从某些部分信息重建连接或度量
Reconstruction of the connection or metric from some partial information
摘要：
在一个给定了特殊的、半测点坐标的（正定）黎曼空间的邻域中，度量张量可以由它在一个适当的超曲面上的值和$(1,3)$型曲率张量在坐标域中的某些分量来计算。半测点坐标是著名的费米坐标的推广，费米坐标在力学和物理学中起着重要的作用，广泛应用于闵可夫斯基空间和一般黎曼空间的微分几何中。在本文中，我们考虑一个更一般的情况。本文引入了特殊的预半测图，并给出了一阶常微分方程初值问题解的存在性和唯一性的Peano 's-Picard' s-Cauchy-like定理的一个形式.然后，我们使用该装置在一个固定的预半测图的流形配备的线性对称联络。我们的目的是重建，或建设，在某些邻域的对称线性连接的知识的“初始条件”：限制连接到一个固定的$(n-1)$维表面$S$和一些组件的曲率张量$R$在“体积”（坐标域）。通过类似的方法，我们从度量对某个非迷向超曲面和体积中的曲率张量的某些分量的已知限制中，在半测点坐标域中检索（或构造）伪黎曼流形的$(0,4)$型度量张量.与其他作者相比，我们基于一阶常微分方程（常微分方程）的经典结果给出了构造性的较短证明。
英文摘要：
In a neighborhood of a (positive definite) Riemannian space in which special, semigeodesic, coordinates are given, the metric tensor can be calculated from its values on a suitable hypersurface and some of components of the curvature tensor of type $(1,3)$ in the coordinate domain. Semigeodesic coordinates are a generalization of the well-known Fermi coordinates, that play an important role in mechanics and physics, are widely used in Minkowskian space, and in differential geometry of Riemannian spaces in general. In the present paper, we consider a more general situation. We introduce special pre-semigeodesic charts characterized both geometrically and in terms of the connection, formulate a version of the Peano's-Picard's-Cauchy-like Theorem on existence and uniqueness of solutions of the initial values problems for systems of first-order ordinary differential equations. Then we use the apparatus in a fixed pre-semigeodesic chart of a manifold equipped with the linear symmetric connection. Our aim is to reconstruct, or construct, the symmetric linear connection in some neighborhood from the knowledge of the "initial conditions": the restriction of the connection to a fixed $(n-1)$-dimensional surface $S$ and some of the components of the curvature tensor $R$ in the "volume" (coordinate domain). By analogous methods, we retrieve (or construct) the metric tensor of type $(0,4)$ of a pseudo-Riemannian manifold in a domain of semigeodesic coordinates from the known restriction of the metric to some non-isotropic hypersurface and some of the components of the curvature tensor in the volume. In comparison to other authors, we give shorter proofs of constructive character based on classical results on first order ODEs (ordinary differential equations).
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