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摘要翻译：
在伪欧氏空间中引入了一类势子流形（每个n维势子流形是2n维伪欧氏空间中一个特殊的平坦无扭转子流形），并证明了每个n维Frobenius流形可以局部表示为n维势子流形。我们证明了所有势子流形在它们的切空间上都具有Frobenius代数的自然特殊结构。这些特殊的Frobenius结构是由相应的子流形的平坦第一基本形式和第二基本形式的集合生成的（实际上，结构常数是由子流形的Weingarten算子的集合给出的）。证明了二维拓扑量子场论的结合方程是伪欧氏空间子流形理论基本非线性方程组的自然约化，并在局部定义了势子流形类。将任意具体Frobenius流形显式化为伪欧氏空间中的势子流形的问题归结为求解一个线性二阶偏微分方程组。对于具体的Frobenius流形，这个实现问题可以在初等函数和特殊函数中显式解决。
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英文标题：
《Frobenius Manifolds as a Special Class of Submanifolds in
Pseudo-Euclidean Spaces》
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作者：
O. I. Mokhov
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Differential Geometry 微分几何
分类描述：Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis
复形，接触，黎曼，伪黎曼和Finsler几何，相对论，规范理论，整体分析
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一级分类：Physics 物理学
二级分类：High Energy Physics - Theory 高能物理-理论
分类描述：Formal aspects of quantum field theory. String theory, supersymmetry and supergravity.
量子场论的形式方面。弦理论，超对称性和超引力。
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一级分类：Physics 物理学
二级分类：Mathematical Physics 数学物理
分类描述：Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Analysis of PDEs 偏微分方程分析
分类描述：Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE's, conservation laws, qualitative dynamics
存在唯一性，边界条件，线性和非线性算子，稳定性，孤子理论，可积偏微分方程，守恒律，定性动力学
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一级分类：Mathematics 数学
二级分类：Mathematical Physics 数学物理
分类描述：math.MP is an alias for math-ph. Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
math.mp是math-ph的别名。这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
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一级分类：Mathematics 数学
二级分类：Symplectic Geometry 辛几何
分类描述：Hamiltonian systems, symplectic flows, classical integrable systems
哈密顿系统，辛流，经典可积系统
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一级分类：Physics 物理学
二级分类：Exactly Solvable and Integrable Systems 精确可解可积系统
分类描述：Exactly solvable systems, integrable PDEs, integrable ODEs, Painleve analysis, integrable discrete maps, solvable lattice models, integrable quantum systems
精确可解系统，可积偏微分方程，可积偏微分方程，Painleve分析，可积离散映射，可解格模型，可积量子系统
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英文摘要：
We introduce a class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanifold. We show that all potential submanifolds bear natural special structures of Frobenius algebras on their tangent spaces. These special Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (in fact, the structural constants are given by the set of the Weingarten operators of the submanifolds). We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and define locally the class of potential submanifolds. The problem of explicit realization of an arbitrary concrete Frobenius manifold as a potential submanifold in a pseudo-Euclidean space is reduced to solving a linear system of second-order partial differential equations. For concrete Frobenius manifolds, this realization problem can be solved explicitly in elementary and special functions.
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PDF链接：
https://arxiv.org/pdf/0710.5860
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