Reconstruction of an Affine Connection in Generalized Fermi Coordinates

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• 作者：J. Mikeš ; A. Vanžurová
• 关键词：Riemannian manifold ; Linear connection ; Metric ; Fermi coordinates ; Semigeodesic coordinates
• 刊名：Bulletin of the Malaysian Mathematical Sciences Society
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：40
• 期：1
• 页码：205-213
• 全文大小：
• 刊物类别：Mathematics, general; Applications of Mathematics;
• 刊物主题：Mathematics, general; Applications of Mathematics;
• 出版者：Springer Singapore
• ISSN：2180-4206
• 卷排序：40

文摘

On a manifold with affine connection, we introduce special pre-semigeodesic charts which generalize Fermi coordinates. We use a version of the Peano’s–Picard’s-Cauchy-like Theorem on the initial values problem for systems of ODSs. In a fixed pre-semigeodesic chart of a manifold with a symmetric affine connection, we reconstruct, or construct, the connection in some neighborhood from the knowledge of the “initial values”, namely the restriction of the components of connection to a fixed surface S and from some of the components of the curvature tensor R in the full coordinate domain. In Riemannian space, analogous methods are used to retrieve (or construct) the metric tensor 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 of type (0, 4) in the ambient space.