arXiv math/06083682006-08math.DG

The Complex Structures on $S^{2n}$

Jianwei Zhou

备注：: 17 pages

英文摘要：

Let $\widetilde{\cal J}(S^{2n})$ be the set of orthogonal complex structures on $TS^{2n}$. We show that the twistor space $\widetilde{\cal J}(S^{2n})$ is a Kaehler manifold. Then we show that an orthogonal almost complex structure $J_f$ on $S^{2n}$ is integrable if and only if the corresponding section $f\colon\; S^{2n}\to \widetilde{\cal J}(S^{2n}) $ is holomorphic. These shows there is no integrable orthogonal complex structure on the sphere $S^{2n}$ for $n>1$. We also show that there is no complex structure in a neighborhood of the space $\widetilde{\cal J}(S^{2n})$. The method is to study the first Chern class of $T^{(1,0)}S^{2n}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

$\widetilde{J}$-tangent affine hypersurfaces with an induced almost paracontact structure

We study real affine hypersurfaces $f\colon M\rightarrow \mathbb{R}^{2n+2}$ with an almost paracontact structure $(φ,ξ,η)$ induced by a $\widetilde{J}$-tangent transversal vector filed, where $\widetilde{J}$ is the canonical paracomplex structure on $\mathbb{R}^{2n+2}$. We give a classification of hypersurfaces for which an induced almost paracontact structure is metric relative to the second fundamental form. Some other properties of such hypersurfaces are also studied.

2017-10-28math.DG

On a characterization of the complex hyperbolic space

Consider a compact Kähler manifold $M^m$ with Ricci curvature lower bound $Ric_M\geq -2(m+1) .$ Assume that its universal cover $% \widetilde{M}$ has maximal bottom of spectrum $λ_1(\widetilde{M}%) =m^2.$ Then we prove that $\widetilde{M}$ is isometric to the complex hyperbolic space $\Bbb{CH}^m.$

2008-02-03math.DG

A generalized integrability problem for G-Structures

Given an $\widetilde n$-dimensional manifold $\widetilde M$ equipped with a $\widetilde G$-structure $\widetildeπ:\widetilde P\rightarrow \widetilde M$, there is a naturally induced $G$-structure $π: P\rightarrow M$ on any submanifold $M\subset\widetilde M$ that satisfies appropriate regularity conditions. We study generalized integrability problems for a given $G$-structure $π: P\rightarrow M$, namely the questions of whether it is locally equivalent to induced $G$-structures on regular submanifolds of homogeneous $\widetilde G$-structures $\widetildeπ:\widetilde P\to \widetilde{H}/\widetilde…

2015-07-09math-phmath.DGmath.MP

Parallel Almost Paracontact Structures on Affine Hypersurfaces

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^{2n+2}\simeq\widetilde{\mathbb{C}}^{n+1}$. We study real affine hypersurfaces $f\colon M\rightarrow \widetilde{\mathbb{C}}^{n+1}$ with a $\widetilde{J}$-tangent transversal vector field. Such vector field induces in a natural way an almost paracontact structure ${(φ,ξ,η)}$ on $M$ as well as the affine connection $\nabla$. In this paper we give the classification of hypersurfaces with the property that $φ$ or $η$ is parallel relative to the connection $\nabla$. Moreover, we show that if $\nablaφ=0$ (respectively $\nablaη…

2018-05-17math.DG

Induced Riemannian structures on null hypersurfaces

Given a null hypersurface $L$ of a Lorentzian manifold, we construct a Riemannian metric $\widetilde{g}$ on it from a fixed transverse vector field $ζ$. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold $(L,\widetilde{g})$ and the vector field $ζ$. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques.

2017-07-25math.DG

Projective structure, $\widetilde{\mathrm{SL}}(3,{\mathbb R})$ and the symplectic Dirac operator

Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective geometry in two real dimensions is $\widetilde{\mathrm{SL}}(3,{\mathbb R})$.

2016-04-15math-phmath.APmath.DG

更多相关论文收起更多论文

Dirac brackets and reduction of invariant bi-Poisson structures

Let $X$ be a manifold with a bi-Poisson structure $\{η^t\}$ generated by a pair of $G$-invariant symplectic structures $ω_1$ and $ω_2$, where the Lie group $G$ acts properly on $X$. Let $H$ be some isotropy subgroup for this action representing the principle orbit type and $X^r_\mathfrak{h}$ be the submanifold of $X$ consisting of the points in $X$ with the stabilizer algebra equal to the Lie algebra $\mathfrak{h}$ of $H$ and with the stabilizer group conjugated to $H$ in $G$. We prove that the pair of symplectic structures $ω_1|_{X^r_\mathfrak{h}}$ and $ω_2|_{X^r_\mathfrak{h}}$ generates an $…

2016-07-15math.DG

非负Ricci曲率、无穷远处的几乎稳定性与基本群结构

Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups

本文研究非负Ricci曲率开流形的基本群，在万有覆盖无穷远切锥截面几乎稳定的条件下，证明基本群有限生成且含有限指数正规阿贝尔子群，并给出指数上界。

2019-01-31math.DG

Frobenius manifolds and a new class of extended affine Weyl groups $\widetilde{W}^{(k,k+1)}(A_l)$

We present a new class of extended affine Weyl groups $\widetilde{W}^{(k,k+1)}(A_l)$ for $1\leq k <l$ and obtain an analogue of Chevalley-type theorem for their invariants. We further show the existence of Frobenius manifold structures on the orbit spaces of $\widetilde{W}^{(k,k+1)}(A_l)$ and also construct Landau--Ginzburg superpotentials for these Frobenius manifold structures.

2019-05-24math-phmath.DGmath.MP

复结构的不可延展性

Non-extendability of complex structures

本文研究了复结构在 $S^6$ 上的延展性问题，证明了局部复结构无法延展为全局可积结构，反驳了Yau问题52的变形策略

2026-01-09math.ATmath.CVmath.DG

A Homotopy-like Class Invariant for Sub-manifolds of Punctured Euclidean Spaces

We consider the $D$-dimensional Euclidean space, $\mathbb{R}^D$, with certain $(D-N)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{singularity manifolds} and represent by $\widetilde{\mathcal{S}}$) removed from it. We define and investigate the problem of finding a homotopy-like class invariant ($χ$-homotopy) for certain $(N-1)$-dimensional compact, closed and orientable sub-manifolds (which we call \emph{candidate manifolds} and represent by $ω$) of $\mathbb{R}^D \setminus \widetilde{\mathcal{S}}$, with special emphasis on computational aspects of the problem.…

2011-03-13math.ATmath.DGmath.GT