Foliations of Codimension One and the Milnor Conjecture

Автор(и)

  • Dmitry V. Bolotov B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine

DOI:

https://doi.org/10.15407/mag14.02.119

Ключові слова:

шарування ковимiрностi один, фундаментальна група, голономiя, кривина Рiччi

Анотація

Ми доводимо, що фундаментальна група шарiв C2-шарування ковимiрностi один невiд'ємної кривини Рiччi замкнутого рiманова многовиду є скiнченно породженою та майже абелевою, тобто мiстить скiнченно породжену абелеву пiдгрупу скiнченного iндексу. Зокрема, ми пiдтверджуємо гiпотезу Мiлнора щодо многовидiв, якi є шарами шарування ковимiрностi один невiд'ємної кривини Рiччi замкнутого рiманова многовиду.

Mathematical Subject Classification: 53A05.

Посилання

S. Adams and G. Stuck, Splitting of non-negatively curved leaves in minimal sets of foliations, Duke Math. J. 71 (1993), 71–92. https://doi.org/10.1215/S0012-7094-93-07104-9

R.L. Bishop, A Relation between Volume, Mean Curvature and Diameter, Euclidean Quantum Gravity (Eds. G.W. Gibbons and S.W. Hawking), World Scientific, Singapore–New Jersey–London–Hong Kong, 1993, 161. https://doi.org/10.1142/9789814539395_0009

D. Burago, Yu. Burago, and S. Ivanov, A Course in Metric Geometry. Graduate Studies in Mathematics, 33, Amer. Math. Soc., Providence, RI, 2001.

S.V. Buyalo, Euclidean planes in open three-dimensional manifolds of nonpositive curvature, Algebra i Analiz 3 (1991), 102–117; Engl. transl.: St. Petersburg Math. J. 3 (1992), 83–96.

J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971/72), 119–128. https://doi.org/10.4310/jdg/1214430220

G. Hector and U. Hirsch, Introduction to the Geometry of Foliations. Part B. Foliations of codimension one. Second edition. Aspects of Mathematics, E3, Friedr. Vieweg & Sohn, Braunschweig, 1987.

H. Imanishi, Structure of codimension one foliations which are almost without holonomy, J. Math. Kyoto Univ. 16 (1976), 93–99. https://doi.org/10.1215/kjm/1250522960

J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. https://doi.org/10.4310/jdg/1214501132

T. Nishimori, Compact leaves with abelian holonomy, Tohoku Math. J. (2) 27 (1975), 259–272. https://doi.org/10.2748/tmj/1178240992

S.P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obšč. 14 (1965), 248–278 (Russian).

J.F. Plante, On the existence of exceptional minimal sets in foliations of codimension one, J. Differential Equations 15 (1974), 178–194. https://doi.org/10.1016/0022-0396(74)90093-X

J.F. Plante and W.P. Thurston, Polynomial growth in holonomy groups of foliations, Comment. Math. Helv. 51 (1976), 567–584. https://doi.org/10.1007/BF02568174

C. Sormani, On loops representing elements of the fundamental group of a complete manifold with nonnegative Ricci curvature, Indiana Univ. Math. J. 50 (2001), 1867– 1883. https://doi.org/10.1512/iumj.2001.50.2048

I. Tamura, Topology of Foliations, Translated from the Japanese by A.A. Bel’skiı̆, Mir, Moscow, 1979 (Russian).

B. Wilking, On fundamental groups of manifolds of nonnegative curvature, Differential Geom. Appl. 13 (2000), 129–165. https://doi.org/10.1016/S0926-2245(00)00030-9

Downloads

Як цитувати

(1)
Bolotov, D. V. Foliations of Codimension One and the Milnor Conjecture. Журн. мат. фіз. анал. геом. 2018, 14, 119-131.

Номер

Том 14 № 2 (2018)

Розділ

Статті

Завантаження

Дані завантаження ще не доступні.