戎小春
微分几何理论
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- 姓名:戎小春
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学术头衔:
博士生导师
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学科领域:
数理逻辑与数学基础
- 研究兴趣:微分几何理论
戎小春:男,1954年生。1982年于首都师范大学数学系本科毕业。1984年于首都师范大学数学系硕士毕业,后留校任教。1986-1990年,戎小春至美国纽约州立大学石溪分校学习并取得博士学位,现为美国Rutgers大学教授。他在留学期间,专心致力于微分几何理论的学习与研究,在微分几何研究方向取得了突出的成绩,成为国际上著名的微分几何学专家,曾应邀在2000年第一届美国Scandinvian国际数学大会(每四年举行一次)作45分钟报告,并获得应邀在2002年国际数学家大会(每四年举行一次)做45分钟报告的殊荣。
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【期刊论文】The second twisted Betti number and the convergence of collapsing Riemannian manifolds
戎小春, Fuquan Fang, ★, Xiaochun Rong, , ★★
,-0001,():
-1年11月30日
Let Mi dGH −→ X denote a sequence of n-manifolds converging to a compact metric space, X, in the Gromov-Hausdorff topology such that the sectional curvature is bounded in absolute value and dim(X)<n. We prove the following stability result: If the fundamental groups of Miare torsion groups of uniformly bounded exponents and the second twisted Betti numbers of Mi vanish, then there is a manifold, M, and a sequence of diffeomorphisms from M to a subsequence of {Mi} such that the distance functions of the pullback metrics converge to a pseudo-metric in C0-norm. Furthermore, M admits a foliation with leaves diffeomorphic to flat manifolds (not necessarily compact) such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics. These results lead to a few interesting applications.
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【期刊论文】Splittings and Cr-structures for manifolds with nonpositive sectional curvature
戎小春, Jianguo Cao, ★, Jeff Cheeger, ★★, Xiaochun Rong, , ★★★
Invent. Math. 144, 139-167 (2001),-0001,():
-1年11月30日
Let Mn denote the universal covering space of a compact Rie-mannian manifold, Mn, with sectional curvature, -1≤Kmn≤O.We show dependent) conditions, determinesan open dense subset of Mn, at every point of which, there exists a local isometric splitting with nontrivial flat factor. Such a coolection, which we call an abelian structure, also gives rise factor. Such a collectionk, which we call an abelian structure, also gives rise to an essentially canonical Cr-stucture in the sense of Buyalo, i.e. an atalas to an essentially canonical Cr-stucture in the sense of Buyalo, i.e an atlas for an injective F-STUCTURE, for which additional conditions hold, It follows in particular that the minimal volume of Mn vanishes. We show that an abelian structure exists if the injectivity radius at all points of Mn is less than ε(n)>O.This yields a conjecture of Buyalo as well as a strength-ened version of the conclusion of Gromov's gap conjecture in our special ituation. In addition, we observe that abelian stuructures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric.
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