Lorentzian geometry as determined by the volumes of small truncated light cones. (English) Zbl 0662.53020

Schimming, Rainer

Arch. Math., Brno 24, No. 1, 5-15 (1988).

Let (M,g) be a Lorentzian manifold and C(y,a) the truncated light cone with vertex \(y\in M\), axis the timelike vector a and altitude \(| a| =g(a,a)^{1/2}.\) Let Vol C(y,a) be the volume of C(y,a) with respect to the invariant Riemannian measure of (M,g). For \(M={\mathbb{R}}^{n+1}\) and g the Minkowski metric, let C(a) be the corresponding truncated light cone with \(y=0\). The main result of the work is the following: “A Lorentzian manifold (M,g) of dimension \(n+1\geq 3\) is flat if and only if \[ Vol C(y,a)=Vol C(a)(1+o(| a|^ 5)) \] for sufficiently small altitudes a of C(y,a). (M,g) is Ricci-flat if and only if \[ Vol C(y,a)=Vol C(a)(1+o(| a|^ 3)).'' \]

Reviewer: V.Cruceanu

MSC:

53B30

Local differential geometry of Lorentz metrics, indefinite metrics

Keywords:

Lorentzian manifold; truncated light cone; volume; flat; Ricci-flat