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

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
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