Sweeping out sectional curvature

Panov, Dmitri¹ ; Petrunin, Anton²

¹ Department of Mathematics, King’s College London, London WC2R 2LS, UK
² Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Geometry & topology, Tome 18 (2014) no. 2, pp. 617-631.

Résumé

We observe that the maximal open set of constant curvature $\kappa$ in a Riemannian manifold of curvature $\ge \kappa$ or $\le \kappa$ has a convexity-type property, which we call two-convexity. This statement is used to prove a number of rigidity statements in comparison geometry.

Détail

DOI : 10.2140/gt.2014.18.617

Classification : 53C24, 53C20
Keywords: rigidity, comparison geometry

Bibliographie

[1] A D Aleksandrov, Vnutrennyaya Geometriya Vypuklyh Poverhnosteĭ, OGIZ, Moscow-Leningrad, (1948) 387

[2] S B Alexander, I D Berg, R L Bishop, Geometric curvature bounds in Riemannian manifolds with boundary, Trans. Amer. Math. Soc. 339 (1993) 703

[3] S B Alexander, V Kapovitch, A Petrunin, Alexandrov geometry

[4] S Brendle, F C Marques, A Neves, Deformations of the hemisphere that increase scalar curvature, Invent. Math. 185 (2011) 175

[5] S V Buyalo, Volume and fundamental group of a manifold of nonpositive curvature, Mat. Sb. 122(164) (1983) 142

[6] J Cheeger, D G Ebin, Comparison theorems in Riemannian geometry, AMS Chelsea Publishing (2008)

[7] M Gromov, Sign and geometric meaning of curvature, Rend. Sem. Mat. Fis. Milano 61 (1991) 9

[8] F Hang, X Wang, Rigidity theorems for compact manifolds with boundary and positive Ricci curvature, J. Geom. Anal. 19 (2009) 628

[9] J Lohkamp, Global and local curvatures, from: "Riemannian geometry" (editors M Lovrić, M Min-Oo, M Y K Wang), Fields Inst. Monogr. 4, Amer. Math. Soc. (1996) 23

[10] J Lott, On the long-time behavior of type-III Ricci flow solutions, Math. Ann. 339 (2007) 627

[11] M Min-Oo, Scalar curvature rigidity of asymptotically hyperbolic spin manifolds, Math. Ann. 285 (1989) 527

[12] M Min-Oo, Scalar curvature rigidity of certain symmetric spaces, from: "Geometry, topology, and dynamics" (editor F Lalonde), CRM Proc. Lecture Notes 15, Amer. Math. Soc. (1998) 127

[13] A Petrunin, W Tuschmann, Diffeomorphism finiteness, positive pinching, and second homotopy, Geom. Funct. Anal. 9 (1999) 736

[14] R Schoen, S T Yau, Proof of the positive mass theorem, II, Comm. Math. Phys. 79 (1981) 231

[15] E Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981) 381

[16] V A Zalgaller, On deformations of a polygon on a sphere, Uspehi Mat. Nauk 11 (1956) 177

[17] W Ziller, Examples of Riemannian manifolds with non-negative sectional curvature, from: "Surveys in differential geometry, Vol. XI" (editors J Cheeger, K Grove), Surv. Differ. Geom. 11, International Press (2007) 63