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Über Volumendefekte und Krümmung in Riemannschen Mannigfaltigkeiten mit Anwendungen in der Relativitätstheorie. (German) Zbl 0551.53037
Due to J. Bertrand and V. Puiseux the Gaussian curvature K of a surface can be determined by geodesic measurements: construct the geodesic circle of radius r to some point P, measure the area F and the defect \(\pi r^ 2-F\), then \(\overset\circ K=12\quad \lim_{r\to 0}\quad (\pi r^ 2- F)/(\pi r^ 2\cdot r^ 2).\) There are similar relations for n- dimensional Riemannian manifolds with positive definite metric. H. Vermeil has proved that the curvature invariant R can be determined from defects of the volume of geodesic balls. In this paper 4-dimensional Riemannian manifolds with signature \((-+++)\) are studied and we find a description of R by the volume of a geodesic part of the light cone and the volume of the light ball at time \(t=T\). Using the field equations we get connections between these defects and physical quantities of the general theory of relativity.

MSC:
53B50 Applications of local differential geometry to the sciences
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
83C15 Exact solutions to problems in general relativity and gravitational theory
83F05 Relativistic cosmology
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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