Riemannian Geometry

Petersen, Peter

doi:10.1007/978-1-4757-6434-5

Cited by 680 publications

(958 citation statements)

References 0 publications

Supporting

Mentioning

940

Contrasting

Unclassified

Order By: Relevance

“…Some general references on this subject are [7,13,15]. Readers who are not familiar with Riemannian geometry are referred to [8] for a pleasant introduction to some of its main concepts and their use in optimization.…”

Section: Notation and Basic Notions Of Riemannian Geometrymentioning

confidence: 99%

See 1 more Smart Citation

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

2006

View full text Add to dashboard Cite

Abstract:We consider the problem of finding a singularity of a vector field X on a complete Riemannian manifold. In this regard we prove a unified result for local convergence of Newton's method. Inspired by previous work of Zabrejko and Nguen on Kantorovich's majorant method, our approach relies on the introduction of an abstract one-dimensional Newton's method obtained using an adequate Lipschitz-type radial function of the covariant derivative of X. The main theorem gives in particular a synthetic view of several famous results, namely the Kantorovich, Smale and Nesterov-Nemirovskii theorems. Concerning real-analytic vector fields an application of the central result leads to improvements of some recent developments in this area.

show abstract

Section: Notation and Basic Notions Of Riemannian Geometrymentioning

confidence: 99%

“…Likewise the tensors can be covariantly derivated (e.g. [15]), if T is (1, q)-tensor its derivative is a (1, q + 1)-tensor denoted T which is given by the following formula…”

Section: Norms and Distances The Riemannian Distancementioning

confidence: 99%

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

2006

View full text Add to dashboard Cite

show abstract

“…Hence K g = o(r −2 ) and diameter is controlled from above and volume growth is controlled from below (from the Sobolev inequality) on each annulus M 2kr − M k −1 r . We can then infer from Anderson-Cheeger harmonic radius' theory [2,3,16,23] that the rescaled annuli (M kri − M k −1 ri , r −2 i g) are covered by a finite (and uniformly bounded) number of balls of uniformly bounded size where the metric coefficients are C 1,α -close to the euclidean metric. Hence one gets volume growth control from above.…”

Section: Vol 77 (2002)mentioning

confidence: 99%

The Huber theorem for non-compact conformally flat manifolds

Carron¹,

Herzlich²

2002

Comment. Math. Helv.

View full text Add to dashboard Cite

Abstract.It was proved in 1957 by Huber that any complete surface with integrable Gauss curvature is conformally equivalent to a compact surface with a finite number of points removed. Counterexamples show that the curvature assumption must necessarily be strengthened in order to get an analogous conclusion in higher dimensions. We show in this paper that any non compact Riemannian manifold with finite L n/2 -norm of the Ricci curvature satisfies Huber-type conclusions if either it is a conformal domain with volume growth controlled from above in a compact Riemannian manifold or if it is conformally flat of dimension 4 and a natural Sobolev inequality together with a mild scalar curvature decay assumption hold. We also get partial results in other dimensions. Mathematics Subject Classification (2000). 53C21, 58J60.

show abstract

“…Therefore, for any w ∈ B(0, δ) lim ε→0 + ε −H Z ε,2 (u) (d) = 0 and Assertion 2(a) follows from Equation (10). (b) Assume that H > 1.…”

Section: Let M 0 ∈ M and Consider The Open Ballmentioning

confidence: 99%