PARAMETRIZED LEGENDRE AND LAGRANGE VARIETIES
PARAMETRIZED LEGENDRE AND LAGRANGE VARIETIES
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参数化勒让德和拉格朗日簇
DOI：
10.2996/kmj/1138040038
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发表时间：
1994
影响因子：
0.6
通讯作者：
G. Ishikawa
中科院分区：
文献类型：
--
作者：
G. Ishikawa
由Goo石川0. Legendre簇和拉格朗日簇在几何光学[A][J2]、Hamilton-Jacobi方程的广义Cauchy问题[G2][13]、射影几何[SI]、微局部分析[P][DP]、复曲面上向量丛的模问题[Y]、本文讨论复解析范畴或C°°范畴中含有参数化的Legendre簇和拉格朗日簇。然后我们的研究符合可微映射奇点理论[AGV][B][D][GWPL][W]的框架。在这种情况下，Nash修改是超曲面所在流形的投影余切丛的正则点集的提升的闭包：投影余切丛与基空间的接触元素（切超平面）的总数相同，并且它具有自然接触结构[A][SI]。前超曲面的正则点的切超平面构成Legendre子流形，即射影余切丛上的接触分布的最大维积分子流形，这种自然提升的闭包可视为Legendre流形。事实上，勒让德簇的一个定义是它包含一个开的稠密勒让德子流形。这样通过Nash修正得到的Legendre簇一般具有奇异性。如果Nash修改是非奇异的，则超曲面是Legendre子流形的投影。前超曲面称为波前集[A][Z1]。注：对于一般的勒让德子流形，投影有限为1。在上面的前超曲面的定义中，我们允许Nash修改的奇异性，并且为了使定义非平凡，我们添加了有限性条件。(See[LT]对于一般理论的限制切空间。我们利用簇的参数化将上面提到的概念表述如下：从n维流形N到n + 1维流形B（例如，类（7°°或复解析））的映射f称为前映射，如果f的正则点集在TV中稠密，并且对于每个点x ∈ TV，正则点的切空间的像收敛到切超平面T
BY Goo ISHIKAWA0. IntroductionLegendre varieties and Lagrange varieties appear in many areas, for instance, geo-metric optics [A][J2], generalized Cauchy problem for Hamilton-Jacobi equations [G2][13], projective geometry [SI], microlocal analysis [P][DP], moduli problem of vectorbundles on complex surfaces [Y], symplectic topology [Gl] and so on.In this survey we treat Legendre and Lagrange varieties admitting some parametriza-tions in complex analytic or C°° category. Then our study fits with the framework ofthe theory of singularities of differentiate mappings [AGV][B][D][GWPL][W].First we introduce the notion of a "front hypersurface" by the property that the Nashmodification projects to the hypersurface itself finitely to one. The Nash modification, inthis case, is the closure of the lifting of the regular points set to the projective cotangentbundle of the manifold where the hypersurface lies in: The projective cotangent bundleis identified with the totality of contact elements (tangent hyperplanes) of the basespace and it has the natural contact structure [A][SI]. The tangent hyperplanes to theregular points of a front hypersurface form a Legendre submanifold, that is, the maximaldimensional integral submanifold of the contact distribution defined over the projectivecotangent bundle and the closure of this natural lifting might be regarded as a Legendrevariety. In fact, a definition of Legendre variety is that it contains an open dense Legendresubmanifold. The Legendre variety thus obtained by Nash modification has singularitiesin general. If the Nash modification is non-singular, then the hypersurface turns out theprojection of a Legendre submanifold. Then the front hypersurface is called a wave frontset [A][Z1]. Remark that, for a generic Legendre submanifold, the projection is finiteto one. In the above definition of front hypersurfaces we allow singularities for Nashmodification, and to make the definition non-trivial, we add the finiteness condition.(See [LT] for the general theory of limits of tangent spaces.)We utilize parametrizations of varieties to formulate the notion above mentionedas follows: A mapping / from an n-dimensional manifold N to an n + 1-dimensionalmanifold B (say, of class (7°° or complex analytic) is called a front mapping if the set ofregular points of / is dense in TV, and, for each point x £ TV, the images of the tangentspaces of regular points converge to a tangent hyperplane T