44 Individual stability in spaces with the analytic RNP- 45 Individual stability in arbitrary Banach spaces- 46 Scalarly integrable semigroups- Notes- 5 Countability of the unitary spectrum- 51 The stability theorem of Arendt, Batty, Lyubich, and V?- 52 The Katznelson-Tzafriri theorem- 53 The unbounded case- 54 Sets of spectral synthesis- 55 A quantitative stability theorem- 56 A Tauberian theorem for the Laplace transform- 57 The splitting theorem of Glicksberg and DeLeeuw- Notes- Append- Al Fractional powers- A2 Interpolation theory- A3 Banach lattices- A4 Banach function spaces- References- Symbols
389 citations
TL;DR: In this article, a metric quasi-Einstein metric is defined, where the m -Bakry-Emery Ricci tensor is a constant multiple of the metric tensor.
Abstract: We call a metric quasi-Einstein if the m -Bakry–Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, it contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some Kahler quasi-Einstein metrics.
275 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 22 |
| 2020 | 19 |
| 2019 | 22 |
| 2018 | 20 |
| 2017 | 18 |