Vladislav V. Goldberg, Valentin V. Lychagin
We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This solves the Blaschke conjecture for 3-webs. As a side result, we show that the number of linearizations in the Gronwall conjecture does not exceed fifteen and give criteria for rigidity of 3-webs.
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
The Blaschke conjecture claims that every compact Riemannian manifold whose injectivity radius equals its diameter is, up to constant rescaling, a compact rank one symmetric space. We summarize the intuition behind this problem, the proof that such manifolds have the cohomology of compact rank one symmetric spaces, and the proof of the conjecture for homology spheres and homology real projective spaces. We also summarize what is known on the diffeomorphism, homeomorphism and homotopy types of such manifolds.
The Blaschke's conjecture asserts that if $\diam(M)=\text{Inj}(M)=\frac\pi2$ (up to a rescaling) for a complete Riemannian manifold $M$, then $M$ is isometric to $\Bbb S^n(\frac12)$, ${\Bbb R\Bbb P}^{n}$, ${\Bbb C\Bbb P}^{n}$, ${\Bbb H\Bbb P}^{n}$ or ${\Bbb Ca\Bbb P}^{2}$ endowed with the canonical metric. In the paper, we prove that the conjecture is true if we in addition assume that $\sec_M\geq1$.
We construct an explicit diffeomorphism taking any fibration of a sphere by great circles into the Hopf fibration, using elementary geometry--indeed the diffeomorphism is a local (differential) invariant, algebraic in derivatives.
In this paper we study the linearizability problem for 3-webs on a 2-dimensional manifold. With an explicit computation based on the theory developed in the paper "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp. 2643-2654), we examine a 3-web whose linearizability was claimed in the same paper. We show that, contrary to the statement of the papers "On the Blaschke conjecture for 3-webs" (arXiv: math.DG/0411460) and "On linearization of planar three-webs and Blaschke's conjecture", (C.R.Acad. Sci. Paris, Ser. I. vol. 341. num 3 (2005)), this particular web is linearizable. …
There are two theories describing the linearizability of 3-webs: one is developed in the article "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp.2643-2654) and another in the article "On the Blaschke conjecture for 3-webs" (J. Geom. Anal. 16, 1 (2006), 69-115). Unfortunately, they cannot be both correct because on an explicit 3-web W they contradict: the first predicts that W is linearizable while the second states that W is not linearizable. The essential question beyond this particular 3-web is: which theory describes correctly the linearizability condition? In this pape…
We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincaré's theorem: a planar 4-web of maximum rank is linearizable. We also find an invariant intrinsic characterization of planar 4-webs of rank two and one and prove that in general such webs are not linearizable. This solves the Blaschke problem ``to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3''. Finally, we f…
In the article "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp.2643-2654), published in 2001, we studied the linearizability problem for 3-webs on a 2-dimensional manifold. Four years after the publication of our article, V.V.Goldberg and V.V.Lychagin in the paper "On linearization of planar three-webs and Blaschke's conjecture" (C.R.Acad. Sci. Paris, Ser. I. vol. 341. num 3 (2005)) obtained similar results by a different method and criticized our article by qualifying the proofs incomplete. However, they obtained false result on the linearizability of a certain web. We pr…
Blaschke Conjecture and Complex Geometry
本文通过引入大二次曲面纤维化，结合bend-and-break论证与Hwang-Mok VMRT识别，证明了在切丛具有适配复结构条件下Blaschke猜想成立。
We study non-flat planar 3-webs with infinitesimal symmetries. Using multi-dimensional Schwarzian derivative we give a criterion for linearization of such webs and present a projective classification thereof. Using this classification we show that the Gronwall conjecture is true for 3-webs admitting infinitesimal symmetries.
We find d - 2 relative differential invariants for a d-web, d \geq 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f (x,y) and g_4 (x,y),...,g_d (x,y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g_4, and d - 4 PDEs of the second order with respect to f and g_4,...,g_d. For d = 4, this result confirms Blaschke's conjecture on the nature of conditions for the linearizabilty o…
We present a projectively invariant description of planar linear 3-webs. For a non-hexagonal 3-web, we introduce family of projective torsion-free Cartan connections, the web leaves being geodesics for each member of the family, and give a web linearization criterion. Finally, we propose an algorithm for resolving the Gronwall conjecture and illustrate this approach by proving the conjecture for 3-webs whose 2 foliations are 2 pencils of lines.
Local invariants of divergence-free webs
本文研究保体积微分同胚几何中的无散度网，引入微分与几何两个局部不变量，证明其平凡性可刻画网芽等价性，并给出平面情形的典范形式与完整不变量集。
In this paper we will explore a way to prove the hundred years old Gronwall's conjecture: if two plane linear 3-webs with non-zero curvature are locally isomorphic, then the isomorphism is a homography. Using recent results of S. I. Agafonov, we exhibit an invariant, the {\sl characteristic}, attached to each generic point of such a web, with the following property: if a diffeomorphism interchanges two such linear webs, sending a point of the first to a point of the second which have the same characteristic, then this diffeomomorphism is locally a homography.
The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjecture states that a Legendrian 3-web admits at most one distinct local linearization, with the only exception when it is locally equivalent to the dual linear Legendrian 3-web of the Legendrian twisted cubic in $\,\PP^3$. We give a partial answer to the conjecture in the affirmative …
Gronwall conjecture states that a planar 3-web which admits more than one distinct linearization is locally equivalent to an algebraic web. We give a partial answer to the conjecture in the affirmative for the class of planar 3-webs with the web curvature that vanishes to order three at a point. The differential relation on the third order jet of web curvature provides an explicit criterion for unique linearization.
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。