On the structure of spaces with Ricci curvature bounded below. II.

*(English)*Zbl 1027.53042The reviewed paper, the sequel of the authors’ article [J. Differ. Geom. 46, 406-480 (1997; Zbl 0902.53034)], is the second in a series devoted to the study of the structure of complete connected Riemannian manifolds, whose Ricci curvature has a definite lower bound and of the Gromow-Hausdorff limits, \(Y\), of sequences of such manifolds. First the authors prove a generalization in the collapsing case, of the volume convergence conjecture of Anderson-Cheeger and deduce some consequences. Next they introduce a family of lower dimensional Hausdorff measures associated to a renormalized limit measure, \(\nu\), on a collapsed limit space and show that for the notion of codimension defined by this family of measures, the strongly singular set satisfies \(\text{ codim}_{\nu}\mathcal S\mathcal S\geqslant 1\). Then the authors investigate connectedness properties of the regular set \(\mathcal R\) of \(Y\), which is the limit of manifolds with Ricci curvature uniformly bounded from below. Using these results they prove that the isometry group of noncollapsed limit space is a Lie group. Finally, the authors show that a limit space which contains a one 1-dimensional piece and which satisfies an additional condition, is itself 1-dimensional.

For Part III, see ibid. 54, 37-74 (2000; Zbl 1027.53043).

For Part III, see ibid. 54, 37-74 (2000; Zbl 1027.53043).

Reviewer: M.Hotloś (Wrocław)

##### MSC:

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

53C20 | Global Riemannian geometry, including pinching |

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |