Complex immersions in Kähler manifolds of positive holomorphic $k$-Ricci curvature

Fang, Fuquan; Mendonça, Sérgio

doi:10.1090/s0002-9947-05-03675-5

Cited by 7 publications

(1 citation statement)

References 19 publications

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“…To prove these connectedness Theorems [7] and [8] apply Morse Theory locally on the ambient space but in this work we apply Morse theory on the space of paths following the work of [6], [4], [5], [14], [21], [22], [23], [26] and [19]. The basic idea of [4] and [5] is to demonstrate that the index of the critical points in the space of paths joining two submanifolds has the appropriate lower bound for a chosen Morse function.…”

Section: Introductionmentioning

confidence: 99%

Theorems of Barth-Lefschetz type and Morse theory on the space of paths in homogeneous spaces

Senapathi

2015

Geom Dedicata

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Homotopy connectedness theorems for complex submanifolds of homogeneous spaces (sometimes referred to as theorems of Barth-Lefshetz type) have been established by a number of authors. Morse Theory on the space of paths lead to an elegant proof of homotopy connectedness theorems for complex submanifolds of Hermitian symmetric spaces. In this work we extend this proof to a larger class of compact complex homogeneous spaces.

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Section: Introductionmentioning

confidence: 99%

Theorems of Barth-Lefschetz type and Morse theory on the space of paths in homogeneous spaces

Senapathi

2015

Geom Dedicata

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Knots in Riemannian manifolds

Fang

Mendonça

2009

Math. Z.

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Abstract. In this paper we study submanifold with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if K ⊂ (S n , g) is a totally geodesic submanifold in a Riemannian sphere with positive sectional curvature where n ≥ 5, then K is homeomorphic to S n−2 and the fundamental group of the knot complement π 1 (S n − K) ∼ = Z. IntroductionIn [Re] the author constructed nontrivial torus knots in S 3 which are totally geodesic with respect to some Riemannian metric with positive curvature. Inspired by this work, it is interesting to ask the following Problem 1: Let (S n , g) be a Riemannian sphere with positive sectional curvature and let i : K → (S n , g) be a codimension 2 totally geodesic submanifold. Could i(K) be a nontrivial knot if n ≥ 2?The problem emerges naturally in the study of transformation group theory acting on manifolds. The famous Smith conjecture asserts that if a cyclic group acting on S 3 has 1-dimensional fixed point set, then the fixed point set must be an unknot. The proof of the conjecture was finally given in 1979 depended on several major advances in 3-manifold theory, in particular the work of William Thurston on hyperbolic structures on 3-manifolds, results by William Meeks and ShingTung Yau on minimal surfaces in 3-manifolds, and work by Hyman Bass on finitely generated subgroups of GL(2, C) (cf. [MB]). The wellknown Thurston's conjecture for 3-orbifold, proved in [BLP], implies that any such an action is topologically conjugate to a linear action. Clearly, every fixed point component of a linear action is a trivial knot and so the latter assertion implies the Smith conjecture.2000 Mathematics Subject Classification. Primary 53C42; Secondary 53C22.

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