拉格朗日乘子的一个几何应用：极值相容线性联络
A geometric application of Lagrange multipliers: extremal compatible linear connections
AI总结：
本文利用拉格朗日乘子法研究广义 Berwald 空间中极值相容线性联络的存在性与内在刻画，先建立黎曼可度量化对应，再逐点求解混合条件极值问题，并给出可解性充要条件及用指示超曲面等距群内在量表示的解。
摘要：
黎曼流形的列维-奇维塔联络是一个度量（相容）线性联络，由其消失的挠率唯一确定。它在每个点处具有最小挠率的意义下是极值的。我们可以将这一思想推广到更一般的空间，其切空间中具有更一般的（未必是二次的）指示超曲面。在这里，基流形上相容线性联络的存在性不再有保证，这需要与极值联络的内在刻画一起加以处理。第一步是给出相容线性联络的黎曼可度量化。这一黎曼环境在线性联络与其挠率张量之间建立了一一对应，同时也给出了一种度量后者长度的方法。第二步是在基流形的每个点处求解一个混合条件极值问题，其所有约束方程（相容性方程）都涉及定义在指示超曲面上的函数。要最小化的目标函数是定义在挠率张量丛的有限维纤维（向量空间）上的二次平方范数函数。我们通过在函数空间上逐点使用拉格朗日乘子法来表达解，给出了可解性的一个充分必要条件，并且该解也用影响指示超曲面线性等距群一致大小的内在量来表示。这完成了对在基流形上容许相容线性联络的微分几何空间（在芬斯勒几何中称为广义 Berwald 空间）的描述。
英文摘要：
The Lévi-Civita connection of a Riemannian manifold is a metric (compatible) linear connection, uniquely determined by its vanishing torsion. It is extremal in the sense that it has minimal torsion at each point. We can extend this idea to more general spaces with more general (not necessarily quadratic) indicatrix hypersurfaces in the tangent spaces. Here, the existence of compatible linear connections on the base manifold is not guaranteed anymore, which needs to be addressed along with the intrinsic characterization of the extremal one. The first step is to provide the Riemann metrizability of the compatible linear connections. This Riemannian environment establishes a one-to-one correspondence between linear connections and their torsion tensors, also giving a way of measuring the length of the latter. The second step is to solve a hybrid conditional extremum problem at each point of the base manifold, all of whose constraint equations (compatibility equations) involve functions defined on the indicatrix hypersurface. The objective function to be minimized is a quadratic squared norm function defined on the finite dimensional fiber (vector space) of the torsion tensor bundle. We express the solution by using the method of Lagrange multipliers on function spaces point by point, we present a necessary and sufficient condition of the solvability and the solution is also given in terms of intrinsic quantities affecting the uniform size of the linear isometry groups of the indicatrices. This completes the description of differential geometric spaces admitting compatible linear connections on the base manifold, called generalized Berwald spaces, in Finsler geometry.
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