R. Albuquerque, Isabel M. C. Salavessa
We give a new proof that the sphere S^6 does not admit an integrable orthogonal complex structure, as in \cite{LeBrun}, following the methods from twistor theory. We present the twistor space of a pseudo-sphere S^{2n}_{2q}=SO_{2p+1,2q}/SO_{2p,2q} as a pseudo-Kähler symmetric space. We then consider orthogonal complex structures on the pseudo-sphere, only to prove such a structure cannot exist.
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
We explicitly describe all SO(7)-invariant almost quaternion-Hermitian structures on the twistor space of the six sphere and determine the types of their intrinsic torsion.
We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration $π:\mathbb{CP}^{3}\to S^{4}$. We prove three results about the topology of the twistor discriminant locus of an algebraic surface in $\mathbb{CP}^{3}$. First of all we prove that, with the exception of two exceptional cases, the real dimension of the twistor discriminant locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible an…
The nearly Kähler structures on the 6-sphere, as a twistor bundle sections are researched. We show that for any point of twistor bundle there exists an 1-parametric family of sections, passing through the point, which give nearly Kähler structures on the round sphere. Some properties of those sections are found.
Let M be a G2-manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy G2 manifold. We interpret G2-instanton bundles as CR-holomorphic bundles on its twistor space.
For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\mathbb{C}^{n+1}$. We use this fact to describe a natural stratification of the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ with $n \geq 3$, in terms of what we have called {\it generalised complex orthogonal Stiefel manifolds} of $\mathbb{C}^{n+1}$. In particular, the twistor space $\mathfrak{Z}(\mathbb{S}^{6})$ is biholomorphic…
We show that given a conformal structure whose holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is always a local metric in the conformal class off a singular set which is Ricci-isotropic and gives rise to a parallel, totally lightlike distribution on the tangent bundle. This naturally applies to parallel spin tractors resp. twistor spinors on conformal spin manifolds and clarifies which twistor spinors are locally equivalent to parallel spinors. Moreover, we study the zero set of a twistor spinor using the curved orbit decomposition for parabolic geomet…
We give the twistor description of harmonic maps of the Riemann sphere into the Hilbert-Schmidt Grassmannian. The study of such maps is motivated by the harmonic spheres conjecture formulated in the beginning of this paper.
In contrast to the classical twistor spaces whose fibres are 2-spheres, we introduce twistor spaces over manifolds with almost quaternionic structures of the second kind in the sense of P. Libermann whose fibres are hyperbolic planes. We discuss two natural almost complex structures on such a twistor space and their holomorphic functions.
We prove that there exist global solutions of the twistor equation on the Fefferman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension and we study their properties.
We consider the generalized Kahler structures (g,J_+,J_-) that arise on a hyperkahler manifold (M,g,I,J,K) when we choose J_+ and J_- from the twistor space of M. We find a relation between semichiral and arctic superfields which can be used to determine the generalized Kahler potential for hyperkahler manifolds whose description in projective superspace is fully understood. We use this relation to determine an S^2-family of generalized Kahler potentials for Euclidean space and for the Eguchi-Hanson geometry. Cotangent bundles of Hermitian symmetric spaces constitute a class of hyperkahler man…
A hyperkähler manifold $M$ has a family of induced complex structures indexed by a two-dimensional sphere $S^2 \cong \mathbb{CP}^1$. The twistor space of $M$ is a complex manifold $Tw(M)$ together with a natural holomorphic projection $Tw(M) \to \mathbb{CP}^1$, whose fiber over each point of $\mathbb{CP}^1$ is a copy of $M$ with the corresponding induced complex structure. We remove one point from this sphere (corresponding to one fiber in the twistor space),and for the case of $M = \mathbb{R}^{4n}$, $n\in\mathbb{N}$, equipped with the standard hyperkähler structure, we construct one quantizat…
We investigate the pseudo-hyperkähler geometry of higher degree rational curves in the twistor space of a hyperkähler $4$-manifold.
As in the case of irreducible holomorphic symplectic manifolds, the period domain $Compl$ of compact complex tori of even dimension $2n$ contains twistor lines. These are special $2$-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a {\em generic} chain of twistor lines. We also prove a criterion of twistor path connectivity of loci in $Compl$ where a fixed second cohomology class stays of Hodge type (1…
We study spin structures on orbifolds. In particular, we show that if the singular set has codimension greater than 2, an orbifold is spin if and only if its smooth part is. On compact orbifolds, we show that any non-trivial twistor spinor admits at most one zero which is singular unless the orbifold is conformally equivalent to a round sphere. We show the sharpness of our results through examples.
Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the `quaternionic-like manifolds'. These contain, as particular subclasses, the CR quaternionic and the $ρ$-quaternionic manifolds. Moreover, the notion of `heaven space' finds its adequate level of generality in this setting: (essentially) any real analytic quaternionic-like manifold admits a (germ) unique heaven space, which is a $ρ$-quaternionic manifold. We, also, give a natural construction of homogeneous complex …
In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Plücker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Plücker quadric by examining the novel contact properties of two-spheres in $S^4,$ generalizing classical Lie geometry in $S^3.$ Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. We define discret…
We study the geometry of the (generalized) twistor triangles $\triangle J_1J_2J_3$ in the period domain of compact complex tori of complex dimension $2n$ by the means of the representation theory of the algebras (of real dimension 8) generated by the complex structures $J_1,J_2,J_3$. Considering the period domain as the homogeneous space for $G=GL_{4n}(\mathbb{R})$, we introduce on it a $G$-invariant pseudometric and define pseudometric invariants, helping us to distinguish triangles from a reasonable class up to $G$-equivalence.
We characterise, in the setting of the Kodaira-Spencer deformation theory, the twistor spaces of (co-)CR quaternionic manifolds. As an application, we prove that, locally, the leaf space of any nowhere zero quaternionic vector field on a quaternionic manifold is endowed with a natural co-CR quaternionic structure. Also, for any positive integers $k$ and $l$, with $kl$ even, we obtain the geometric objects whose twistorial counterparts are complex manifolds endowed with a conjugation without fixed points and which preserves an embedded Riemann sphere whose normal bundle is $l$ times the line bu…
We characterize Willmore tori in the 4-sphere with nontrivial normal bundle as Twistor projections of elliptic curves in complex projective space or as inverted minimal tori (with planar ends) in Euclidean 4-space.
We study a natural functional on the space of holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We give a link to a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. Moreover, we prove that for a certain class of real holomorphic sections of the Deligne-Hitchin moduli space, the functional is basically given by the Willmore energy of corresponding (equivariant) conformal map to the 3-sphere. As an application we use the func…
In this work a proposal for definition of twistors on generic curved spaces is exposed and investigated. We consider superpositions of nearly autoparallel and nearly geodesic maps (nearly conformal maps, nc-maps) of (pseudo-)Riemannian spaces as generalizations of conformal transforms. We introduce the nearly autoparallel twistor equations being compatible on nc-flat spaces and study nearly autoparallel twistor structures generating curved spaces and vacuum Einstein equations.
On the twistor spaces of ALE gravitational instantons of type $A_{\rm odd}$
研究了toric ALE引力瞬子类型$A_{2n-1}$的扭矢空间及其关联的非标准minitwistor空间，通过分析线性系统的基础点，确定了某些特殊扭矢线的像，描述了由瞬子产生的三维实minitwistor线家族。
In recent papers math.DG/0701278 and arXiv:0705.0060, we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called minitwistor spaces). These include the twistor spaces studied in the papers as very special cases. Our source of the result is a series of self-dual metrics with torus action constructed by D. Joyce. Actually, for arbitrary Joyce metrics and U(1)-subgroups of the torus which fix…
We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic fields.
The tangent bundle to the $n$--dimensional sphere is the space of oriented lines in $\R^{n+1}$. We characterise the smooth sections of $TS^n\to S^n$ which correspond to points in $\R^{n+1}$ as gradients of eigenfunctions of the Laplacian on $S^n$ with eigenvalue $n$. The special case of $n=6$ and its connection with almost complex geometry is discussed.
In this paper we obtain several curvature properties of the twistor and reflector spaces of a paraquaternionic Kähler manifold and prove the existence of both positive and negative mixed 3-Sasakian structures in a principal SO(2,1)-bundle over a paraquaternionic Kähler manifold.
The twistor space of the sphere S^{2n} is an isotropic Grassmannian that fibers over S^{2n}. An orthogonal complex structure on a subdomain of S^{2n} (a complex structure compatible with the round metric) determines a section of this fibration with holomorphic image. In this paper, we use this correspondence to prove that any finite energy orthogonal complex structure on R^6 must be of a special warped product form, and we also prove that any orthogonal complex structure on R^{2n} that is asymptotically constant must itself be constant. We will also give examples defined on R^{2n} which have i…
The aim of our article is the study of solution space of the symplectic twistor operator $T_s$ in symplectic spin geometry on standard symplectic space $({\mathbb R}^{2n},ω)$, which is the symplectic analogue of the twistor operator in (pseudo)Riemannian spin geometry. In particular, we observe a substantial difference between the case $n=1$ of real dimension 2 and the case of ${\mathbb R}^{2n}$, $n>1$. For $n>1$, the solution space of $T_s$ is isomorphic to the Segal-Shale-Weil representation.
Classification of Willmore $2$-spheres in $S^n$
本文通过构造 harmonic 序列，对 $S^n$ 中 Willmore 2 球面进行了分类，证明其要么完全 isotropic，要么严格 $k$-isotropic。
The spaces of harmonic maps of the projective plane to the four-dimensional sphere are investigated in this paper by means of twistor lifts. It is shown that such spaces are empty in case of even harmonic degree. In case of harmonic degree less than 6 it was shown that such spaces are path-connected and an explicit parameterization of the canonical representatives was found. In addition, the last section summarizes known results for harmonic maps of the two-dimensional sphere to the four-dimensional sphere of harmonic degree less than 6.
Four-dimensional quaternion-Kahler metrics, or equivalently self-dual Einstein spaces M, are known to be encoded locally into one real function h subject to Przanowski's Heavenly equation. We elucidate the relation between this description and the usual twistor description for quaternion-Kahler spaces. In particular, we show that the same space M can be described by infinitely many different solutions h, associated to different complex (local) submanifolds on the twistor space, and therefore to different (local) integrable complex structures on M. We also study quaternion-Kahler deformations o…
We find geometric conditions on a Hermitian-Weyl manifold under which the complex structure is a pseudo-harmonic map in the sense of G. Kokarev \cite{K09} from the manifold into its twistor space. This is done under the assumption that the dimension of the manifold is four or the Hermitian-Weyl structure is locally conformally Kähler.
In the present paper we use twistor theory in order to solve two problems related to harmonic maps from surfaces to Euclidean spheres $\mathbb{S}^n$. First, we propose a new approach to isoperimetric inequalities based on energy index. Using this approach we show that for any positive $k$, the $k$-th non-zero eigenvalue of the Laplacian on the real projective plane endowed with a metric of unit area, is maximized on the sequence of metrics converging to a union of $(k-1)$ identical copies of round sphere and a single round projective plane. This extends the results of P. Li and S.-T. Yau for $…
The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $ω$. This article studies limits of manifolds for which $ω$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperkähler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorph…
Invariant distributions and the transport twistor space of closed surfaces
本文研究闭定向黎曼曲面单位切丛上的输运方程，建立其与输运扭量空间的关联，证明不变纤维全纯分布构成有单位元代数，给出其与扭量空间上全纯函数的双射对应，并分类Anosov曲面上光滑至边界的全纯线丛，得到Flaminio结果的定量版本。
We study the Penrose transform for the `quaternionic objects' whose twistor spaces are complex manifolds endowed with locally complete families of embedded Riemann spheres with positive normal bundles.
Natural metric structures on the tangent bundle and tangent sphere bundles $S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many important unsolved problems. Admitting metric connections on $M$ with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations of reducibility of $TM$ to the almost Hermitian category. Our purpose is the study of the natural contact structure on $S_rM$ and the $G_2$-twistor space of any oriented Riemannian 4-manifold.
We consider sphere bundles P and P' of totally null planes of maximal dimension and opposite self-duality over a 4-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product PP' of P and P' is found to be appropriate for the encoding of both the selfdual and the Einstein-Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical objects on PP'. The formulation is suitable for both complex- and real-valued metrics. It unifies results for all three possible real signatures. In the purely Riemannian positive d…
In this paper we look at the question of integrability, or not, of the two natural almost complex structures $J^{\pm}_\nabla$ defined on the twistor space $J(M,g)$ of an even-dimensional manifold $M$ with additional structures $g$ and $\nabla$ a $g$-connection. We also look at the question of the compatibility of $J^{\pm}_\nabla$ with a natural closed $2$-form $ω^{J(M,g,\nabla)}$ defined on $J(M,g)$. For $(M,g)$ we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection $\nabla$. In all cases $…
Firstly we give a condition to split off the K"ahler factor from a nearly pseudo-K"ahler manifold and apply this to get a structure result in dimension 8. Secondly we extend the construction of nearly K"ahler manifolds from twistor spaces to negatively curved quaternionic K"ahler manifolds and para-quaternionic K"ahler manifolds. The class of nearly pseudo-K"ahler manifolds obtained from this construction is characterized by a holonomic condition. The combination of these results enables us to give a classification result in (real) dimension 10. Moreover, we show that a strict nearly pseudo-K"…
We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification is obtained, while in other cases the solutions are described by an explicit system of partial differential equations.
Let M be a compact pseudo-umbilical submanifold of the unit sphere S. In the present note, it is shown that if the normal curvature, scalar curvature S and square of the length of second fundamental form satisfy certain conditions, then M is totally geodesic.
We prove that the envelope of meromorphy of any imbedded symplectic sphere in $CP^2$ coincides with the whole $CP^2$. As a tool for the proof we use the Gromov theory of pseudo-holomorphic curves. Several results in this subject, such as adjunction formula, smoothness of moduli space in the neighborhood of a cusp-curve are improved. We introduce a natural holomorphic structure on the pulled back tangent bundle, in which the differential of a ps.-hol. map in an analytic morphism and describe cusps of ps.-hol. curves using Bennequin index.
We give a local representation for the pseudoholomorphic surfaces in Euclidean spheres in terms of holomorphic data. Similar to the case of the generalized Weierstrass representation of Hoffman and Osserman, we assign such a surface in $\Sf^{2n}$ to a given set of $n$ holomorphic functions defined on a simply-connected domain in $\C$.
In this paper we develop the theory of Willmore sequences for Willmore surfaces in the 4-sphere. We show that under appropriate conditions this sequence has to terminate. In this case the Willmore surface either is the twistor projection of a holomorphic curve into complex projective space or the inversion of a minimal surface with planar ends in 4-space. These results give a unified explanation of previous work on the characterization of Willmore spheres and Willmore tori with non-trivial normal bundles by various authors.
On tangent sphere bundles with contact pseudo-metric structures
本文在切球丛上构造切触伪度量结构，并证明其成为(κ, μ)-切触伪度量流形或K-切触流形分别等价于基流形具有常截面曲率或常曲率ε。
In this paper, we use Clifford algebra and the spinor calculus to study the complex structures on Euclidean space $R^8$ and the spheres $S^4,S^6$. By the spin representation of $G(2,8)\subset Spin(8)$ we show that the Grassmann manifold G(2,8) can be looked as the set of orthogonal complex structures on $R^8$. In this way, we show that G(2,8) and $CP^{3}$ can be looked as twistor spaces of $S^6$ and $S^4$ respectively. Then we show that there is no almost complex structure on sphere $S^4$ and there is no orthogonal complex structure on the sphere $S^6$.
In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculated. In contrast to the usual Riemann sphere $\mathbb{S}^6$, there exist (integrable) complex structures and para-complex structures on the pseudospheres under consideration.
We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type operators parameterized over the complexified dual torus. The kernel line bundle of this family over the spectral curve describes the generalized Darboux transforms of the conformally immersed torus. If the spectral curve has finite genus the kernel bundle can be extended to the compa…
In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere $\mathbb{S}^4_s(1)$ with index s, $s=1, 2$, and having harmonic pseudo-spherical Gauss map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere $\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s$ with 1-type pseudo-spherical Gauss map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space $\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1$ with 1-type p…
A GL(2, R) structure on an (n+1)-dimensional manifold is a smooth pointwise identification of tangent vectors with polynomials in two variables homogeneous of degree n. This, for even n=2k, defines a conformal structure of signature (k, k+1) by specifying the null vectors to be the polynomials with vanishing quadratic invariant. We focus on the case n=6 and show that the resulting conformal structure in seven dimensions is compatible with a conformal G_2 structure or its non-compact analogue. If a GL(2, R) structure arises on a moduli space of rational curves on a surface with self-intersectio…
We study the space of (orthogonal) almost complex structures on closed six-dimensional manifolds as the space of sections of the twistor space for a given metric. For a connected six-manifold with vanishing first Betti number, we express the space of almost complex structures as a quotient of the space of sections of a seven-sphere bundle over the manifold by a circle action, and then use this description to compute the rational homotopy theoretic minimal model of the components that satisfy a certain Chern number condition. We further obtain a formula for the homological intersection number o…
Inhomogeneity of Isoparametric Hypersurfaces of OT-FKM-type in the Pseudo-Sphere
该研究针对伪黎曼空间形式中的等参超曲面，显式构造任意符号的Clifford系统，并证明满足特定符号条件的伪球面OT-FKM型等参超曲面及其焦集M₊的连通分支均具有非齐性。
Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a product structure and symmetric with respect to g, then r induces a pseudo Riemannian product structure on M. Sometimes the integrability condition is expressed by the closedness of an associated two-form: if j is almost complex on M and ω(x, y) = g(jx, y) is symplectic, then M is almost pseudo Kähler. Now, product, complex and symplectic structures on M ar…
We initiate the study of the generalized quaternionic manifolds by classifying the generalized quaternionic vector spaces, and by giving two classes of nonclassical examples of such manifolds. Thus, we show that any complex symplectic manifold is endowed with a natural (nonclassical) generalized quaternionic structure, and the same applies to the heaven space of any three-dimensional Einstein-Weyl space. In particular, on the product $Z$ of any complex symplectic manifold $M$ and the sphere there exists a natural generalized complex structure, with respect to which $Z$ is the twistor space of …
Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a 2-sphere of complex structures, but they no longer behave like unit imaginary quaternions. We still require, however, that the 2-sphere of complex structures determines a decomposition of the complexified tangent space as tensor product of C^{2n} and C^2. Among interesting properties of pluricomplex manifold is the existence of a canonical torsion free connec…
We show that a twistor construction of Hitchin and Ward can be adapted to study unitons (harmonic spheres in a unitary group). Specifically, we show that unitons are equivalent to holomorphic bundles with extra structure over a rational ruled surface with energy given by Chern class. This equivalence allows us to confirm the conjecture of Wood that unitons are rational. We also construct an example of a two uniton in U(3) using this construction.
Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as well as the twistor spaces over a compact quaternion-Kahler manifold. Differing from Fateev's examples most of our examples are non-collapsed. The construction of this paper, different from the ad hoc anastz of Fateev, is systematic, generalizing (as well as unifying) the previous constructions of Einstein metrics by Bourguignon-Karcher, Jensen, and Ziller in…
In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the calculus of uniformly supported pseudo-differential operators that is analogous to the calculus with bounds defined on manifolds with boundary. We derive a Fredholmness criterion for operators on the Bruhat sphere, and prove that their parametrices up to compact operators lie inside…
n this paper we define an invariant of a pair of 6 dimensional symplectic %optional manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of the bounding cochains (solution of A infinity version of Maurer-Cartan equation of the filtered A infinity algebra associated to the Lagrangian submanifold). In the case when the Lagrangian submanifold is a rational homology sphere, it becomes a numerical invariant. This invariant depends on the choice of almost complex structure. The way…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。