Let M be a compact n-manifold of positive sectional curvature.We will review classical results on the fundamental group of M,a motivation on the c(n)-cyclic conjecture that the fundamental group of M contains a cyclic...Let M be a compact n-manifold of positive sectional curvature.We will review classical results on the fundamental group of M,a motivation on the c(n)-cyclic conjecture that the fundamental group of M contains a cyclic subgroup of index bounded above by c(n),a constant depending only on n,and we will survey partial results(up to date)on the c(n)-cyclic conjecture.展开更多
Let M be a closed n-manifold of positive sectional curvature. Assume that M admits an effective isometrical T1× Zpk-action with p prime. The main result of the article n＋1 for n 〉 5, then there exists a positiv...Let M be a closed n-manifold of positive sectional curvature. Assume that M admits an effective isometrical T1× Zpk-action with p prime. The main result of the article n＋1 for n 〉 5, then there exists a positive constant p（n）, is that ifk=lforn=3or k〉 n＋1/4 for n≥5,then there exists a positive constant p（n）,depending only on n, such that π1 （M） is cyclic if p ≥ p（n）.展开更多
et Mn （n ≥ 3） be a complete Riemannian manifold with secM ≥ 1, and let Mni^ni （i = 1, 2） be two complete totally geodesic submanifolds in M. We prove that if n1 ＋ n2 = n - 2 and if the distance ｜M1M2｜≥π/2, ...et Mn （n ≥ 3） be a complete Riemannian manifold with secM ≥ 1, and let Mni^ni （i = 1, 2） be two complete totally geodesic submanifolds in M. We prove that if n1 ＋ n2 = n - 2 and if the distance ｜M1M2｜≥π/2, then Mi is isometric to s^ni/Zh, CP^m/2, or CP^ni/2/Z2 with the canonical metric when ni 〉 0, and thus, M is isometric to Sn/Zh, CPn/2, or CPn/2/Z2 except possibly iso when n = 3 and M1 （or M2） ≌ S1/Zh with h ≥ 2 or n iso= 4 and M1 （or M2） iso ≌ RP^2展开更多
We depict recent developments in the field of positive sectional curvature, mainly, but not exclusively, under the assumption of isometric torus actions. After an elaborate introduction to the field, we shall discuss ...We depict recent developments in the field of positive sectional curvature, mainly, but not exclusively, under the assumption of isometric torus actions. After an elaborate introduction to the field, we shall discuss various classification results, before we provide results on the computation of Euler characteristics. This will be the starting point for an examination of more involved invariants and further techniques. In particular, we shall discuss the Hopf conjectures, related decomposition results like the Wilhelm conjecture, results in differential topology and index theory as well as in rational homotopy theory, geometrically formal metrics in positive curvature and much more. The results we present will be discussed for arbitrary dimensions, but also specified to small dimensions. This survey article features mainly depictions of our own work interest in this area and cites results obtained in different collaborations; full statements and proofs can be found in the respective original research articles.展开更多
文摘Let M be a compact n-manifold of positive sectional curvature.We will review classical results on the fundamental group of M,a motivation on the c(n)-cyclic conjecture that the fundamental group of M contains a cyclic subgroup of index bounded above by c(n),a constant depending only on n,and we will survey partial results(up to date)on the c(n)-cyclic conjecture.
文摘Let M be a closed n-manifold of positive sectional curvature. Assume that M admits an effective isometrical T1× Zpk-action with p prime. The main result of the article n＋1 for n 〉 5, then there exists a positive constant p（n）, is that ifk=lforn=3or k〉 n＋1/4 for n≥5,then there exists a positive constant p（n）,depending only on n, such that π1 （M） is cyclic if p ≥ p（n）.
文摘et Mn （n ≥ 3） be a complete Riemannian manifold with secM ≥ 1, and let Mni^ni （i = 1, 2） be two complete totally geodesic submanifolds in M. We prove that if n1 ＋ n2 = n - 2 and if the distance ｜M1M2｜≥π/2, then Mi is isometric to s^ni/Zh, CP^m/2, or CP^ni/2/Z2 with the canonical metric when ni 〉 0, and thus, M is isometric to Sn/Zh, CPn/2, or CPn/2/Z2 except possibly iso when n = 3 and M1 （or M2） ≌ S1/Zh with h ≥ 2 or n iso= 4 and M1 （or M2） iso ≌ RP^2
文摘We depict recent developments in the field of positive sectional curvature, mainly, but not exclusively, under the assumption of isometric torus actions. After an elaborate introduction to the field, we shall discuss various classification results, before we provide results on the computation of Euler characteristics. This will be the starting point for an examination of more involved invariants and further techniques. In particular, we shall discuss the Hopf conjectures, related decomposition results like the Wilhelm conjecture, results in differential topology and index theory as well as in rational homotopy theory, geometrically formal metrics in positive curvature and much more. The results we present will be discussed for arbitrary dimensions, but also specified to small dimensions. This survey article features mainly depictions of our own work interest in this area and cites results obtained in different collaborations; full statements and proofs can be found in the respective original research articles.
